Concepts

• Abel summation 1 essay
• Abelian group 5 essays — A group in which the order of combining never matters: a times b always equals b times a. The whole numbers under addition and the symmetries of turning a wheel are examples, and every finite one is a product of cyclic groups.
• Absolute convergence 1 essay
• Absorbing state 2 essays — A state of a Markov chain that cannot be left once entered, so every walk reaching it stays there. Their presence replaces the question of where a chain settles with two others — how long until it stops, and where — and both have exact answers rather than limits.
• Absorption 1 essay
• Abstraction 1 essay
• Abundance 6 essays — The ratio of the sum of a number's divisors to the number itself. It is exactly 2 for a perfect number, above 2 for an abundant one, and it multiplies across coprime factors, which is why its average over all numbers can be computed at all.
• Abundancy 1 essay
• Accessibility 3 essays — The relation between worlds of a Kripke model saying which are visible from which, and the whole of what a modal operator quantifies over. Every classical modal axiom turns out to hold on exactly the frames whose accessibility relation has a stated property.
• Accumulation 1 essay
• Additive number theory 1 essay
• Additivity 1 essay
• Adjoint 2 essays — The quantity carried backwards through a computation in reverse-mode differentiation: how much the final output changes per unit change of an intermediate value. Propagating adjoints from the end gives every partial derivative at once, for a small multiple of the cost of the computation itself.
• Affine function 1 essay
• Affine plane 2 essays — A projective plane with one line and its points removed, so that lines split into classes of parallels that never meet. Over a field of q elements it is the grid of q² points with coordinates, and its q² + q lines are the solutions of linear equations.
• Aggregation 5 essays — Combining several individuals' positions into one collective position. Whether it can be done coherently depends on the logical structure of what is being aggregated, and the impossibility theorems say when it cannot.
• Airport problem 1 essay
• Alabama paradox 1 essay
• Aleph 1 essay
• Alexander polynomial 2 essays — A polynomial in one variable computed from a knot diagram's crossings, unchanged by every way of redrawing the knot. Its value at −1 is the determinant that the colourings count, and two values of it are enough to tell apart the small knots that random loops tie.
• Algebra 1 essay
• Algebra tiles 1 essay
• Algebraic identity 4 essays — An equation that holds for every value of the letters in it rather than for particular ones, so that both sides are two descriptions of the same object. Proving one usually means transforming one side into the other, and a counting identity is a claim that two ways of counting the same collection agree.
• Algebraic integer 8 essays — A number that is a root of a polynomial with whole coefficients and leading coefficient 1. Such numbers are closed under sums and products, and in a field like the one generated by √5 they form the ring that number theory uses in place of the integers.
• Algebraic number 9 essays — A number that is a root of some polynomial with whole-number coefficients. They include every fraction and every root of a whole number, they arrive in finite batches when sorted by the size of the polynomial that catches them, and so they can be listed — which is why almost every real number is not one.
• Algorithm 10 essays — A finite procedure of precisely stated steps that takes an input and is guaranteed to stop with an answer. What usually matters about one is how many steps it needs as the input grows, which decides whether it can be run on large cases at all.
• Aliasing 3 essays — The confusion of one frequency with another when a signal is read only at separate instants or points. Every frequency that differs from the true one by a multiple of the sampling rate gives identical readings, which is why sampling too slowly makes fast waves look slow.
• Allocation 4 essays — A way of dividing a fixed total between a fixed set of recipients, given as one number for each. It is the object every rule for sharing produces, and what distinguishes one rule from another is which allocations it is prepared to return.
• Almost everywhere 2 essays — A property holds almost everywhere when the points where it fails form a set of measure zero. It is the sense in which a function can equal another, or a sequence converge, while ignoring a set too thin to have any length, area or probability.
• Almost prime 1 essay
• Almost surely 2 essays — With probability one: true of every outcome except a set of outcomes whose total chance is nought. A random series converging almost surely may still diverge for some sign patterns, and those exceptions can be uncountably many while carrying no probability at all.
• Alternating group 7 essays — The group of permutations of a finite set that can be built from an even number of swaps, exactly half of all of them. It is the rotation group of the tetrahedron at four letters and of the icosahedron at five, and on five or more letters it cannot be broken into smaller pieces.
• Alternating knot 1 essay
• Alternating series 6 essays — A series whose terms change sign at every step, so its partial sums land on either side of the total in turn. Its partial sums therefore bracket the total, so stopping anywhere gives an error no larger than the first term left out.
• Altitude 3 essays — The line from a corner of a triangle perpendicular to the opposite side, and the segment of it inside. The three altitudes of any triangle meet at a single point, and each is the shortest distance from its corner to the opposite side.
• Amenability 1 essay
• Analytic continuation 3 essays — Extending an analytic function beyond where its defining formula works. A new series is built at a point inside the current region and used where the old one has nothing to say, and the extension is unique wherever the region swept out has no hole - which is what lets the result be called the function rather than an extension of it.
• Analytic function 5 essays — A function that equals its own Taylor series on a neighbourhood of every point of its domain. Analytic implies infinitely differentiable and the converse fails on the real line, which is the whole difference between a flexible category and a rigid one: an analytic function vanishing on any interval vanishes everywhere.
• Angle 3 essays — The amount of turning between two rays from a common point, measured as a fraction of a full turn or in degrees or radians. It is what a rotation is measured by, and the quantity that trigonometric functions take as their input.
• Angle bisector 2 essays — An angle bisector is the line that splits an angle into two equal halves. Every point on it is equally far from the two sides of the angle, which is why a circle touching both sides has its centre there, and why the three bisectors of a triangle meet.
• Angle defect 3 essays — The amount by which the faces meeting at a corner of a polyhedron fall short of closing up flat, measured as an angle. Summed over every corner of a convex polyhedron it comes to a full seven hundred and twenty degrees, whatever the solid.
• Angle excess 1 essay
• Angle sum 1 essay
• Angle trisection 5 essays — Dividing a given angle into three equal parts. It cannot be done with straightedge and compass alone, proved in 1837, but the trisectors themselves are perfectly good lines, and some of geometry's best theorems are about where they meet.
• Annulus 1 essay
• Antichain 5 essays — A collection of elements of an order no two of which are comparable — none is above or below another. The largest one measures how much of an order is undecided, and for the subsets of a set it turns out to be a whole layer of them, which is Sperner's theorem.
• Antiderivative 2 essays — A function whose derivative is a given function, determined only up to an added constant. By the fundamental theorem of calculus, an antiderivative evaluated at the two ends of an interval gives the area under the curve between them.
• Antipodal map 2 essays — The map sending each point of a sphere to the point diametrically opposite it. It moves every point, so the quotient by it is again a surface, and that quotient is the projective plane — the standard way a one-sided surface is built out of a two-sided one.
• Antipodal pair 7 essays — Two points of a sphere directly opposite each other, joined by a line through the centre. Many existence theorems concern them: any continuous map from a sphere to a space of its own dimension sends some antipodal pair to the same point.
• Aperiodicity 1 essay
• Apollonian gasket 1 essay
• Apollonius problem 1 essay
• Apportionment 9 essays — The problem of turning shares that are not whole numbers into whole numbers adding to the same total. It arises whenever seats are divided between regions in proportion to their populations, and every rule for doing it breaks one of three obviously desirable properties — which one is a decision rather than a calculation.
• Approximation 39 essays — A value put in place of another, together with a stated bound on how far apart the two can be. What makes one usable is the bound rather than the value: without a stated error it says nothing about the quantity it replaces.
• Approximation algorithm 1 essay
• Arc length 8 essays — The total distance travelled along a curve, defined as the supremum of the lengths of polygons inscribed in it. That supremum is finite only for some curves, and it belongs to a parametrisation rather than to the set of points, since a route traced twice is twice as long.
• Archimedean solid 2 essays — A convex solid whose faces are regular polygons of more than one kind, with every corner surrounded by the same faces in the same order. There are thirteen, apart from the prisms and antiprisms, and eleven of them come from reflecting one point in the mirrors of a regular solid.
• Archimedean spiral 1 essay
• Archimedes 4 essays — Archimedes of Syracuse, whose method of exhaustion pinned a curved quantity between two sequences of straight-sided ones and ruled out every other value. He proved the disc's area, the sphere's volume and the parabolic segment that way, and computed the first good bounds on pi from ninety-six-sided polygons.
• Arcsine distribution 2 essays — The probability law with density proportional to 1/√(x(1 − x)) on an interval, piling up at both ends. It is where a steadily turning angle spends its time when only its sine is recorded, and it governs the last lead change in a fair coin game.
• Arcsine law 1 essay
• Area 39 essays — The amount of surface a shape covers, measured against a chosen unit square. It is additive over pieces that do not overlap, and it is what a dissection proof compares before and after the pieces are moved.
• Area preservation 1 essay
• Area-preserving map 2 essays — A map of the plane or a surface that keeps the area of every region. Nothing can settle onto an attractor under such a map, so its orbits either lie on closed curves or wander through a chaotic sea, usually both in one picture.
• Argument 1 essay
• Arithmetic mean 1 essay
• Arithmetic progression 4 essays — A sequence of numbers each a fixed step above the last. Which of them contain infinitely many primes is decided entirely by whether the step and the starting value share a factor, and the answer is that every progression not obviously ruled out does.
• Arithmetisation 1 essay
• Arrangement 1 essay
• Arrow's theorem 2 essays — Arrow's theorem of 1951: with three or more candidates, every ranking rule that respects unanimity and decides each pair from that pair alone is a dictatorship. Its proof uses the finiteness of the electorate once, and with infinitely many voters the dictator can disappear.
• Assignment 11 essays — A way of giving each of several people exactly one of several tasks, with no task handed out twice. The count of them grows as a factorial, so which one is best is settled by structure — a matching condition, or the corners of a polytope — rather than by trying them all.
• Associativity 7 essays — The condition that bracketing does not matter: combining three things as (a·b)·c gives the same result as a·(b·c). It is what separates a group from a bare table of products, and among all the multiplication tables of a given small size it holds for a vanishing fraction of them.
• Astrolabe 2 essays — An instrument that maps the sky onto a flat brass plate by stereographic projection, with a rotating star map laid over a plate engraved for one latitude. Because circles project to circles, it answers questions of spherical astronomy, such as the time of sunrise, by lining up circles.
• Asymptotic 4 essays — A description of how a quantity behaves in the limit, ignoring everything that stays comparatively small. Two quantities are asymptotic when their ratio approaches one, which permits an error growing without bound so long as it grows more slowly.
• Asymptotic series 2 essays — A series whose partial sums approximate a function better and better as a parameter grows, even though the series itself may diverge. Its truncation is useful up to a best stopping point, as in Stirling's formula and the Euler–Maclaurin corrections.
• Asymptotics 16 essays — The behaviour of a quantity in the limit, stated as a ratio: two expressions are asymptotic when their quotient tends to one. That says nothing about their difference, which can grow without bound while the ratio settles, so an asymptotic formula needs an error term before it can be used at a particular size.
• Atlas 1 essay
• Attractor 7 essays — A set that nearby trajectories approach and stay near, and which contains no smaller set with that property. It is what an orbit settles onto once the transient has died away, and its shape is what distinguishes steady behaviour from chaotic.
• Auction 2 essays — A rule for selling an object by collecting bids and choosing a winner and a price from them. Different rules give bidders different incentives, and comparing what they raise is a central question of the economics of information.
• Autocorrelation 1 essay
• Automorphism 5 essays — A symmetry of a structure: a relabelling of its elements that preserves every relation the structure has. The automorphisms of an object form a group under composition, and that group is the precise statement of what the object's symmetry is.
• Average 1 essay
• Avoided crossing 1 essay
• Axiom 9 essays — A condition imposed on a rule or a structure in advance, from which its behaviour is then derived rather than argued about. Stating a set of them turns a matter of taste into a theorem, because anybody who dislikes the conclusion must name the condition to drop.
• Axiom of choice 13 essays — The assumption that one member can be picked from every set in a family at once, with no rule given for the picking. It is independent of the other axioms of set theory, and a surprising number of ordinary theorems — that a countable union of countable sets is countable, among them — quietly need it.
• Axiomatic characterisation 1 essay
• Axiomatic set theory 2 essays — The foundation in which every mathematical object is a set and every construction is licensed by a short list of axioms. The usual list is Zermelo and Fraenkel's with choice, and its models can disagree about questions it leaves open, such as the size of the continuum.

• Back and forth 3 essays — A method for proving two countable structures the same by building a matching one element at a time, alternately extending it from one side and then the other. Cantor used it to show every countable dense order without ends is the rationals.
• Backtracking 4 essays — A search that builds a solution one choice at a time and steps back to the last choice whenever it reaches a dead end. Its record of choices is a tree, and how large that tree is decides whether the search finishes in seconds or never.
• Backward induction 9 essays — Solving a sequential decision by starting at the end, where the answer is forced, and working towards the present. Each step compares acting now against a value already computed for everything later, which is what makes the calculation finite rather than a search over strategies.
• Baire category 2 essays — Baire's theorem: a complete space is not a countable union of nowhere-dense sets, so a countable list of small sets cannot exhaust it. It turns questions about typical behaviour into questions about which sets are meagre, and produces objects — continuous functions nowhere differentiable, for instance — without constructing one.
• Balancedness 3 essays — A condition on a game's values saying that no schedule of overlapping coalitions, each player fully occupied, earns more than the whole group working together. It holds exactly when some division of the winnings survives every coalition's objection, which is what makes it a finite test for a question about infinitely many divisions.
• Ballot problem 3 essays — The question of how likely the eventual winner of a count is to lead at every moment of it. Its answer is the difference of the two vote totals divided by their sum, and the reflection argument that gives it is the same one that counts lattice paths above a diagonal.
• Banzhaf index 2 essays — A measure of a voter's power in a weighted vote: the share of all coalitions in which that voter's switch would change the outcome. It differs from the Shapley value in counting coalitions rather than orderings, and the two can rank the same voters differently.
• Barbier's theorem 1 essay
• Bargaining 2 essays — Two or more parties dividing a gain between them by negotiation rather than by a market price. Its theory asks which divisions are reachable, which are stable, and what is lost when each side knows things the other does not.
• Barycentric coordinates 2 essays — Coordinates that locate a point by the weights it would need at a triangle's corners to balance there, normalised to add to one. They make lines, centres and ratios in a triangle linear, and they are how triangle centres are tabulated.
• Base point 2 essays — The point at which loops in a space are required to start and finish, so that one can be run after another. Changing it gives an isomorphic fundamental group, but only up to conjugation by the path used, which is why base-point-free statements are about conjugacy classes.
• Base rate 1 essay
• Basin of attraction 10 essays — The set of starting points whose orbits all end up at the same place. Iterating a map divides the space into basins, one for each attractor, and the boundaries between them can be extremely intricate.
• Basis 19 essays — A smallest set of vectors from which every other vector in a space can be built, and built in exactly one way. Choosing one turns abstract vectors into lists of coordinates, and the number of vectors in it is the dimension of the space.
• Bayes factor 3 essays — The ratio of how probable the observed data are under one hypothesis to how probable they are under another, each averaged over its own unknowns. Multiplying prior odds by it gives posterior odds, and under the hypothesis in the denominator it averages exactly one.
• Bayes' theorem 11 essays — The rule for turning a probability around: how likely a cause is, given that its effect has been observed. It is computed from the prior chance of the cause and how likely the effect is under each cause, and is most easily drawn as two rectangles.
• Bernoulli convolution 2 essays — The distribution of the random sum ±1 ± λ ± λ² ± …, with a fair coin for every sign. Below λ = 1/2 it lives on a Cantor set and at 1/2 it is flat; above 1/2 it has a density for almost every λ, and none when 1/λ is a Pisot number such as the golden ratio.
• Best reply 11 essays — The option that pays a chooser most, given what everybody else is doing. An equilibrium is exactly a state where every chooser's option is one, which is why marking every best reply on a payoff matrix and boxing the cells carrying both is a complete search for equilibria.
• Best response 1 essay
• Beta distribution 3 essays — A family of distributions on the interval from 0 to 1, with a density proportional to x^(a−1)(1 − x)^(b−1). It is the usual model for an unknown probability, and it stays in the family when coin tosses are used to update it.
• Beta expansion 1 essay
• Bias 5 essays — A persistent lean in a quantity that is equal in the limit, showing up in a lower-order term rather than in the leading one. The primes leaving remainder 3 on division by 4 lead those leaving 1 at almost every bound, and the two classes still have the same density.
• Bifurcation 15 essays — A parameter value at which a system's long-run behaviour changes kind rather than merely changing size. The parameter values at which they occur are what a bifurcation diagram plots, and the sequence of them can be a route to chaos.
• Bijection 34 essays — A pairing that matches every member of one collection with exactly one member of another, leaving nothing over on either side. Exhibiting one is how two collections are shown to be the same size, without either of them being counted.
• Billiards 9 essays — The motion of a point bouncing inside a region, leaving each wall at the angle it arrived. The shape of the region decides the long-run behaviour completely, from paths that repeat forever to paths that come arbitrarily close to every point.
• Binary 13 essays — Notation in which every number is written with two digits only, each place worth twice the one to its right. It is the notation in which a number's digits record which powers of two it is built from, one for each place.
• Binary expansion 10 essays — A number written as a sum of powers of two, so that it is a string of noughts and ones. Doubling shifts the string one place, which is why maps that double are exactly readable from the expansion and why finite arithmetic loses them after fifty-odd steps.
• Binary trees 5 essays — Trees in which every node has a left child and a right child, either of which may be absent. They are counted by the Catalan numbers, and the count is where several apparently unrelated families of objects turn out to be the same family.
• Binomial coefficient 31 essays — The number of ways of choosing k things from n without regard to order, and the entry in row n of Pascal's triangle. It counts lattice paths, subsets and coefficients of an expanded power alike, which is why the same numbers appear in unrelated problems.
• Binomial distribution 9 essays — The distribution of how many successes come out of a fixed number of independent trials with the same chance each. Its mean is the number of trials times the chance of success, and its spread grows like the square root of that number.
• Bipartite 1 essay
• Bipartite graph 13 essays — A graph whose vertices split into two sides, every edge joining one side to the other. It models pairings — applicants and posts, rows and columns — and its matchings are governed by Hall's condition on groups of vertices.
• Bipartite matching 1 essay
• Birthday problem 12 essays — The question of how many random draws from a set of possibilities make a repeat likely. Among 365 equally likely birthdays the answer is 23, and in general a repeat is likely after about the square root of the number of possibilities.
• Bisection 3 essays — Halving an interval repeatedly and keeping the half in which a sign change, a target value or some other property persists. Each step costs one evaluation and halves the uncertainty, so precision grows by one binary digit a step.
• Bisimulation 2 essays — A relation between two labelled graphs matching worlds that carry the same labels and can answer each other's moves along arrows, forever. Two worlds related by one satisfy exactly the same modal formulas, of every depth.
• Block design 2 essays — A collection of blocks drawn from a set of points, arranged so that every pair of points lies in the same number of blocks. The balance is what makes it a design rather than a list: no pair is favoured, which is why it is the arrangement an experiment wants when every treatment must be compared with every other equally often.
• Block stacking 1 essay
• Blocking pair 7 essays — Two people on opposite sides of a matching who each prefer the other to the partner they were given. A matching with none of them is stable, so hunting for one is how stability is checked.
• Blocking set 1 essay
• Bonferroni inequalities 1 essay
• Boolean circuit 1 essay
• Boolean formula 1 essay
• Boolean function 6 essays — A function taking a string of true-false values to a single true or false. There are 2^(2^n) of them on n inputs, they are the objects a truth table describes, and most questions about computation are questions about which ones are cheap to evaluate.
• Borda count 3 essays — A voting rule in which each ballot ranks the candidates and gives a candidate one point for every rival ranked below it, the highest total winning. It uses the whole of every ranking, and it can defeat a candidate whom a majority prefers to each rival.
• Borromean rings 1 essay
• Borsuk ulam 1 essay
• Borsuk ulam theorem 3 essays — The theorem that any continuous map from a sphere to space of one dimension fewer sends some pair of opposite points to the same place. Its relatives, built on other symmetries, prove that fair divisions, colourings and intersections must exist without constructing them.
• Bound 7 essays — A number known to lie above or below a quantity nobody can compute exactly. Its value is in being provable rather than sharp, and a bound met exactly somewhere is worth far more than one that is merely close everywhere.
• Boundary 10 essays — The edge of a region — the points lying in neither its inside nor its outside. What a set does at its boundary decides whether it is open or closed, and both fixed-point and extreme-value arguments turn on it.
• Bounded gaps 1 essay
• Boundedness 2 essays — The property of a set or sequence that it stays within some fixed distance of a point. Whether an orbit stays bounded or escapes to infinity is often the one question about it that can be settled, and it separates the inside of the Mandelbrot set from the outside.
• Bounds 1 essay
• Box counting 2 essays — A way to measure a set's fractal dimension: cover it with boxes of side ε, count how many N(ε) are needed, and see how the count grows as ε shrinks. If N grows like ε^(−d), the set has box-counting dimension d.
• Box dimension 7 essays — The exponent relating how many boxes of side epsilon are needed to cover a set to how small epsilon is. It is a growth rate rather than a size, and its value is that it can be measured on any bounded set at all - including the attractor of a system nobody designed, where no construction rule exists to compute a dimension from.
• Braess paradox 1 essay
• Branching 2 essays — A point at which an argument or a construction splits into cases that must all be handled. In a proof tree it is where a formula forces one of two things rather than both, and it is what makes the tree grow in width rather than in depth.
• Branching process 1 essay
• Breadth-first search 1 essay
• Bridge 1 essay
• Brouwer 10 essays — The Dutch topologist whose fixed-point theorem says that every continuous map of a disc to itself leaves some point exactly where it was. He also founded intuitionism, under which the non-constructive proof he gave of his own theorem would not count as a proof at all.
• Brunnian link 1 essay
• Buffon's needle 1 essay
• Burnside lemma 3 essays — A formula that counts arrangements up to symmetry by averaging, over the symmetries, the number of arrangements each one leaves unchanged. It turns a count of equivalence classes, which is hard, into counts of fixed points, which are usually easy.
• Busy beaver 3 essays — The largest number of steps, or of marks left on the tape, achieved by any halting Turing machine with a given number of states started on a blank tape. It is known for up to five states and grows faster than any function a program can compute.

• Cancellation 5 essays — The disappearance of terms of opposite sign from a sum, leaving fewer than were written down. It is what makes a determinant computable by elimination and a signed count exact, and its absence is exactly why the same sum with the signs deleted is far harder to evaluate.
• Canonical form 1 essay
• Cantor function 1 essay
• Cantor set 21 essays — The set left in the unit interval after the middle third of every remaining piece is discarded, for ever. It has no interval inside it and total length zero, and yet as many points as the interval it came from, which makes it the standard example of a set that measure and cardinality describe quite differently.
• Cap set 2 essays — A set of points in (Z/3)ⁿ with no three on a line — equivalently, no three distinct points adding to nought, and no three-term arithmetic progression. In four dimensions it is a hand of SET cards with no set; the largest possible size is known only up to dimension six.
• Capacity 4 essays — The most a road, pipe or edge of a network can carry per unit of time. Capacities bound every flow, and the cheapest set of roads separating two places, added up by capacity, is exactly the most that can travel between them.
• Cardinal arithmetic 1 essay
• Cardinality 12 essays — The size of a collection, measured by whether its members can be matched one to one with another collection's. It is compared by matching rather than by counting, which is what makes it usable for collections with no last member.
• Carmichael number 3 essays — A composite number n for which a to the power n − 1 leaves remainder one modulo n for every a sharing no factor with n. It passes Fermat's primality test for every base, and Korselt's criterion says it is squarefree with p − 1 dividing n − 1 for each prime p dividing it.
• Case analysis 1 essay
• Catalan numbers 17 essays — The sequence 1, 1, 2, 5, 14, 42 and onward, counting bracket sequences, polygon triangulations and binary trees alike. Each term is computed from the ones before it by a convolution, and the same sequence counts a dozen apparently unrelated families.
• Categorical statement 1 essay
• Categoricity 3 essays — A theory is categorical in a given size when all its models of that size are isomorphic, so the axioms pin the structure down completely. The theory of dense orders without ends is categorical for countable models and not for larger ones.
• Cauchy schwarz 6 essays — The inequality saying that an inner product is never larger in size than the product of the two lengths. Equality holds exactly when one vector is a multiple of the other, and the bound is what makes the ratio of a product to two lengths a cosine, so that an angle can be defined in a space with no picture.
• Caustic 3 essays — A curve that every trajectory of a family stays tangent to and never crosses. In a billiard table the caustics bound the regions a path cannot enter, and a table filled with them is one whose motion is completely organised rather than chaotic.
• Cayley graph 9 essays — A group drawn as a map: one dot per element and one arrow per generator, so that multiplying becomes walking. It depends on which generators are chosen, and what survives that choice — growth, the shape at large scales — is what the geometric theory of groups is about.
• Cayleys formula 5 essays — The count n^(n−2) of trees that can be drawn on n labelled points. It was stated by Cayley in 1889 and has many proofs, including Prüfer's code and a bijection with parking functions.
• Cellular automaton 11 essays — A row or grid of cells updated together, each one deciding its next state from its immediate neighbours. The rule is a lookup table on the neighbourhood, and the interest is in how much behaviour a very small table can produce.
• Central angle 1 essay
• Central limit 1 essay
• Central limit theorem 15 essays — The statement that a standardised sum of many independent contributions is approximately bell-shaped. It describes a window of width one over the square root of the count around the mean, and says nothing about probabilities further out than that.
• Centroid 3 essays — The average position of the points of a region, weighted by area. It is the point about which the mean squared distance to the region is smallest, and the parallel-axis identity is what makes that true.
• Certificate 13 essays — A short piece of data accompanying an answer that lets anybody verify it without repeating the computation. A set of dual prices certifies an assignment's optimality in a quadratic number of comparisons, against a factorial search.
• Cesaro summation 1 essay
• Chain 4 essays — A collection of elements of an order every two of which are comparable, so that they can be listed one above another. Chains are the opposite extreme from antichains, and the two are related by a min-max theorem: the fewest chains covering an order equals the size of its largest antichain.
• Chain decomposition 1 essay
• Chain rule 2 essays — The rule for differentiating a composition: the derivative of f(g(x)) is the derivative of f, taken at g(x), times the derivative of g. Applied step by step through a long computation it is how derivatives are accumulated automatically, forwards or backwards through the steps.
• Change of basis 1 essay
• Change of variable 1 essay
• Channel capacity 3 essays — The greatest rate, in message bits per transmitted symbol, at which a noisy channel can carry information with an error probability as small as desired. Shannon computed it in 1948 and proved it is both achievable and impossible to beat, which fixed a hard limit on every coding scheme.
• Chaos 20 essays — Behaviour that is fully determined by its rule yet unpredictable in practice, because nearby starts separate exponentially. It requires sensitivity to the starting point together with mixing, and it is compatible with the rule being simple and fully known.
• Character table 1 essay
• Character theory 1 essay
• Characterisation 1 essay
• Characteristic 1 essay
• Characteristic function 3 essays — The average of cos(tX) + i·sin(tX) for a random quantity X, viewed as a function of the frequency t. It turns sums of independent pieces into products, and it dies away as t grows whenever X has a density — which is how the absence of a density is sometimes proved.
• Characteristic polynomial 5 essays — The polynomial whose roots are the eigenvalues of a square matrix, got as the determinant of that matrix with a variable taken off the diagonal. Its coefficients are the trace, the determinant and the quantities between them, so it records only what a change of basis leaves alone.
• Chart 1 essay
• Chebyshev inequality 1 essay
• Chebyshev nodes 1 essay
• Chebyshev polynomial 8 essays — The polynomial Tₙ defined by Tₙ(cos θ) = cos nθ, which swings between −1 and 1 exactly n + 1 times on the interval [−1, 1]. Scaled to lead with 1, it is the polynomial that stays closest to zero there, and it underlies near-best approximation.
• Check digit 3 essays — An extra digit appended to an identifying number, computed from the others so that common copying mistakes give a number that fails the check. Which errors a scheme catches depends on the arithmetic used to compute it.
• Chinese remainder theorem 3 essays — The fact that remainders modulo coprime numbers can be chosen independently and determine one remainder modulo their product. It lets a computation modulo a composite number be split into smaller computations modulo its factors and reassembled.
• Chirality 2 essays — The property of an object that cannot be moved onto its own mirror image, as a left hand cannot become a right. The trefoil knot has it and the figure-eight knot does not, and invariants such as the Jones polynomial and the signature tell the two hands apart.
• Choice function 4 essays — A function that picks one element out of each set in a given family of non-empty sets. The axiom of choice asserts that one always exists, even when the family is infinite and no rule describes which element to pick.
• Chord 5 essays — A straight segment joining two points of a circle or other curve. On a circle its length is fixed by the angle it subtends, and the products of the pieces of chords through one point are all equal, which is the power of that point.
• Chromatic number 10 essays — The fewest colours a graph can be properly coloured with, so that no edge joins two vertices of one colour. It is at most four for any map drawn in the plane, and computing it in general means searching rather than evaluating a formula.
• Chromatic polynomial 2 essays — The polynomial that counts, for each number of colours k, the ways to colour a graph's vertices so that neighbours differ. It is a relative of the forest and spanning-tree counts, and its coefficients obey inequalities that took decades to prove.
• Circle 26 essays — The set of points at a fixed distance from a centre, which is the shape a fixed length sweeps out as it turns. It is the shape enclosing the most area for its perimeter, and the unit circle is where the trigonometric functions are read off.
• Circle area 3 essays — The space enclosed by a circle, which is half its circumference times its radius and therefore pi times the square of the radius. The half-base-times-height form is the one the ring dissection establishes, and it makes no reference to pi at all — the constant arrives only when the circumference is written out.
• Circle map 7 essays — A rule advancing a point around a circle by an amount depending on where it is. The average advance per step is its rotation number, and for a range of parameters that average sticks at a rational value because a periodic orbit attracts.
• Circle packing 1 essay
• Circle-preserving 3 essays — A property of a map: every circle and every straight line is carried to a circle or a straight line. Inversion has it, and it is why lines and circles are one family rather than two — a line being the circle that passes through the point at infinity.
• Circuit complexity 2 essays — The fewest and, or and not operations needed to compute a truth function, when a result may be used as often as it is needed. Almost every function needs exponentially many, yet no function anyone can name is proved to need more than a few times its number of letters.
• Circumcircle 6 essays — The unique circle passing through the three corners of a triangle, centred where the three perpendicular bisectors meet. Whether it is empty of other points is the condition defining the Delaunay triangulation.
• Circumradius 1 essay
• Class group 3 essays — The group formed by the classes of quadratic forms, or of ideals, of one discriminant under composition. Its size is the class number; its structure says how factorisation fails, and a group with every element of order two marks Euler's convenient numbers.
• Class number 5 essays — The number of classes of quadratic forms, or of ideals, sharing one discriminant, counted by reducing each form to a smallest representative. It is one exactly when factorisation into primes is unique, and it grows roughly like the square root of the discriminant.
• Classification 9 essays — A complete list of the objects of some kind, together with a proof that nothing else qualifies. What distinguishes it from a catalogue is the second half: a classification says what does not exist, which is the part no amount of searching supplies.
• Clause 1 essay
• Clique 3 essays — A set of vertices in a graph every two of which are joined by an edge, so that together they form a complete graph. Finding the largest clique is a standard hard problem, and forbidding cliques of a given size is the setting of Turán's theorem.
• Clone 1 essay
• Closed curve 8 essays — A curve that returns to where it began, so that it has no loose ends anywhere along it. It divides the plane into an inside and an outside, which is obvious to look at and requires a real theorem to establish.
• Closed form 1 essay
• Closed surface 2 essays — A surface with no edge and no boundary, finite in extent. Every one of them is a sphere with some number of handles or with some number of cross-caps, and which of the two lists it is on is decided by whether a sense of turning survives being carried round every loop.
• Closure 7 essays — The smallest set containing a starting collection and shut under a stated set of operations, so applying one never leaves it. It is what makes a definition well posed: without it, applying an allowed operation could take an object outside the set it was defined in.
• Coalition 14 essays — A subset of the players in a cooperative game, together with the value that subset could secure acting alone. Every question the theory asks is settled by comparing a proposed division against what each coalition could take by walking out.
• Cobweb 3 essays — A staircase drawn between a map's graph and the diagonal, which traces an orbit on the picture of the map itself. Its spiralling in or out shows at a glance whether a fixed point attracts, and the slope at the crossing is what decides.
• Code rate 1 essay
• Codimension 1 essay
• Coefficient 4 essays — One of the numbers multiplying the powers of the variable in a polynomial, listed in order of that power. A great deal about the roots — their sum, their product, whether two polynomials share one — is a computation on the coefficients with no root ever found.
• Cofinality 1 essay
• Collatz 6 essays — The rule that halves an even number and sends an odd one to three times itself plus one. Whether every starting value reaches one is unproved, though almost every value is known to fall below its own start.
• Collatz conjecture 3 essays — The claim that halving every even number and tripling every odd one and adding one always leads to one. Every number anybody has tried does, no proof exists, and the two ways it could fail — a cycle or an escape to infinity — need completely different arguments.
• Collinear 1 essay
• Collision 10 essays — Two things landing in the same place, which is what a counting argument produces when there are more things than places. Bounding how many places there are is therefore a way of proving that two things must coincide, without saying which two.
• Colour refinement 2 essays — An algorithm that colours a graph's points by their degree, then by the multiset of their neighbours' colours, and repeats until nothing splits. It is the first thing every practical isomorphism test does, and it is exactly as strong as a two-pebble game — so the pairs it cannot separate are the pairs two variables cannot describe apart.
• Combinations 1 essay
• Combinatorial design 1 essay
• Combinatorial explosion 1 essay
• Combinatorial proof 2 essays — An argument establishing an identity by exhibiting the collection both sides count, rather than by manipulating expressions. It usually generalises where an algebraic verification does not, and it says where the identity fails, because a bijection announces the case it cannot handle.
• Commitment 2 essays — A move made and made binding before the other side chooses, which changes what the other side's best reply is. Its value comes from the other side's behaviour rather than from the move itself, so reducing one's own options can raise one's payoff — and everything it buys depends on the binding being real.
• Commutativity 6 essays — The condition that the order of two operands does not matter. It holds for ordinary addition and multiplication and fails for composition, for matrices and for the arithmetic of order types, where a step placed in front of infinitely many is absorbed and a step placed after them is not.
• Commutator 6 essays — The element aba⁻¹b⁻¹ of a group, which is the identity exactly when a and b commute. It measures how far two elements are from commuting, and the subgroup all the commutators generate is the smallest one whose quotient is abelian.
• Compactness 10 essays — The property that a statement about an infinite object follows from the same statement about all its finite parts. It turns finite combinatorial theorems into infinite ones and back, and is what makes a colouring of every finite stretch of the integers extend to a colouring of all of them.
• Comparison 1 essay
• Compass 6 essays — An instrument that draws a circle of a given radius about a given point, and thereby carries a length from place to place. Removing the straightedge entirely, or fixing its opening, changes neither what it reaches nor what its partner reaches.
• Compass and straightedge 3 essays — The classical construction rules: draw a line through two known points or a circle about a known centre, and keep the intersections. The lengths they reach are exactly those built from rational ones by arithmetic and square roots.
• Competitive ratio 2 essays — The worst share of the best possible result that a rule acting without knowledge of the future is guaranteed to achieve. It compares an online rule with an offline one that sees the whole input in advance, taking the worst input for the rule.
• Complementary counting 4 essays — Counting what is wanted by counting everything and subtracting what is not wanted. It is the shorter route whenever the unwanted cases have more structure than the wanted ones, and the count of derangements is the standard example.
• Complementary slackness 5 essays — The condition that every constraint a solution uses is tight in its dual, and every dual variable that is positive names a constraint met with equality. It is what turns two matching optima into a proof of each other, and it restricts an optimal solution to a much smaller set of candidates.
• Complete bipartite 3 essays — A graph whose vertices are split into two sides, with every vertex on one side joined to every vertex on the other and no edges within a side. It contains no triangle, and with equal sides it is the densest graph that avoids one.
• Complete graph 13 essays — A graph in which every pair of vertices is joined, so nothing is left unconnected. It has one edge for every pair, so its edge count is a binomial coefficient and it is the worst case for any drawing without crossings.
• Complete theory 1 essay
• Completeness 12 essays — The property of a proof system that every statement true in all models of its axioms can be derived in it. With soundness it makes derivability and truth-in-every-model the same relation, which is what makes searching for a derivation worth doing.
• Completing the square 7 essays — Rewriting a quadratic as a squared term plus a constant, so that x squared plus bx becomes the square of x plus b/2, less b squared over four. It turns a quadratic into a shifted square, which gives the quadratic formula and locates the vertex of a parabola.
• Complex exponential 1 essay
• Complex numbers 25 essays — Numbers with a real and an imaginary part, which multiply by turning and scaling in a plane. Multiplying by one adds angles and multiplies lengths, which is what makes them the natural setting for rotations and for roots of polynomials.
• Complex plane 1 essay
• Complexity 17 essays — How the cost of a computation grows with the size of its input, ignoring constant factors. It is what separates a method that is usable at a million items from one that is usable at a thousand, and the separations are usually about which structure the method exploits.
• Component 3 essays — A maximal set of points of a graph reachable from each other. A sparse random graph has all of them small; past a threshold it has one holding a definite fraction of everything and all the rest still small.
• Composite 5 essays — A whole number above one that is the product of two smaller ones, and therefore not prime. Every composite has a prime factor no larger than its own square root, which is what makes sieving to that bound sufficient.
• Compound-interest 1 essay
• Comprehension 1 essay
• Computation 8 essays — A process that transforms an input into an output by a finite sequence of mechanical steps, each fixed by a rule. Models of computation such as Turing machines and cellular automata make precise which problems such a process can solve and at what cost.
• Computational irreducibility 1 essay
• Computer-assisted proof 4 essays — A proof in which some steps are computations too long to check by hand, carried out by a program. It is as rigorous as the program is correct, which is why such proofs are written to be rerun independently and checked with exact or interval arithmetic.
• Concave function 1 essay
• Concentration inequality 8 essays — A bound on how much probability can sit far from a distribution's average. What separates one from another is how much it assumes — a variance, a bounded range, an exponential moment — and correspondingly how fast the bound falls away.
• Concentration of measure 4 essays — The tendency of a function of many independent inputs, each of small influence, to stay close to its average. The deviation probability falls exponentially in the square of the distance, which is why large random systems behave almost deterministically.
• Condition number 1 essay
• Conditional convergence 2 essays — The convergence of a series whose terms add to a finite total only because positive and negative terms cancel, while their absolute values sum to infinity. Riemann showed that such a series can be rearranged to sum to any number at all.
• Conditional expectation 1 essay
• Conditional probability 18 essays — The chance of one event given that another is known to have happened. It is the quantity every updating rule is written in terms of, and it is computed as a ratio of two probabilities on the same space.
• Conditioning 1 essay
• Condorcet cycle 10 essays — A set of candidates each of which loses to another, so following the majorities round arrives back where it started. Its existence is why a majority verdict between pairs need not assemble into any ranking at all.
• Condorcet jury theorem 4 essays — The result that a majority of independent voters, each right more often than not, is right more often than any one of them. As the group grows the majority is almost surely right, and the theorem fails when the votes are correlated or each voter is more often wrong.
• Condorcet paradox 2 essays — The fact that majority preferences can run in a cycle, A beating B, B beating C and C beating A. It happens although every voter's own ranking is consistent, and it is the simplest reason a group's will need not be well defined.
• Condorcet winner 3 essays — A candidate who would beat every other candidate in a two-way majority vote. One need not exist, because majorities can go in a circle, and a rule that elects the Condorcet winner whenever there is one is called Condorcet-consistent.
• Conductance 1 essay
• Confidence 1 essay
• Confidence interval 1 essay
• Configuration 1 essay
• Conformal 3 essays — Preserving angles: two curves crossing at some angle have images crossing at the same angle. A conformal map is a similarity at each point separately, scaling by a factor that varies from place to place, so it keeps shapes small-scale and distorts sizes.
• Conformal map 10 essays — A map that keeps the angle at which two curves cross, while changing lengths and areas freely. Any hypothesis about tangency or perpendicularity survives one, which is what lets a configuration be moved to an easier position and moved back.
• Congestion 2 essays — The situation in which a resource gets worse the more it is used, so what an option is worth depends on how many others take it. Games of this shape always have a pure equilibrium, and a search that lets participants move one at a time to whatever is currently better always finds one.
• Congruence 17 essays — The relation between two figures of the same shape and size, so that one can be carried onto the other by a rigid motion. It is what a dissection proof asserts about its pieces, and it is checked by comparing edge lengths rather than by looking.
• Congruent number 1 essay
• Conic 13 essays — A curve cut from a cone by a plane — a circle, an ellipse, a parabola or a hyperbola, according to the angle of the cut. The kind is decided by the angle of the cut against the cone's own slope, and all four share one description in terms of a focus and a directrix.
• Conic section 1 essay
• Conjecture 35 essays — A statement believed true and not proved, usually on the strength of computation or of a pattern holding in every case anybody has checked. Its value is in what it organises, and the proof of a good one generally needs machinery invented for the purpose.
• Conjugacy 9 essays — A change of coordinate carrying one map onto another, so that the two are the same map differently labelled. It preserves everything defined from orbits and neighbourhoods — cycles, entropy, chaos — and destroys everything defined from distances or derivatives.
• Conjugacy class 9 essays — The set of elements of a group that are the same operation seen from different positions, obtained from one of them by composing with every element and its inverse. Classes partition a group, and a subgroup that is a union of them is exactly one that can be quotiented by.
• Conjugate 8 essays — The function recording, for each slope, how far the line of that slope must be pushed to touch a given curve. It is convex whatever it came from, and applying the construction twice returns the original exactly when the original was convex.
• Conjugate partition 2 essays — The partition obtained by reading a partition's diagram of dots by columns instead of rows. Its largest part is the original's number of parts, so conjugation swaps two statistics and proves at a stroke that they are equally distributed.
• Connected component 1 essay
• Connected sum 2 essays — The knot made by cutting two knots open and joining their loose ends, so that one is tied after the other on a single loop. Genus and the determinant behave simply under it, and long random loops are mostly connected sums of small knots.
• Connectedness 11 essays — The property of a set that cannot be split into two separated pieces. It is what the intermediate value theorem needs, and losing it is one of the ways a fixed-point argument fails.
• Connective 2 essays — A word or symbol that builds a compound statement out of simpler ones, taking their truth values to a truth value. There are sixteen of them on two inputs, and a single well-chosen one is enough to build every other.
• Connectivity 5 essays — The property of a graph that every point is reachable from every other. For a random graph it arrives much later than a giant component does, and what holds it up is the last point with no edges at all.
• Conservation 1 essay
• Consistency 12 essays — The property of a set of assumptions from which no statement and its own negation can both be derived. It is what a model establishes, by exhibiting a world in which every assumption holds at once.
• Constant width 7 essays — A convex shape that measures the same across between parallel supporting lines whatever direction they take. The circle is not the only one: a Reuleaux triangle rolls as smoothly under a plank and is not round.
• Constraint 1 essay
• Constructible number 22 essays — A length reachable from a unit segment by straightedge and compass, which is exactly one expressible with the four operations and square roots. The characterisation is what settles the classical impossibilities, since doubling a cube and trisecting an angle need cube roots.
• Construction 24 essays — A figure produced by a stated sequence of permitted operations, so that what is drawn is reachable rather than merely plausible. Which figures are constructible depends entirely on which operations are permitted, so the operation set is part of the problem.
• Constructive proof 3 essays — An argument that establishes something exists by exhibiting one, rather than by ruling out its absence. It is the stronger of the two: it settles existence and hands over an example, at the cost of usually being harder to find.
• Continued fraction convergent 2 essays — A fraction obtained by cutting a continued fraction short, which is the best approximation available for the size of its denominator. The convergents of an irrational number are exactly the fractions that beat every fraction with a smaller denominator.
• Continued fractions 32 essays — A number written as a whole part plus one over a whole part plus one over another, nested as far as it goes. The expansion terminates exactly when the number is rational, and it is periodic exactly when the number is a quadratic irrational.
• Continuity 37 essays — The property of a function whose output moves only a little when its input moves a little — a graph with no jumps in it. It is what makes a sign change force a crossing, and it is far weaker than differentiability — a continuous curve need have no tangent anywhere.
• Continuum hypothesis 3 essays — The statement that there is no size between the whole numbers and the real numbers. It can neither be proved nor disproved from the usual axioms of set theory, so both it and its negation may consistently be assumed, and the size of the continuum is left undetermined by them.
• Contraction 2 essays — A map that brings every pair of points closer together by at least a fixed factor smaller than one. Such a map has exactly one fixed point, and applying it over and over from any start converges to that point, with the error shrinking by the factor at every step.
• Contraction mapping 1 essay
• Contradiction 1 essay
• Convergence 55 essays — The behaviour of a sequence or a sum that settles on a single value rather than growing or oscillating without end. What it does not say is how fast: a rate is a separate result, and usually a harder one than the convergence itself.
• Convergence rate 42 essays — How quickly a sequence closes on its limit, measured by how the remaining error shrinks each step. It is what decides whether an iteration is worth running, and it can be geometric, quadratic or as slow as one over a square root.
• Convergent 3 essays — A fraction obtained by cutting a continued fraction off after some number of terms. The convergents of a number approach it from alternate sides, and each is closer to it than any fraction with a smaller denominator.
• Converse 2 essays — The statement obtained from an if-then statement by swapping its hypothesis and its conclusion. A theorem's converse is a separate claim that may be true or false, and proving it is often what turns a property into a test.
• Convex game 1 essay
• Convex hull 14 essays — The smallest convex set containing a given collection of points, which is what an elastic band snapped round them would enclose. Its corners are exactly the points that are not mixtures of the others, and finding them is the basic problem of computational geometry.
• Convex position 2 essays — A set of points is in convex position when every one of them is a corner of their convex hull, so that none lies inside the polygon the others form. Asking how many points force a subset in convex position is the Erdős–Szekeres problem.
• Convexity 52 essays — The property of a shape that contains the whole straight line between any two of its own points. For the unit ball of a distance it is exactly the triangle inequality, and it is what makes an optimum reachable by going downhill.
• Convolution 6 essays — The sequence whose n-th term sums the products of terms of two sequences whose indices add to n. It is what a product of generating functions computes, which is why a recurrence involving a split becomes an algebraic equation.
• Cooperative game 10 essays — A list of players together with a number for every subset of them, saying what that group could secure on its own. It carries no moves and no order of play; what is asked of it is how the value of the whole group should be divided.
• Coordinates 2 essays — Numbers that locate a point by its position relative to a chosen frame, such as distances along perpendicular axes or a radius and an angle. Different frames suit different problems: confocal ellipses and hyperbolas give coordinates whose natural origin is a pair of foci.
• Coprime 1 essay
• Core 12 essays — The set of ways of dividing what a group is worth that no smaller coalition can beat by walking out. It can be a polygon, a single point or empty, and its emptiness on games as simple as three-player majority is what forces rules such as the nucleolus to be defined instead.
• Correlated equilibrium 3 essays — A distribution over the cells of a game such that neither party, told privately which of their own options was drawn, prefers to do something else. It generalises Nash equilibrium by dropping the requirement that the two randomisations be independent, and the set it defines is a polytope rather than a fixed point.
• Correlation 8 essays — A measure of how two random quantities move together: positive when one tends to be high as the other is, negative when high goes with low. It is the covariance divided by both standard deviations, so it lies between −1 and 1 and ignores units.
• Correspondence theory 1 essay
• Coset 6 essays — One of the equal-sized shifted copies a subgroup cuts the whole group into. Their equal size is what forces a subgroup's order to divide the group's, which is Lagrange's theorem in one line.
• Cosine 5 essays — The horizontal coordinate of a point travelling round the unit circle, as a function of the angle turned through. It is the derivative of sine, and the law of cosines is the Pythagorean relation with the term it supplies restored.
• Cost-sharing 1 essay
• Countability 17 essays — The property of a collection whose members can be arranged in a single numbered list. The rationals have it and the reals do not, which is what the diagonal argument establishes — and the two facts together are what make most numbers unnameable.
• Counterexample 74 essays — A single case satisfying a claim's hypotheses and failing its conclusion, which is enough to refute the claim entirely. One is enough, which is the asymmetry between proving a general statement and refuting it.
• Counting 7 essays — Determining how many members a finite collection has without listing them. The two standard routes are a pairing with something already counted, and a decomposition into pieces whose counts add or multiply.
• Counting argument 91 essays — A proof that establishes a fact by counting a collection rather than by producing an example. It is what pigeonhole and parity arguments are made of, and it establishes existence without locating anything.
• Counting two ways 48 essays — Counting one collection by two different routes and reading off the identity that the two answers are forced to agree on. It is the standard route to a combinatorial identity, and the identity is forced rather than verified term by term.
• Coupon collector 5 essays — The problem of how long it takes to see every one of n equally likely kinds when sampling at random with replacement. The answer is about n ln n draws, and the wait is dominated by the last few kinds, which is why covering everything takes far longer than seeing most things.
• Coverage 2 essays — The fraction of a region covered by a union of randomly placed shapes. For shapes dropped by a Poisson process it is one minus e to the minus the average number of shapes covering a point, because a point is bare exactly when no shape lands near it.
• Covering 6 essays — A collection of sets whose union contains the set in question. Both definitions of dimension are about covers - box counting fixes their size and takes a count, Hausdorff's allows any sizes and takes an infimum - and the difference between the two answers is exactly what that freedom buys.
• Covering space 18 essays — A space mapping onto another so that each point below has a neighbourhood whose preimage is a stack of separate copies. Paths lift to it uniquely, which turns a question about which loops shrink into a question about where a lifted path ends.
• Covolume 1 essay
• Coxeter diagram 1 essay
• Cramér's model 2 essays — A random imitation of the primes in which each number n is declared prime with chance one over its logarithm, independently. It predicts the primes' density and the size of their largest gaps well, but ignores divisibility by small primes.
• Credible interval 1 essay
• Critical clause 1 essay
• Critical exponent 1 essay
• Critical point 7 essays — A critical point of a function is a place where its slope is zero in every direction: a peak, a pit or a saddle. Counted with signs, plus for peaks and pits and minus for saddles, the critical points of a function on a closed surface always add up to its Euler characteristic.
• Cross product 1 essay
• Cross-cap 2 essays — A cross-cap is a Möbius band sewn into a hole cut in a surface, the one-sided counterpart of a handle. Every one-sided closed surface is a sphere with some number of cross-caps, and each one lowers the Euler characteristic by one.
• Cross-ratio 2 essays — A number computed from four points, as a ratio of two ratios of differences between them. It is unchanged by every map that carries circles to circles and preserves angles, and it is real exactly when the four points lie on one circle or one line.
• Crossing number 11 essays — The fewest pairs of edges that must cross in any drawing of a graph in the plane. It is zero exactly for planar graphs, and the first two graphs with crossing number one are the obstructions Kuratowski named.
• Cryptography 2 essays — The study of what can be kept secret from an adversary with bounded resources. Its constructions are conditional by necessity: an unconditional one would separate complexity classes nobody has separated.
• Cubic 6 essays — An equation of degree three, whose roots are the numbers a compass and straightedge cannot reach and an angle trisector can. Adding any instrument that solves one — a marked ruler, a drawn conic, a paper fold — extends the reachable numbers by exactly that degree.
• Cubic graph 1 essay
• Curse of dimensionality 3 essays — The collective name for the ways space behaves against intuition when it has many dimensions, making grids, searches and samples exponentially costly. Nearly all of a high-dimensional ball's volume lies near its surface and near every equator, so random points are far from the centre and from each other.
• Curvature 11 essays — How sharply a curve or a surface bends away from being straight or flat, measured at a point. It is what distinguishes a sphere's geometry from the plane's, and it is why a triangle drawn on a globe has angles summing past a straight line.
• Curve 1 essay
• Cut 6 essays — A division of a network's places into two sides, priced by the total capacity of the roads that cross between them. Everything travelling from one side to the other is bounded by a cut's price, so the cheapest cut is the tightest bound there is.
• Cut elimination 4 essays — The theorem that any sequent-calculus proof using the cut rule, which applies a proved lemma, can be rewritten as one that never does. The lemma-free proof may be far longer, but every formula in it is part of what it proves.
• Cycle 9 essays — A closed walk in a network that returns to its start without reusing an edge; in a permutation, objects each sent to the next and the last back to the first. Cycles are what Euler's circuits are built from, and every permutation is a collection of disjoint ones.
• Cyclic group 21 essays — A group generated by repeating one operation, which is the abstract form of a dial. Every group of prime order is one, and the whole numbers on a dial of any size are the standard example.
• Cyclic order 1 essay
• Cyclic polygon 1 essay
• Cyclic word 1 essay
• Cyclic-quadrilateral 3 essays — A four-sided figure whose corners all lie on one circle. Its opposite angles add to a half-turn, and Ptolemy's theorem ties its sides to its diagonals; either property is a test for whether four points are concyclic.
• Cyclotomic polynomial 7 essays — The polynomial whose roots are the n-th roots of unity of exact order n. Its coefficients are whole numbers, it is irreducible over the rationals, and its degree counts the residues coprime to n — which is what decides whether a regular polygon can be constructed.
• Cylinder 1 essay

• Dandelin spheres 2 essays — Two spheres inscribed in a cone on either side of a cutting plane, touching that plane at the section's two foci. They turn the definition by a cut through a cone into the definition by two foci, with no algebra anywhere in the argument.
• De bruijn sequence 7 essays — A cyclic sequence of letters in which every word of a fixed length appears exactly once, and in a cycle just long enough to hold them all. One exists for every alphabet and every word length, because the overlaps between consecutive windows form a graph in which every vertex has as many edges leaving as arriving.
• De bruijn torus 1 essay
• De gua theorem 1 essay
• Decidability 6 essays — Whether some procedure settles every instance of a question in finite time. A logic is decidable when a non-theorem always fails in a small enough model to be found by search, which propositional systems manage and systems with quantifiers generally do not.
• Decided by exhaustion 1 essay
• Decimal expansion 2 essays — The writing of a real number as a sequence of digits in base ten, or in any other base. Its digits record repeated multiplication by the base, which is why questions about digits are often questions about a dynamical system.
• Decision procedure 23 essays — A method guaranteed to halt on every input and report a correct yes or no. Whether one exists is a separate question from whether the answer is determined, and for some questions it provably does not.
• Decisive set 2 essays — A set of voters whose agreement on a pair of candidates forces a voting rule's outcome on that pair, whatever everyone else submits. The decisive sets are the working object of the proof of Arrow's theorem, and for any rule with Arrow's properties they form an ultrafilter.
• Deck transformation 3 essays — A symmetry of a covering space that leaves the covering map unchanged, permuting the sheets over each point. The collection of them is a group whose size is the number of sheets, and it records what the covering remembers that the base has forgotten.
• Decomposition 2 essays — Writing an object as a combination of simpler pieces — a graph as a tree of overlapping bags, a number as a product of primes, a function as a sum of waves. A good decomposition turns a hard question about the whole into easy questions about the pieces and a rule for putting their answers together.
• Defective matrix 2 essays — A square matrix with fewer independent eigenvectors than its size, so that no change of basis makes it diagonal. The shortfall is always at a repeated eigenvalue, and the nearest available form keeps ones just above the diagonal to record what is missing.
• Deferred acceptance 8 essays — A construction in which one side proposes down its lists while the other holds the best offer so far and releases the rest. It always terminates with a stable matching, and which side proposes decides which side gets its best stable outcome.
• Deficiency 3 essays — How far a set overshoots what it can be matched to: the size of a set of vertices less the number of vertices its edges reach. Its largest value over all such sets is exactly how many vertices a largest matching must leave out, which turns a claim about every arrangement into a set anybody can count.
• Definability 2 essays — The property of a set or relation that some formula of a given language picks out exactly its members. Whether a set is definable depends on the language, and counting shows most sets of numbers are definable in none, since formulas can be listed and sets cannot.
• Deformation 2 essays — A continuous movement of one object into another, with no cutting and no gluing along the way. What survives every deformation is what topology studies, and a quantity unchanged by all of them is called an invariant.
• Degeneracy 1 essay
• Degenerate conic 2 essays — The solution set of a quadratic that is a point, a line, a pair of lines or nothing at all rather than a curve. These are the sections of a cone by planes through its apex, and the discriminant names their type while a second determinant detects the degeneracy.
• Degree 19 essays — The highest power appearing in a polynomial, or the number of edges meeting a vertex in a graph. In both senses it bounds what the object can do: a polynomial has at most that many roots, and a graph's degrees sum to twice its edge count.
• Degree of a map 1 essay
• Degree of an extension 8 essays — The dimension of one field regarded as a space of vectors over a smaller one inside it. It multiplies along a tower of extensions, which is what turns a question about constructibility into arithmetic on powers of two.
• Dehn twist 1 essay
• Delaunay triangulation 4 essays — The triangulation of a set of points in which no triangle's circumcircle contains any other point. It is the dual of the Voronoi diagram, it maximises the smallest angle, and it contains the shortest spanning tree of the same points.
• Deletion channel 1 essay
• Deletion contraction 1 essay
• Deliberation 1 essay
• Delta y exchange 1 essay
• Dense orbit 3 essays — A sequence of points under a repeated map that comes arbitrarily close to every point of the space. An irrational rotation of a circle has one from every starting point, and density is the first step towards the stronger property of equidistribution.
• Dense order 1 essay
• Dense set 5 essays — A set meeting every interval of the line, however short, so that no gap in it can be found. Being dense says nothing about how much room a set takes up: the rationals are dense and have measure nought.
• Density 22 essays — How thinly a set is spread through the whole numbers, measured as the fraction of them it occupies up to a bound. It is what a counting theorem estimates, and it constrains a total while saying nothing about how the members are arranged inside it.
• Derangement 9 essays — An arrangement of a set of objects that leaves none of them in its original place. The proportion of all arrangements that are derangements settles almost immediately on the reciprocal of the number e, because the count is a truncated series for it.
• Derivative 34 essays — The instantaneous rate at which a quantity is changing, and the slope of the tangent to its graph. In more than one variable it is a linear map rather than a number, and the number was always the one-dimensional case of that.
• Derived series 1 essay
• Descartes rule of signs 2 essays — The rule that a polynomial has at most as many positive roots as its coefficients have sign changes, and fewer by an even number. It bounds roots by the number of terms rather than the degree, and it is exact when every root is real.
• Descent 8 essays — Fermat's method: from a supposed solution in whole numbers, construct a strictly smaller one, and conclude there was none. It works because a strictly decreasing sequence of positive whole numbers cannot go on forever, which is the whole of the machinery.
• Detailed balance 2 essays — The condition that a Markov chain's traffic along each edge is equal in the two directions, one edge at a time. It is strictly stronger than being stationary, and where it holds the long-run shares can be read off the edge weights with no system to solve.
• Determinant 41 essays — The factor by which a linear map multiplies area or volume, together with the sign saying whether it turns the space over as well. It is computed from the entries and measured off the drawn image, and it is zero exactly when the map collapses the space.
• Determinism 6 essays — The property of a rule whose next state is a function of the present one, with nothing left to chance. It is compatible with unpredictability: a fully determined rule can separate nearby starts fast enough that no measurement pins down the future.
• Diagonal argument 7 essays — The construction that reads one entry from each row of a table and changes it, producing something that cannot be any row. It is what shows that no list holds every real number, and the same construction underlies the incompleteness and halting results.
• Diagonal dominance 1 essay
• Diagonalisation 5 essays — Rewriting a linear map against a basis of its own eigenvectors, so that it becomes a stretch along each axis and mixes nothing. It makes powers of the map cost two multiplications rather than many, and is available exactly when the eigenvectors span the space.
• Diameter 2 essays — The greatest distance between two points of a set. It measures how wide a shape is at its widest, and questions about cutting a shape into narrower pieces, or fitting the most area or perimeter inside a given width, are questions about it.
• Dictatorship 3 essays — A rule for combining several people's choices that always returns one fixed person's choice. It satisfies every consistency condition trivially, which is why impossibility theorems in voting and judgement single it out as the escape.
• Difference quotient 4 essays — The change in a function divided by the change in its input, which is the slope of a chord and the thing a derivative is a limit of. Its behaviour as the two points come together is the whole question: it may settle on a number, on two different numbers, or on none.
• Differentiability 4 essays — The property of having a tangent at a point, meaning the chord slopes settle on a single value as the two points come together. It is strictly stronger than continuity, and it is what licenses replacing a curve by a straight line near a point.
• Differential equation 7 essays — An equation relating a quantity to its own rate of change, whose solutions are functions rather than numbers. Solving one means finding every function satisfying it, and the useful ones pin down a single solution once a starting value is given.
• Diffusion 3 essays — The spreading of a quantity from where there is more of it to where there is less, at a rate set by how sharply it varies. It sends a quantity's spread up like the square root of time rather than in proportion to it, which is why it is so bad at covering distance.
• Dihedral angle 3 essays — The angle between two faces of a solid, measured across the edge they share. A cut through a polyhedron either preserves such an angle or splits it into two that add back to it, which is why it appears in the invariant that blocks a dissection.
• Dihedral group 11 essays — All the rotations and reflections that leave a regular polygon occupying the same place. It has twice as many elements as the polygon has sides, half of them turns and half of them flips.
• Dilworth theorem 1 essay
• Dimension 26 essays — The number of independent directions in a space, and so the number of coordinates a point in it needs. It is what a basis counts, and it is preserved by every invertible linear map, which is why it can be used to tell spaces apart.
• Diophantine approximation 10 essays — How closely a real number can be approached by fractions with small denominators. The continued fraction supplies the best approximations and nothing else comes near, which turns questions about real numbers into questions about a short list of fractions.
• Diophantine equation 3 essays — A polynomial equation whose solutions are required to be whole numbers, such as x³ + y³ + z³ = n. Asking for whole-number solutions turns easy algebra into hard number theory, and no single method decides every such equation.
• Directrix 1 essay
• Dirichlet 3 essays — Peter Gustav Lejeune Dirichlet, whose name is on both the pigeonhole principle and the theorem that every irrational has unreasonably good rational approximations. Both results carry the same signature: a counting bound forcing something to exist without exhibiting it.
• Dirichlet kernel 1 essay
• Dirichlet theorem 4 essays — The statement that every arithmetic progression whose step and start share no factor contains infinitely many primes. Its proof abandons arithmetic for analysis, and what it establishes is stronger than infinitude: the admissible classes hold equal shares.
• Dirichlet's function 1 essay
• Disc 1 essay
• Discharge 2 essays — In natural deduction, the step that withdraws an assumption once a conclusion has been reached from it. It is what turns a derivation of B from A into a proof of the implication A → B that depends on nothing.
• Discounting 1 essay
• Discrepancy 5 essays — A measure of how unevenly a set of points fills a region: the largest gap, over all boxes, between a box's share of the points and its share of the volume. Low discrepancy is what makes a point set good for numerical integration.
• Discrete logarithm 2 essays — The exponent solving g to the power x equals h in a finite group, when one exists. It is easy to state, easy to check, and believed to be hard to compute in the groups cryptography uses — which is what a great deal of public-key cryptography rests on.
• Discriminant 16 essays — A number computed from a polynomial's coefficients that vanishes exactly when the polynomial has a repeated root. It is the resultant of the polynomial and its own derivative, and its sign decides how many of the roots are real.
• Discursive dilemma 1 essay
• Disjunction property 2 essays — The property of a logic that whenever it proves A or B, it proves A or it proves B. The constructive system has it and classical logic does not, since it proves p or not-p without proving either half.
• Dissection 19 essays — A cutting of a shape into pieces that reassemble into another shape with no gap, no overlap and nothing left over. It is the site's flagship kind of proof, and the assertion behind one is that the pieces tile the target with no gap and no overlap.
• Dissipation 2 essays — The contraction of volume by a map or flow, so that a region of starting points shrinks as it is carried along. It is what forces an attractor to have no volume, and its rate is the sum of the Lyapunov exponents, which is the logarithm of the determinant of the derivative.
• Distance 1 essay
• Distance function 1 essay
• Distribution 2 essays — A description of how a quantity's values are spread: which values occur, and how often or with what probability. It can be given by a histogram, a density or a cumulative function, and two quantities with the same distribution cannot be told apart by any statistic.
• Divergence 6 essays — The failure of a sequence or a sum to settle on a value, whether by growing without bound or by never coming to rest. It comes in two kinds worth separating: growth without bound, and oscillation that never settles on anything.
• Divergent series 1 essay
• Divide and choose 4 essays — The two-person rule under which one person cuts the thing in two and the other takes whichever part they prefer. It guarantees each of the two a share they value at half or more, and it needs no agreement about what the parts are worth.
• Divisibility 13 essays — The relation between two whole numbers when one is a whole multiple of the other. It is the relation the whole of number theory is built on, and it is what a factorisation records.
• Division algebra 4 essays — An algebra in which every non-zero element can be divided by. Over the real numbers there are exactly four with a multiplicative length — dimensions one, two, four and eight — and each step up the list gives up a law the one below it kept.
• Divisor 3 essays — A whole number dividing another without remainder. Sorting a number's divisors, or counting the numbers sharing each one with it, turns a question about multiplication into a partition — which is how several counting identities in number theory are proved.
• Divisor function 9 essays — The count of a number's divisors, which comes out as the product of one more than each exponent in its factorisation. Its multiplicativity is what makes it computable from the factorisation rather than by trial, and it is odd exactly for square numbers.
• Divisor method 7 essays — A rule for apportionment that divides every population by a common number and rounds the result. The divisor is adjusted until the rounded shares add to the right total, and the five classical methods differ only in where the rounding boundary sits between two whole numbers.
• Divisor sum 9 essays — The total of all a number's divisors, the number itself included. Comparing it with twice the number itself is what sorts numbers into deficient, perfect and abundant, and Euclid's rule builds a perfect one from every Mersenne prime.
• Domain restriction 1 essay
• Dominant strategy 4 essays — An option that pays its owner more than every alternative, whatever the other chooser does. When one exists the game needs no reasoning about the opponent, which is what makes it the strongest kind of solution.
• Domination 1 essay
• Dot product 4 essays — The number obtained from two vectors by multiplying matching coordinates and adding. It is the signed length of the shadow one casts on the other times the length of that other, so it is zero exactly at right angles, and every notion of angle in a space without a picture is defined from it.
• Double counting 1 essay
• Double cover 2 essays — A space that maps onto another so that every point below has exactly two points above it, arranged consistently nearby. The circle wrapping twice round itself is the simplest; rotations of space are covered twice by unit quaternions.
• Double negation 3 essays — Denying a denial, which classically returns the original statement and constructively does not. In a model made of open sets the first negation loses the boundary and the second hands it back, so the double negation is larger than what it started from.
• Doubling map 8 essays — The map sending x to 2x with the whole part discarded, on the numbers from 0 to 1. It shifts binary digits one place left, which makes its orbits exactly as varied as the digit sequences, and it is the simplest chaotic system that can be analysed completely.
• Doubling the cube 1 essay
• Doubly stochastic 1 essay
• Drift 3 essays — The average step of a random walk or process: how far it moves per unit of time once its fluctuations are averaged out. When the step depends on where the walk stands, the drift is an average over the places it visits, weighted by how often it visits them.
• Dual graph 1 essay
• Dual lattice 1 essay
• Duality 47 essays — A correspondence turning each object of one kind into an object of another and back again, with their roles exchanged. It halves the work whenever it holds, since every theorem proved on one side arrives free on the other.
• Dummy player 2 essays — A player whose arrival adds nothing to any coalition, whoever is already present. Any fair division gives such a player nothing, and that requirement is one of the four conditions that single out the Shapley value.
• Durfee square 2 essays — The largest square of dots that fits in the top-left corner of a partition's diagram. Cutting it out leaves two smaller partitions, one to its right and one below, and summing over the square's size is how several product formulas for partitions are proved.
• Dynamic programming 4 essays — A method that solves a problem by solving its smaller subproblems once each and storing the answers in a table. It turns searches that would revisit the same subproblem exponentially often into computations of polynomial size.

• e, the number 20 essays — The number whose exponential curve has slope equal to its own height everywhere, near enough 2.71828. It is the base at which the exponential is its own derivative, and it is the limit of compounding a hundred per cent interest ever more often.
• Eccentricity 2 essays — The fixed ratio of a conic's distance from a focus to its distance from a directrix. It is below one for an ellipse, exactly one for a parabola and above one for a hyperbola, so a single number produces the whole family from one focus and one line.
• Eckmann hilton 1 essay
• Edge count 4 essays — The number of edges a graph has, the simplest measure of how dense it is. Extremal questions ask how large the edge count can be before some structure is forced, and degree sums give it as half the total of all degrees.
• Edge density 1 essay
• Edge-disjoint paths 1 essay
• Efficiency 5 essays — The property of an allocation that no alternative makes someone better off without making someone else worse off. It is also called Pareto optimality, and it is the condition that fairness rules are most often forced to trade against.
• Egyptian fraction 1 essay
• Ehrenfeucht fraisse game 6 essays — A two-player game on a pair of structures whose outcome decides which sentences can tell them apart. Spoiler picks a point and Duplicator answers in the other structure, and surviving k rounds means no sentence of quantifier depth k separates them.
• Ehrhart polynomial 1 essay
• Eigenfunction 2 essays — A function that an operation such as differentiation merely rescales, as a matrix rescales an eigenvector. Sines are eigenfunctions of the second derivative, which is why a vibrating string or a cooling rod is solved by breaking its shape into sine waves.
• Eigenvalue 33 essays — The factor by which a linear map stretches one of the directions it does not turn. It is a root of the characteristic polynomial, and its size decides whether repeated application of the map grows or shrinks that direction.
• Eigenvector 12 essays — A direction a linear map leaves pointing the same way, changing only its length. The directions a map leaves alone are what make its long-run behaviour legible, since repeated application simply scales each of them.
• Electrical network 1 essay
• Elementary equivalence 5 essays — The relation holding between two structures when every sentence of first-order logic true of one is true of the other. It is weaker than isomorphism — a structure can have elementarily equivalent companions of every infinite size — and games are how it is established.
• Elimination 1 essay
• Ellipse 12 essays — The closed conic section, and the set of points whose distances to two fixed points add to a constant. It is the cut of a cone by a plane meeting every generator, and its two foci are where the inscribed spheres touch.
• Elliptic curve 5 essays — A smooth curve of degree three, such as y² = x³ + ax + b, whose points can be added: a line through two of them meets the curve at a third. That addition turns collinearity into arithmetic, and it governs counts of solutions modulo primes.
• Elliptic function 1 essay
• Embedding 3 essays — A way of placing one shape inside another without the copy crossing itself, so that the shape keeps its own form in the larger space. How much room a shape needs to embed is a measure of how twisted it is.
• Emergence 1 essay
• Empirical distribution 1 essay
• Encoding 2 essays — A rule attaching to each object of one kind a distinct object of another, usually a string of symbols. It turns comparison and computation on the originals into operations on the strings, and its cost is the length those strings need.
• Ends of a group 2 essays — The number of separate ways a group's picture runs off to infinity: remove a large finite piece of its Cayley graph and count the infinite pieces left. It is always 0, 1, 2 or infinitely many, and it records whether the group is finite, plane-like, a line up to finite fuzz, or splits along a finite subgroup.
• Entropy 11 essays — A number measuring how spread out a distribution is, computed as the average logarithm of one over each outcome's probability. It is the shortest average length any code for those outcomes can achieve, in the units the logarithm is taken in.
• Enumeration 1 essay
• Envelope 2 essays — The curve that touches every member of a family of lines or curves, each at one point. It is the shape a family draws without anyone drawing it, as the chords of a times table on a circle trace out a cardioid.
• Envy matrix 1 essay
• Envy-freeness 7 essays — The property of a division that nobody values another's share above their own. It is a stronger requirement than each person getting a fair share by their own reckoning, and for three or more people it is much harder to achieve.
• Epicycloid 1 essay
• Epsilon nought 2 essays — The smallest ordinal that cannot be written out of smaller ones using addition, multiplication and powers of omega. It is the limit of the tower ω, ω raised to ω, and so on, and it measures precisely how much induction ordinary arithmetic is able to carry out.
• Equicontinuity 1 essay
• Equidistribution 15 essays — The property of a sequence that the share of its terms landing in an interval tends to that interval's length. It is the strongest sense in which a deterministic sequence imitates a random one, and repeated rotation by an irrational angle has it.
• Equilateral triangle 5 essays — A triangle with three equal sides, and therefore three angles of 60 degrees. It is the shape that appears whenever a construction is forced to be symmetric under a third of a turn, which is why it answers questions about angle trisectors, about tilings and about the densest packing of circles.
• Equilibrium selection 7 essays — The question of which equilibrium happens when a game has several and the definition prefers none. The standard tie-breakers - preferring the one that pays more, or the one that is safer against an unknown opponent - can disagree, and the safer one usually has the larger basin under any best-reply rule.
• Equioscillation 1 essay
• Equivalence 5 essays — A relation that pools things into classes: reflexive, symmetric and transitive, so no thing is in two classes. It is what licenses working with classes instead of members, which is how modular arithmetic and quotient groups are built.
• Equivalence class 3 essays — The set of all objects that a given equivalence relation counts as the same as one particular object. The classes split the whole collection into disjoint pieces, and many constructions treat each class as a single new object.
• Equivalence relation 6 essays — A relation that is reflexive, symmetric and transitive, and therefore cuts a set into classes with every element in exactly one. It is the standard way of declaring two things the same for a stated purpose without them being identical.
• Erasure 3 essays — A transmission error in which a symbol arrives marked as missing rather than as a wrong value. Erasures are easier to correct than errors, since the location is known and only the value has to be recovered, and a code that corrects any e errors handles 2e erasures.
• Erdos ko rado 1 essay
• Erdos szekeres 1 essay
• Ergodic theorem 3 essays — The theorem that, for a thoroughly mixing map, the share of time an orbit spends in a region equals the region's measure. It holds for almost every starting point, and it turns averages over time into averages over space.
• Ergodicity 5 essays — The property that a single trajectory spends time in each region in proportion to that region's size. It makes a long-run average along one orbit equal to an average over the whole space, which is the assumption that statistical mechanics rests on and which is hard to prove in any particular case.
• Error analysis 5 essays — Accounting for how far a computed answer is from the exact one, and where the discrepancy came from. It separates what the mathematics guarantees from what the arithmetic delivers, which are routinely different: a construction can be exact on paper and lose every significant digit in floating point.
• Error bound 1 essay
• Error term 3 essays — The difference between a quantity and the approximation used for it, usually written as a bound on its size. Much of analytic number theory consists of shrinking error terms, where the main term is easy and the error decides everything.
• Error-correcting code 19 essays — A chosen set of words far enough apart that a corrupted one can be repaired to the nearest. How many errors it can repair is decided by its minimum distance, which is the smallest disagreement between two of its words.
• Escape-time 2 essays — The number of steps an orbit takes to leave a bounded region, used to shade the plane around a set. It is what colours a picture of the Mandelbrot set or of a Newton basin, and the boundary is where the count stops being stable.
• Estimate 4 essays — A bound or approximate value for a quantity that cannot conveniently be computed exactly, with a reason to trust it. A good estimate says how far off it can be, so that it can settle the questions the exact value would settle.
• Estimator 5 essays — A rule that turns observed data into a guess at an unknown quantity, such as a population's size or the number of kinds a further sample will reveal. It is judged by its bias, the average error, and by its spread from one sample to the next.
• Estimator bias 4 essays — The difference between what a random estimate gives on average and the quantity it is meant to estimate. An estimator with none is centred on the answer however noisy it is, which is why unbiasedness and precision are separate questions with separate remedies.
• Euclid 1 essay
• Euclid lemma 2 essays — The statement that a prime dividing a product must divide one of the factors. It is the step that proves factorisation into primes unique, and it fails exactly where an irreducible number is not prime, as in the numbers of the form 4k + 1.
• Euclidean algorithm 10 essays — Repeatedly replacing a pair of numbers by the smaller and the remainder, until nothing is left over. It finds the greatest common divisor, and a ring in which it can be run has unique factorisation as a consequence.
• Euler characteristic 53 essays — Vertices minus edges plus faces, which comes out the same for every way of dividing a given surface up. It is a topological invariant, so two surfaces with different values cannot be deformed into one another.
• Euler formula 12 essays — For a connected graph drawn in the plane, corners minus edges plus faces is two, counting the region outside as a face. It is what forbids a sixth regular solid and what makes the two obstructions to planarity obstructions.
• Euler relation 1 essay
• Eulerian circuit 2 essays — A closed walk through a graph that uses every edge exactly once. It exists in a connected graph exactly when every vertex has even degree, or, for arrows, as many arrows in as out at every vertex.
• Eulerian number 1 essay
• Eulerian orientation 1 essay
• Eulerian path 5 essays — A walk through a graph using every edge exactly once. One exists precisely when the graph is connected and at most two of its vertices have an odd number of edges — the first theorem of graph theory, and the reason de Bruijn sequences exist.
• Evolutionary stability 1 essay
• Exact arithmetic 14 essays — Computation carried out in whole numbers or fractions, with no rounding anywhere. It is what a claim of strict inequality requires, because an arithmetic that has already rounded a quantity to the edge of an interval cannot say whether it was strictly inside.
• Exact-cover 1 essay
• Exceptional object 1 essay
• Excess 2 essays — How much better a coalition could do alone than the division on offer gives it. Sorting every coalition's excesses and minimising the list in dictionary order picks exactly one division, whether or not any division satisfies all of them.
• Exchange algorithm 1 essay
• Exchangeability 2 essays — A sequence is exchangeable when reordering its terms never changes its probability. Such sequences need not be independent, and de Finetti and Kingman showed that each one is a mixture drawn from some random limiting distribution.
• Excluded middle 5 essays — The axiom that every statement holds or fails, which the constructive reading refuses because a proof of a disjunction has to say which side. Dropping it leaves a coherent logic whose models are the open sets of a space, where a set and its negation miss the boundary between them.
• Exclusive-or 2 essays — The logical operation that is true when exactly one of two inputs is true — addition of bits with no carry. It is the operation linear codes are built from, it cannot be computed by any single threshold, and that failure is the standard example of what one layer of a network cannot do.
• Exhaustive search 227 essays — Settling a question by generating every candidate and testing each one. It needs no cleverness and gives a definite answer, and it is available only when the candidates can be listed and the list is short enough to finish.
• Existence proof 55 essays — An argument establishing that something exists without producing it. Pigeonhole and fixed-point arguments are of this kind, and they are usually far shorter than any construction of the object.
• Existential import 1 essay
• Expectation 60 essays — The average of a quantity's values, each weighted by how likely it is. It need not exist at all: for a heavy-tailed quantity the defining sum or integral diverges, and every argument using it fails.
• Expected value 5 essays — The average of a random quantity, weighting each possible value by its probability. Expectations add even when the quantities depend on each other, which is why many counting arguments in probability go through them rather than through whole distributions.
• Exponential 2 essays — The function whose rate of growth equals its current value, written e to the power x. It turns addition into multiplication, describes compounding and decay, and on imaginary arguments it traces the unit circle.
• Exponential constant 1 essay
• Exponential decay 2 essays — A quantity decays exponentially when each step multiplies it by the same factor below one. On a logarithmic scale it falls along a straight line, and the slope is the logarithm of that factor, which is how such rates are usually read from data.
• Exponential growth 2 essays — Growth by a fixed factor in every equal step of time, so that the quantity doubles at regular intervals. It eventually overtakes every polynomial, and a count or cost that grows this way is usually the sign of a search that cannot be finished.
• Exponential time 2 essays — Running time that grows like a fixed number raised to the size of the input, such as 2 to the n. Methods for hard search problems are compared by that base, and whether some problems need exponential time at all is open.
• Expressive power 11 essays — What a language can and cannot distinguish, measured by the equivalence between structures it respects rather than by a list of what it can say. Naming that equivalence settles every question of the form *can this be expressed* at once.
• Extension property 2 essays — The property that for any two finite disjoint sets of vertices there is a vertex joined to everything in the first and nothing in the second. It characterises the infinite random graph, and finite graphs can have it for small sets.
• Extremal combinatorics 1 essay
• Extremal configuration 7 essays — An arrangement achieving the largest or smallest possible value of some count, given a constraint. Finding one settles half of an extremal problem; the other half is proving that nothing does better, and the two halves are usually different arguments.
• Extremal example 3 essays — An object attaining a bound exactly, which is what turns a true inequality into the best one available. For a bound derived from stated moments the extremal object sits on as few points as there are constraints, and its support is where the proof's inequalities become equalities.
• Extremal graph 5 essays — A graph with the largest number of edges possible while still avoiding some forbidden subgraph. Extremal graph theory asks how many edges force a pattern to appear, and which graphs sit exactly at the threshold without containing it.
• Extremal graph theory 1 essay
• Extremal problem 8 essays — A question asking how large or small a structure can be while keeping some property. The answer usually has two halves that are proved separately: a bound saying nothing can beat some number, and a construction reaching it.
• Extreme point 1 essay

• Factorial 8 essays — The product of all the whole numbers up to a given one, written with an exclamation mark. It counts the orderings of that many things, and it outgrows every fixed base raised to the same power — which is what makes it the multiplier in almost every squeeze argument.
• Factoring 3 essays — Writing a whole number as a product of primes, which is unique but can be hard to find. The difficulty of factoring large numbers underlies public-key cryptography, and the fastest methods look for a collision or a congruence of squares rather than dividing.
• Factorisation 5 essays — A way of writing an object as a product of smaller objects of the same kind. Whether the decomposition is unique is a property of the surrounding system rather than of the object, and uniqueness fails in arithmetics only slightly larger than the whole numbers.
• Fair division 13 essays — The problem of splitting one resource among people who value its parts differently, judged against a property the split must have. The properties usually asked for are proportionality and envy-freeness, and the difficulty is that they are not the same requirement.
• Fairness 11 essays — A stated condition a division or a rule must meet for nobody to have a complaint of a specified kind. Several such conditions conflict with one another, so choosing which to insist on decides which procedures remain available.
• False positive 1 essay
• Fano plane 5 essays — The smallest projective plane, with seven points and seven lines, three points on each line and three lines through each point. It is the smallest example of a finite geometry, and its incidence structure is also the multiplication table of the octonions.
• Farey sequence 6 essays — Every fraction in the unit interval with denominator at most a given bound, listed in increasing order. Adjacent entries always satisfy a determinant condition of one, which is what makes their circles tangent and what keeps every entry in lowest terms.
• Farkas lemma 1 essay
• Fast growing hierarchy 1 essay
• Feasible region 3 essays — The set of points satisfying every constraint of a program, which is where any optimum has to be found. It is an intersection of half-planes and so convex, which is why a linear objective attains its optimum at a corner of it.
• Feature attribution 1 essay
• Feigenbaum constant 5 essays — The number 4.6692…, the limiting ratio between successive intervals of a period-doubling cascade. It is the same for every smooth one-humped map with a quadratic top, a universality explained by a renormalisation that rescales the map onto itself.
• Fermat last theorem 1 essay
• Fermat prime 1 essay
• Fermats little theorem 12 essays — Fermat's little theorem: for a prime p and any a not divisible by it, a^(p−1) leaves remainder 1 on division by p. It is the basis of the fast tests that sift composite numbers from prime candidates, and the starting point for the certificates that prove primality outright.
• Ferrers diagram 3 essays — A picture of a partition of a whole number as rows of dots, the largest part on top and each row no longer than the one above. Reading the dots by columns instead of rows gives another partition of the same number, which is how many identities about partitions are proved.
• Fewnomial 1 essay
• Fibonacci 9 essays — The sequence in which each term is the sum of the two before it — 1, 1, 2, 3, 5, 8 and onward. Consecutive terms have ratio approaching the golden ratio, and they are the denominators of that number's continued-fraction convergents.
• Fibonacci numbers 10 essays — The sequence 1, 1, 2, 3, 5, 8, 13, in which each term is the sum of the two before. Their ratios approach the golden ratio, and reduced modulo any whole number they repeat with a period called the Pisano period.
• Fibration 2 essays — A way of filling a space with copies of one smaller space, such as circles, so that every point lies on exactly one copy and nearby copies sit side by side. Hopf filled the three-sphere with circles this way, every two of them linked once.
• Field automorphism 1 essay
• Field extension 17 essays — A field containing a smaller one, so that the larger one's elements can be built from the smaller one's. Adjoining a root of a polynomial is the standard way to build one, and its degree is what decides what can be constructed.
• Figurate numbers 3 essays — The counts of dots that fill a regular shape of growing size — triangles, squares, pentagons and on, or pyramids in three dimensions. Each is a running total of an arithmetic sequence, and which numbers are sums of a few of them has exact answers that Fermat began to state.
• Finite automaton 9 essays — A machine with finitely many states that reads a word one letter at a time and accepts or rejects at the end. The languages it recognises are exactly the ones a sentence with set quantifiers defines, so it is the computational face of a logical fragment rather than a separate subject.
• Finite branching 1 essay
• Finite check 1 essay
• Finite field 38 essays — A finite set with addition, multiplication and division that obey the ordinary rules, which exists exactly when its size is a prime power. Its multiplicative group is cyclic, which is what makes discrete logarithms and Reed-Solomon codes possible.
• Finite geometry 2 essays — A geometry with finitely many points, whose lines and planes are finite sets obeying incidence rules. Built from a finite field or from a combinatorial design, it supplies exact structures for error-correcting codes and experimental plans.
• Finite intersection 1 essay
• Finite model property 2 essays — The property of a logic that every sentence with a model has a finite one. With a bound on the size needed, it turns the question of whether a sentence is satisfiable into a search that is certain to end.
• First moment 4 essays — The expected number of copies of a structure, used to prove none exists. A count taking whole values and having a small mean is usually zero, which needs no independence at all and gives one direction of a threshold immediately.
• First moment method 1 essay
• First return 2 essays — The first moment at which a walk or an orbit comes back to the place it started from. On the line it happens with certainty and takes infinitely long on average, which is a pairing worth holding on to.
• First-passage percolation 1 essay
• Fisher information 1 essay
• Fixed field 1 essay
• Fixed point 34 essays — A point that a map leaves exactly where it was. Whether one must exist is decided by the shape of the set and the continuity of the map, and the theorems saying so are the deepest in the subject.
• Flip 2 essays — The move that replaces one diagonal of a convex quadrilateral inside a triangulation by the other diagonal. Any two triangulations of a polygon are joined by a chain of flips, and the length of the shortest chain is a distance between them.
• Floating-point 4 essays — The way computers store real numbers: a fixed number of significant binary digits and an exponent. Every operation rounds to the nearest representable value, so long calculations can lose accuracy through cancellation even when each step is correct.
• Flow 7 essays — An assignment of amounts to the roads of a network that stays within every road's capacity and passes on at each junction exactly what arrives there. Its value is what leaves the source, and the largest value possible is certified by a cut of the same capacity.
• Fluctuations 2 essays — How far a random quantity typically strays from its expected value. For sums of independent terms the spread grows like the square root of their number; for growth processes and random matrices it can grow at other rates.
• Focus 8 essays — A point that a conic section is defined in terms of, and to which the curve reflects rays from its partner or from infinity. An ellipse has two of them with distances summing to a constant, a parabola one, and the reflection property follows in each case.
• Forcing 1 essay
• Ford circles 1 essay
• Formal power series 3 essays — An infinite sequence of coefficients written with powers of a variable as position markers, added and multiplied by the usual rules. Nothing is ever substituted for the variable, so convergence never arises, and each coefficient of a product is settled by finitely many terms.
• Formal system 3 essays — A set of axioms and rules under which proofs are finite objects a machine could check. What such a system can say about its own proofs is itself a mathematical question, and the answers are the incompleteness theorems.
• Formula size 1 essay
• Fourier analysis 9 essays — The decomposition of a function into sines and cosines, whose amounts are found by projecting the function onto each one in turn. It turns convolution into multiplication, which is why it settles questions about sums of independent quantities and about differential equations alike.
• Fourier series 15 essays — A periodic function written as a sum of sines and cosines at whole-number multiples of one frequency. It turns a condition on a shape into a condition on which frequencies may appear, and differentiating becomes multiplication by the frequency.
• Fourier transform 2 essays — The recipe that rewrites a signal as a sum of waves of every frequency, recording how much of each is present. It turns convolution into multiplication and is how sampling, filtering and smoothing are analysed.
• Fourth power 1 essay
• Fractal 9 essays — A set with detail at every scale, whose measured dimension need not be a whole number. Its dimension is measured by how its content scales with the ruler used, and the answer is generally not a whole number.
• Fractal dimension 11 essays — An exponent saying how a set's content scales with the scale it is measured at, which need not be a whole number. It is computed by counting boxes of shrinking size, and it distinguishes sets that have the same topology and different roughness.
• Fractional brownian motion 1 essay
• Frame 5 essays — An ordered list of independent directions at a point, one for each dimension. Two frames at the same point are compared by the determinant of the matrix carrying one to the other, and the sign of that determinant is the whole of what an orientation is.
• Framing 1 essay
• Free action 1 essay
• Free group 13 essays — The group of all reduced words in a set of letters and their inverses, with no relations beyond a letter cancelling against its own inverse. It is the fundamental group of a wedge of circles, and every group generated by that many elements is a quotient of it.
• Free product 1 essay
• Frequency 3 essays — How many times a repeating pattern completes a cycle per unit of time or distance. A signal is described by the frequencies it contains, and sampling it too slowly folds a high frequency onto a low one.
• Frobenius number 1 essay
• Function space 1 essay
• Functional completeness 2 essays — A set of logical connectives is functionally complete when every truth function can be built from them alone. Post's theorem says a set fails exactly when it lies inside one of five maximal families, such as the connectives that preserve truth.
• Functional equation 1 essay
• Fundamental group 18 essays — The loops in a space based at one point, counted up to deformation, composed by running one after another. It separates spaces that no count of pieces can separate, and for a ring it is the whole numbers under addition.
• Fundamental solution 6 essays — The smallest solution in positive whole numbers of Pell's equation x² − Dy² = 1, leaving aside x = 1, y = 0. Every other solution comes from its powers, multiplying x + y√D by itself, so this one pair determines the whole infinite family.
• Fundamental theorem 6 essays — The result naming the central fact of a subject: for calculus, that integration and differentiation undo each other. The calculus one is what makes areas computable by antidifferentiation, and it is why the two halves of the subject are one subject.
• Fundamental theorem of algebra 1 essay

• Galois group 13 essays — The group of symmetries of the roots of a polynomial that preserve every relation among them with rational coefficients. Its subgroups match the fields between the rationals and the roots, which is how it decides what can be solved by radicals.
• Galton board 1 essay
• Gamblers ruin 2 essays — The chance that a gambler betting fixed stakes loses everything before reaching a target. For a fair game it is the distance to the target as a share of the whole range, and an unfavourable game makes ruin nearly certain.
• Game of life 1 essay
• Gamma function 2 essays — The function Γ(x) that extends the factorial to real and complex numbers, with Γ(n) = (n − 1)! for whole n. It appears wherever a volume, a probability density or a count involves factorials of non-integer size, such as the volume of a ball in any dimension.
• Gauss bonnet 2 essays — The theorem that the total curvature of a closed surface equals 2π times its Euler characteristic. Curvature can be moved about by bending, and the total cannot change, so a purely local quantity adds up to a topological one.
• Gauss lemma 2 essays — The rule that the Legendre symbol of a is minus one to the number of the multiples a, 2a, … that fold down from the top half of the residues. It is the count that reaches the two supplements, which the lattice-point proof of reciprocity cannot.
• Gauss sum 3 essays — A sum of roots of unity weighted by a character, such as the quadratic symbol, over all remainders modulo a prime. Its size is exactly the square root of the prime, which is what bounds character sums and proves reciprocity.
• Gaussian integers 9 essays — The complex numbers with whole real and imaginary parts, forming a ring with unique factorisation and four units. A prime of the ordinary integers stops being prime there exactly when it is a sum of two squares.
• General position 2 essays — Said of points none of which satisfy a coincidence they need not: no three on a line, no four on a circle, no three lines through a point. Statements proved for points in general position hold for almost every arrangement, and the exceptions have to be built on purpose.
• Generating function 24 essays — A series whose coefficients are the terms of a counting sequence. Multiplying two of them convolves the coefficients, so combining independent choices becomes multiplication and a recursion becomes an equation that can be solved.
• Generating set 4 essays — A collection of elements whose products reach every element of a group. The choice is not part of the group, so a picture drawn from it carries features that are artefacts — the degree and the diameter — beside features that are not, such as the growth rate.
• Generator 2 essays — An element of a group from which, together with its fellows, every other element can be built by multiplication and inversion. A group given by generators and relations is presented rather than described, and almost every question about it is then a question about words.
• Genus 22 essays — The number of holes in a surface, and the single number deciding which surfaces can be deformed into one another. It is one half of the classification of closed surfaces, the other being whether the surface is orientable.
• Genus theory 1 essay
• Geodesic 2 essays — The shortest path between two points along a surface, staying on the surface the whole way. On a sphere they are great circles, which is why the shortest air route between two cities looks bent on a flat map.
• Geometric mean 6 essays — The nth root of the product of n positive numbers, which is the average of their logarithms exponentiated. It is never larger than the arithmetic mean, with equality only when every number is the same, and the gap between the two is the logarithm's concavity.
• Geometric probability 2 essays — Probability computed from lengths, areas or volumes, when points, lines or arcs are placed at random in a region. It turns a question of chance into one of measuring a shape, and its classic results include Buffon's needle and the chance that random arcs cover a circle.
• Geometric series 22 essays — A sum whose terms shrink by a fixed ratio each step, finite in total exactly when that ratio is under one. Its total is the first term divided by one minus the ratio, and the picture of that is a square dissected into ever smaller pieces.
• Gerechte design 1 essay
• Gershgorin discs 1 essay
• Gibbs' phenomenon 3 essays — The overshoot near a jump in a Fourier partial sum, which narrows as terms are added and never shortens. It is why a truncated Fourier series of a square wave rings at the edges, and the overshoot settles at about nine per cent of the jump.
• Girth 1 essay
• Glider 2 essays — A small pattern in a cellular automaton that reproduces itself one step over after a few generations, so it travels across the grid. In Conway's Game of Life the five-cell glider moves diagonally one cell every four generations and carries signals in larger constructions.
• Gluing 2 essays — Building a space by identifying parts of simpler pieces, such as pairing the edges of a polygon so that each pair becomes one seam. The pairing word fixes the result up to topology, and every closed surface arises from a single polygon in this way.
• Gluing diagram 10 essays — A polygon with its edges paired and arrowed, describing the surface those identifications produce. Every closed surface has one, and reading the edge word off it computes the surface's characteristic and orientability.
• Gnomon 2 essays — The piece that, added to a figure, produces a larger figure of the same shape: an L of 2n + 1 dots turns an n by n square into the next square. Adding gnomons one at a time is the oldest way of proving that sums of simple sequences are squares, triangles and other shapes.
• Golay code 1 essay
• Goldbach 1 essay
• Golden ratio 31 essays — The proportion in which the whole is to the larger part as the larger part is to the smaller, which makes it one more than its own reciprocal. Its continued fraction is all ones, which makes it the number rational fractions approximate worst.
• Goodstein sequence 1 essay
• Gradient 4 essays — The vector of a function's partial derivatives, pointing in the direction of steepest increase. It vanishes at every minimum and at a great many other places, which is why a vanishing gradient certifies an optimum only under convexity.
• Gram schmidt 3 essays — The procedure that turns any independent set of vectors into a perpendicular one by subtracting from each vector its shadows on those already built. It preserves the space the vectors span, it reports dependence by returning a vector of length zero, and the coefficients it discards form the triangular half of a QR factorisation.
• Graph 41 essays — A set of points with a set of connections between them and nothing else — no distances, no angles, no positions. Everything else about a picture has been thrown away, which is what makes graph statements survive any redrawing.
• Graph colouring 18 essays — An assignment of colours to a graph's vertices in which no edge joins two of the same colour. The fewest colours needed is the chromatic number, and four suffice for every graph that can be drawn without crossings.
• Graph decomposition 1 essay
• Graph isomorphism 2 essays — The question of whether two graphs are the same graph with the points relabelled. It has no known polynomial algorithm and no proof of hardness, which places it in an unusual middle ground; the best known method is quasi-polynomial and the natural refinement approach is provably not enough.
• Graph labelling 1 essay
• Graph minor 3 essays — A graph obtained from another by deleting vertices and edges and contracting edges into points. Many properties survive taking minors, and each such property is decided by a finite list of forbidden minors.
• Gray code 5 essays — An ordering of the binary words in which each step changes exactly one place. It is a Hamiltonian cycle on the cube of binary words, and it is used wherever two bits changing at once would be read as a third value.
• Greatest common divisor 13 essays — The largest whole number dividing two others, which is also the side of the largest square that tiles a rectangle with those sides. It is computed by the Euclidean algorithm in a number of steps proportional to the digits, and never by factorising.
• Greedy algorithm 18 essays — A rule that repeatedly makes the choice which looks best at the moment and never reconsiders. It gives the true optimum for problems with a matroid structure, such as the minimum spanning tree, and only an approximation for most others.
• Greens theorem 1 essay
• Group 13 essays — A set with a way of combining any two of its elements that is associative, has an identity, and gives every element an inverse. It is the standard object for describing symmetry, because the motions that leave something unchanged always form one.
• Group action 18 essays — A group applied to a collection, each element permuting it, so that composing the elements composes the permutations. Counting orbits under one is what Burnside's lemma does, by averaging how much each element leaves fixed.
• Group order 1 essay
• Group presentation 1 essay
• Group representation 1 essay
• Group table 1 essay
• Growth model 1 essay
• Growth rate 14 essays — How fast the count of elements within a given distance of the identity rises in a group. It survives a change of generating set up to rescaling the distance, so polynomial and exponential growth are properties of the group, and polynomial growth forces a nilpotent structure.
• Gumbel distribution 1 essay

• Hadamard matrix 2 essays — A square matrix of entries +1 and −1 whose rows are mutually perpendicular, so that its determinant is as large as any sign matrix of that size allows. Such matrices exist only in sizes 1, 2 and multiples of 4, and whether every multiple of 4 has one is unknown.
• Half-life 1 essay
• Halls condition 1 essay
• Halting problem 4 essays — The question whether a given program, run on a given input, eventually stops. Turing proved that no algorithm answers it correctly for every program, by a diagonal argument, and most other impossibility results about programs are reduced to it.
• Ham sandwich theorem 1 essay
• Hamiltonian cycle 6 essays — A closed path in a graph that visits every point exactly once. Deciding whether one exists is hard in general; complete graphs have many, and they are the cycles that seating plans, travelling salesman tours and knotted paths through points are made of.
• Hamiltonian path 1 essay
• Hamming code 5 essays — A code of sixteen seven-bit words whose balls of radius one fill the whole space exactly, correcting any single error. It is perfect in the technical sense that its correction balls fill the space exactly, leaving no word unaccounted for.
• Hamming distance 14 essays — The number of places in which two strings of the same length disagree. It is city-block distance restricted to the corners of a cube, and it is what decides how many errors a code can survive.
• Handedness 1 essay
• Hardy littlewood conjecture 1 essay
• Harmonic conjugate 1 essay
• Harmonic measure 1 essay
• Harmonic series 25 essays — The sum of the reciprocals of the whole numbers, which grows without bound even though its terms shrink to nothing. Its partial sums track the natural logarithm, and its divergence is what forces the sum over the primes to diverge too.
• Harmonics 5 essays — The whole-number multiples of a base frequency, and the components a periodic wave is built from. How much of each one a wave holds is its spectrum, and that list determines the wave as completely as its shape over time.
• Hash collision 1 essay
• Hasse diagram 3 essays — A drawing of an ordering in which each element sits directly above those it covers, with the implied steps left out. It is the smallest drawing from which the whole ordering can be recovered, since the implied relations are exactly the paths upward.
• Hausdorff dimension 8 essays — The exponent at which a set's measure stops being infinite and becomes zero. The measure is defined by covering the set with pieces of any sizes at all and adding the s-th powers of their diameters, and the freedom in the sizes is why this dimension is at most the box one and is the notion theorems are stated about.
• Hausdorff distance 1 essay
• Heat equation 2 essays — The equation saying that temperature at a point changes at a rate proportional to how far it differs from the average of its surroundings. Each sine component of the starting profile decays at its own rate, fastest for the finest ripples, so sharp features are smoothed away first.
• Heavy tails 14 essays — A distribution whose extreme values are common enough to dominate averages, and whose variance or mean may not exist at all. The largest of a sample is then comparable to the whole sum, which is why averaging such quantities buys so little.
• Heawood number 1 essay
• Height 1 essay
• Henon map 4 essays — A map of the plane that stretches, folds and squeezes, sending (x, y) to (1 − ax² + y, bx). Iterated at its classic settings it traces a strange attractor, and it is the simplest invertible map known to have one.
• Hensel lemma 2 essays — The rule that a root of a polynomial modulo a prime, at which the derivative is not divisible by that prime, extends to a root modulo every power of it. Each step solves a linear congruence for one more digit, which is Newton's method carried out in p-adic arithmetic.
• Hereditary base 2 essays — A number written in a base with its exponents written in that base too, and their exponents likewise, until every digit visible is smaller than the base. Replacing the base by a larger one then leaves the shape of the expression unchanged, which is what a Goodstein sequence is built on.
• Heuristic 13 essays — An argument that predicts an answer without establishing it, usually by replacing a determined object with a random one that resembles it. A good one gets the constants right and fails on exceptional cases, which is exactly what the proof would have to handle.
• Hexagonal packing 1 essay
• Heyting algebra 6 essays — A lattice with an operation standing in for implication, taking the largest element whose meet with the antecedent lies below the consequent. The open sets of any space form one, and the constructive propositional logic proves exactly what holds in every such structure.
• Hierholzer's algorithm 3 essays — The procedure that builds a closed walk using every edge of a graph exactly once, by walking until stuck and splicing in detours from vertices with edges to spare. It is a proof as well as a method: the splicing fails only when the degree condition does.
• Higher homotopy 1 essay
• Hilbert hotel 2 essays — The illustration in which a hotel with a room for every whole number is full and can still take another guest. It makes the shifting bijection vivid and it is not itself the argument, since the correspondence is a single function rather than a sequence of moves in time.
• Hilberts tenth problem 1 essay
• Hinged dissection 2 essays — A dissection of one shape into pieces that are joined at hinges, so that the pieces swing as a chain from one shape into another. Dudeney's four-piece dissection of a triangle into a square is the best-known example.
• Hoeffding inequality 3 essays — A bound on how far a sum of independent bounded quantities strays from its average. For n terms each between 0 and 1, the chance of straying by t falls exponentially in t squared over n; it uses only each term's range, so it is simple and often far from sharp.
• Holder continuity 1 essay
• Homeomorphism 8 essays — A continuous correspondence between two spaces whose inverse is continuous too. It is the notion of sameness topology uses, and it preserves exactly what is defined from open sets — connectedness, holes, dimension — and nothing defined from distance.
• Homology 3 essays — An algebraic count of the holes in a space, built from cycles of edges, faces and higher cells modulo those that bound. Its first group is the group of loops made commutative, and it can be computed by linear algebra.
• Homomorphism 2 essays — A map from one structure to another that preserves its operations or relations, such as a map of graphs sending edges to edges. Counting homomorphisms of a small graph into a large one measures how often the small pattern appears.
• Homotopy 9 essays — A continuous family of maps interpolating between two of them, and the relation of being joined by one. Objects related this way are indistinguishable to any invariant defined by continuity alone, which is most of what topology computes.
• Hopf fibration 1 essay
• Horseshoe 1 essay
• Hurst exponent 1 essay
• Hyperbola 11 essays — The conic section obtained by cutting a cone at an angle steeper than its own side, giving two separate branches. Its points are those whose distances to two foci differ by a fixed amount, and the whole-number points on one branch are what Pell's equation asks for.
• Hyperbolic geometry 6 essays — The geometry in which, through a point off a line, infinitely many lines miss it, and the angles of a triangle add to less than half a turn. It has constant negative curvature, and most surfaces with two or more handles carry it.
• Hypercube 18 essays — The shape whose corners are all the ways of choosing true or false for each of several variables, with edges between corners differing in one. Its edges join words differing in one place, so a path along them is a sequence of single changes and its diameter is the word length.
• Hyperplane 1 essay
• Hypotenuse 4 essays — The side of a right triangle opposite the right angle, and the longest of its three sides. It is the side whose square the other two account for, and the one every parametrisation of a triple treats separately.

• Icosahedron 1 essay
• Ideal 1 essay
• Identity 1 essay
• Identity element 1 essay
• Idoneal number 1 essay
• Image 2 essays — The image of a map is the set of everything it actually reaches: all outputs, as the input ranges over its whole domain. For a linear map it is a subspace, and its dimension, the rank, plus the dimension of what is sent to zero equals the dimension of the domain.
• Imaginary unit 2 essays — A square root of minus one, multiplication by which turns the plane through a quarter turn. Adjoining it to the reals gives a field in which every polynomial has a root, which is what the fundamental theorem of algebra says.
• Immersion 6 essays — A smooth map whose derivative never collapses, so that every small patch of the source is a faithful patch of the image. It is weaker than an embedding, which is also one to one, and it is what allows a Klein bottle to be drawn in three dimensions at the cost of a crossing.
• Implication 4 essays — The logical connective written A → B, false only when A holds and B fails. It is proved by assuming A and deriving B, and used by combining it with a proof of A to obtain B, the step called modus ponens.
• Importance sampling 2 essays — A Monte Carlo method that draws samples more often where they matter and weights each by the inverse of how much more often it was drawn. The weighting keeps the estimate unbiased, and a good choice of where to sample can shrink its error enormously.
• Impossibility 18 essays — A theorem stating that no object of a described kind exists, as against a report that none has been found. What makes one is an argument covering every candidate at once, usually a quantity a candidate would have to change and provably cannot.
• Impossibility theorem 4 essays — A theorem showing that no procedure can satisfy a given list of requirements at once, rather than that nobody has yet found one. Arrow's and Gibbard's are the famous examples in voting; each derives a contradiction from the requirements, which is why the response is always to give one of them up.
• Imputation 5 essays — A division of what a group earns that hands out the whole of it and gives no member less than it could earn alone. The divisions satisfying every coalition's objection form a region inside that set, and the region is sometimes empty.
• Incidence 23 essays — The relation saying which points lie on which lines, taken as the whole of a geometry's structure. Taking it as the whole structure is what makes finite geometries possible, since no notion of distance or angle is required.
• Incircle 4 essays — The incircle of a triangle is the largest circle that fits inside it, touching all three sides. Its centre is where the three angle bisectors meet, and its radius is the triangle's area divided by half its perimeter.
• Inclusion exclusion 12 essays — Counting a union by adding the parts, subtracting the overlaps, adding back the triple overlaps, and continuing. It is exact and has one term per subset, so every practical use of it is a truncation — and truncating alternates between over- and under-counting.
• Incommensurability 5 essays — The property of two lengths with no common measure — no unit, however small, dividing both a whole number of times. Its discovery is what forced the Greeks to separate number from magnitude, and it is what the Euclidean algorithm failing to terminate detects.
• Incompleteness 4 essays — The fact that any consistent formal system strong enough for arithmetic has true sentences it cannot prove, and cannot prove its own consistency. Both halves follow from one modal axiom about what provability implies.
• Incremental cost 1 essay
• Independence 33 essays — The relation between two events when knowing that one happened says nothing about the chance of the other. It is what makes variances add, and it is a hypothesis that fails invisibly — no picture of a sample shows whether it holds.
• Independence of axioms 1 essay
• Independence of irrelevant alternatives 3 essays — The condition that a verdict between two candidates depends only on how the voters ranked those two. It is one of the conditions Arrow's theorem shows cannot all be met at once, and it is the one most rules give up.
• Independent set 2 essays — A set of vertices in a graph no two of which are joined by an edge. Counting them by size gives the independence polynomial, and finding the largest is one of the standard hard problems of combinatorial optimisation.
• Index 6 essays — The number of blocks a subgroup cuts its group into, which for finite groups is the group's size divided by the subgroup's. It counts the sheets of the corresponding covering space and the degree of the corresponding field extension.
• Indivisible goods 3 essays — Items that must go whole to one person, such as a house, a painting or a chore. They make exact envy-freeness impossible in general, so fairness for them is stated with a tolerance, such as envy removable by taking away one item.
• Induction 4 essays — A method of proof in which a statement about every whole number follows from the first case and from each case implying the next. Its strength is that a single argument covers infinitely many claims, and its cost is that the conclusion is only as sharp as the step.
• Inequality 8 essays — A statement that one quantity is at least another. Most named ones are a single supporting line at a well-chosen point of a convex function, which is why collecting convex functions is a way of collecting inequalities.
• Infimum 1 essay
• Infinite descent 4 essays — An argument that from any solution in whole numbers a strictly smaller one can be produced, which is impossible because the whole numbers do not descend forever. It proves an impossibility by exhibiting a procedure, and it survives into settings where no polynomial is available to argue about.
• Infinite path 1 essay
• Infinite product 1 essay
• Infinite set 1 essay
• Infinity 4 essays — The size of a collection that never runs out, such as the whole numbers or the points of a line. Arguments that work step by step for finite collections can fail for infinite ones, and repairing them usually needs a finiteness condition somewhere else.
• Information 2 essays — The reduction in uncertainty that learning something provides, measured in bits: an outcome of probability p carries the base-2 logarithm of 1/p of them. Averaged over outcomes it is entropy, and it sets the least number of yes-or-no questions that can identify an answer.
• Information dimension 1 essay
• Initial condition 1 essay
• Injection 1 essay
• Inner product 12 essays — The operation pairing two vectors or functions into a number, from which length and the angle between them are defined. Only one of the p-norms comes from one, which is why angles and projections exist for ordinary distance and for no other exponent.
• Inscribed angle 8 essays — The angle a chord subtends at a point of the circle, the same wherever on its arc that point sits. It is exactly half the central angle on the same arc, which is why a triangle on a diameter always has a right angle.
• Inscribed square 1 essay
• Instability 1 essay
• Instant runoff 1 essay
• Integer 1 essay
• Integer program 1 essay
• Integrability 3 essays — The property of a dynamical system that enough conserved quantities exist to confine each trajectory to a low-dimensional set. An integrable system's orbits are orderly and describable in closed form; the typical system has no such quantities.
• Integral 25 essays — It is the limit of sums of rectangles under a curve, and it measures accumulated total rather than instantaneous rate. The fundamental theorem says that it undoes differentiation, which is what makes areas computable from antiderivatives.
• Integral geometry 8 essays — The study of geometric quantities recovered from averages over random lines, positions or orientations. Buffon's needle is its first result: a probability computed by averaging over positions and orientations turns out to contain the circle constant.
• Integrality 6 essays — The fact that a rational root of a polynomial whose leading coefficient is one must be a whole number. It turns an infinite search over fractions into a finite search over the divisors of the constant term, and it is what decides which roots of whole numbers are irrational.
• Integrality gap 2 essays — How much better the best fractional solution of a problem is than its best whole-number solution. When there is no gap the linear program's prices certify the whole answer; when there is one, no prices can, and the size of the gap bounds how well rounding can do.
• Integration by parts 1 essay
• Interaction 1 essay
• Interlacing 3 essays — The pattern in which the roots of two real-rooted polynomials alternate along the line, one root of each between consecutive roots of the other. It is what deleting a vertex does to a matching polynomial and what removing a row and column does to a symmetric matrix's eigenvalues.
• Interleaving 1 essay
• Intermediate value theorem 9 essays — A continuous quantity that takes two values also takes every value between them. It is what makes bisection work, and it is the one-dimensional ancestor of every fixed-point theorem in this collection.
• Intermittency 1 essay
• Interpolation 10 essays — Finding a function of a chosen kind that passes exactly through given data points. For polynomials there is exactly one of degree below the number of points, and how good it is depends far more on where the points were placed than on how many there are.
• Interpretation 1 essay
• Intersecting family 3 essays — A collection of sets in which every two members share at least one element. The Erdős–Ko–Rado theorem bounds how large such a family of k-sets can be, and in a Kneser graph the families are exactly the sets of vertices that one colour may take.
• Intersection number 1 essay
• Interval 4 essays — A stretch of the line between two endpoints, with or without them. It is the object measure is defined against — every other set's length is an infimum over coverings by intervals — and a set containing none of them is nowhere dense.
• Interval arithmetic 2 essays — Computation in which each number is replaced by an interval guaranteed to contain it, with every result rounded outward. A computer can then prove inequalities despite rounding error, which is how computer-assisted proofs in dynamics are made rigorous.
• Intractability 1 essay
• Intrinsic knotting 1 essay
• Intrinsic property 1 essay
• Intrinsically linked 1 essay
• Intuitionistic logic 6 essays — Classical logic without the axiom that every statement holds or fails, so that a proof of a disjunction must name a side. Its models are Heyting algebras and stages of knowledge, and classical reasoning embeds inside it under a translation inserting double negations.
• Invariance 1 essay
• Invariance of domain 2 essays — The theorem that a continuous one-to-one map from an open piece of n-dimensional space into n-dimensional space has an open image. Its consequence is that spaces of different dimensions are never homeomorphic, which is what makes dimension a property of a space rather than of a way of describing it.
• Invariant 66 essays — A quantity computed from an object that does not change under the transformations being allowed. Finding one is how impossibility is proved: if a quantity never changes, no sequence of moves reaches a position where it differs.
• Invariant density 2 essays — A density on the states of a dynamical system that the map carries to itself. When it exists it gives the long-run fraction of time a typical orbit spends in each region, and for the logistic map at its top parameter it is the arcsine law.
• Invariant direction 3 essays — A line through the origin that a linear map sends into itself, stretched but never turned. It is an eigenvector by another name, and the factor by which the map stretches it is the eigenvalue.
• Invariant measure 6 essays — A way of assigning sizes to sets that a map leaves unchanged: the preimage of every set has the same size as the set. It is what turns long-run questions about an orbit into questions about areas, and for a chaotic map it gives the share of time spent in each region.
• Invariant set 2 essays — A set that a map or flow sends into itself, so that a point starting inside never leaves. The interesting ones are much smaller than the space they sit in, and the dynamics restricted to them is what a system's long-run behaviour is.
• Invariant subspace 1 essay
• Inverse 5 essays — The element that undoes another under an operation, returning the identity. Modulo a whole number it exists exactly for the residues coprime to the modulus, and the extended Euclidean algorithm finds it in a few steps.
• Inverse function 1 essay
• Inversion 8 essays — The map that sends each point out along its own ray from a fixed centre, to the distance whose product with the original is the square of a fixed radius. It carries circles and lines to circles and lines and preserves angles, which is what turns a hard tangency question into an easy one.
• Invertibility 1 essay
• Involution 6 essays — An operation that undoes itself, so that applying it twice returns what it started from. Pairing objects by an involution that reverses a sign is the standard way to make a signed sum collapse: the paired terms cancel and only what the pairing cannot touch survives.
• Irrational rotation 12 essays — Turning a circle repeatedly by an angle that is not a rational part of a full turn, so no point is ever revisited. Its orbit fills the circle densely, which is why the three-distance theorem and equidistribution both concern it.
• Irrationality 13 essays — The property of a number that no fraction of whole numbers equals it. It is proved either by a descent, by a finite search over candidates, or by a squeeze in which something forced to be a whole number is shown to lie strictly between nought and one.
• Irrationality measure 2 essays — The exponent saying how closely a number can be approached by fractions relative to their denominators. Rational numbers have measure one and every irrational number at least two; almost every number has exactly two, and a proof of irrationality usually bounds the measure from above too.
• Irreducible 3 essays — An object with no smaller one of its kind hiding inside it. A representation is irreducible when it has no invariant subspace, and the irreducible ones are the pieces every other is assembled from.
• Irreducible polynomial 12 essays — A polynomial that cannot be written as a product of two smaller ones with coefficients from the same field. It is what plays the role of a prime in a polynomial ring, and quotienting by one is how a field extension is built.
• Irreversibility 2 essays — The property of a process that it cannot be run backwards to recover where it started. Heat flow smooths every profile and forgets its fine detail, so running it in reverse would have to amplify ripples without limit, and for almost every starting state it cannot be done.
• Irrevocable decision 8 essays — A choice that cannot be revisited once made, so that a rejected option is gone and an accepted one ends the process. It is the hypothesis that makes a stopping problem hard, and no amount of information about the alternatives removes the cost it imposes.
• Isodiametric inequality 1 essay
• Isogonal conjugate 1 essay
• Isolated vertex 2 essays — A point of a graph with no edges. In a random graph their expected number is the point count times e to the minus the average degree, and their disappearance is exactly when the graph becomes connected.
• Isomorphism 3 essays — A one-to-one correspondence between two structures that carries every relation and operation of one onto those of the other. Isomorphic structures differ only in the names of their elements, so any property expressible in their language is shared.
• Isoperimetric inequality 4 essays — The statement that a closed curve of length L encloses at most L²/4π of area, with equality only for the circle. Its relatives say the same of other quantities — the lowest eigenvalue of a region among them — and symmetrisation proves most of them.
• Isoperimetric problem 1 essay
• Isosceles triangle 1 essay
• Isotopy 1 essay
• Iterated function system 5 essays — A finite list of contracting maps, whose attractor is the unique set equal to the union of its own images under them. It is the standard way of specifying a fractal, and the dimension it forces can be read off the maps' contraction ratios without covering anything.
• Iteration 38 essays — Applying a rule to its own output, over and over, and asking what the resulting sequence does. What the sequence does is decided by the slope at the fixed point, and the same rule can settle, cycle or wander depending on a parameter.
• Iterative scaling 1 essay
• Itinerary 3 essays — The sequence of labels an orbit collects, one for each region of the space it lands in, step by step. It turns the motion of a point into a string of symbols, so that questions about trajectories become questions about words.

• Jacobian 1 essay
• Jensen inequality 1 essay
• Jones polynomial 3 essays — A polynomial in t attached to every knot by Vaughan Jones in 1984, unchanged by every deformation of the knot. Unlike the Alexander polynomial it can tell a knot from its mirror image, and its span gives the exact crossing number of every alternating knot.
• Jordan curve 6 essays — A closed curve in the plane that never crosses itself. The theorem named after it says the complement of such a curve falls into exactly two pieces, which is obvious for the curves anyone draws and needs real machinery for the ones the hypotheses actually allow.
• Judgement aggregation 9 essays — Combining several people's yes-or-no verdicts on a set of logically connected propositions into one collective verdict. No rule that responds to the votes is both decisive on every proposition and free of self-contradiction.
• Julia set 4 essays — The boundary between the starting points whose orbits under a fixed polynomial map escape to infinity and those whose orbits stay bounded. For z² + c it is connected exactly when c lies in the Mandelbrot set, and otherwise it is a dust of points.

• Karnaugh map 2 essays — A grid of a formula's truth values laid out so that squares which touch differ in exactly one variable. Adjacent squares differ in one variable, so a block of them is a single simplified term and minimising a formula becomes covering a picture.
• Kauffman bracket 2 essays — A polynomial in a variable A computed from a knot diagram by smoothing every crossing in both possible ways and counting the circles each result leaves. It is unchanged by two of the three Reidemeister moves, and corrected by the writhe it becomes the Jones polynomial.
• Kempe chain 1 essay
• Kepler poinsot 1 essay
• Kernel 9 essays — The set of inputs a linear map sends to nothing, which is a subspace of the space it started from. Its dimension is the nullity, and it plus the rank is the dimension of the source — so what a map destroys and what it keeps always add to what it was given.
• Kidney exchange 1 essay
• Kirchhoff 1 essay
• Kirkman 1 essay
• Kissing number 2 essays — The largest number of equal non-overlapping balls that can all touch one central ball of the same size. It is 6 in the plane, 12 in space and 24 in four dimensions, and only a handful of other dimensions have known values.
• Klein bottle 5 essays — The one-sided closed surface made by gluing a square's edges with one pair joined straight and one flipped. It cannot be embedded in three dimensions without passing through itself, and cutting it in half gives two Mobius bands.
• Knapsack 1 essay
• Kneading sequence 1 essay
• Kneser graph 1 essay
• Knot 21 essays — A closed curve in space, considered up to deformations that never pass one strand through another. Telling two apart means finding an invariant, since no amount of failed rearranging proves that a rearrangement does not exist.
• Knot determinant 7 essays — The absolute value of a knot's Alexander polynomial at −1, a whole number that every diagram of the knot gives alike. It counts colourings of the knot with a prime number of colours in the sense that a prime dividing it permits non-trivial ones.
• Koch curve 3 essays — The curve obtained by repeatedly replacing the middle third of every segment with two sides of a triangle. It has infinite length and no area, four pieces at a third the size, and a dimension of log 4 over log 3 - which is the standard first example of a dimension that is not a whole number.
• Konig lemma 1 essay
• Kripke frame 1 essay
• Kripke model 10 essays — A set of worlds with arrows between them, together with an assignment saying which statements hold at each world. It gives the modal operators their meaning: necessary at a world means true at every world that world points to.
• Kummer's theorem 3 essays — The statement that a prime divides a binomial coefficient exactly as many times as there are carries when the two lower indices are added in that prime's base. It turns a question about an enormous number into a question about column addition, and Lucas' theorem is its case of no carries.

• Labelled graph 1 essay
• Labelled tree 7 essays — A tree — a connected network with no cycles — whose vertices carry distinct names, so two trees are the same only if the same pairs of names are joined. There are n^(n−2) of them on n names, Cayley's formula, and the labels are what make that count finite and exact.
• Lagrange theorem 5 essays — The rule that the size of any subgroup divides the size of the group containing it. It follows from the cosets all having the same size, and it is what limits which subgroups a group can have at all.
• Lagrangian relaxation 1 essay
• Lamé's theorem 1 essay
• Langford pairing 1 essay
• Langtons ant 1 essay
• Lap number 1 essay
• Laplacian 1 essay
• Large deviations 4 essays — The study of how fast the chance of a fixed-size departure from the typical falls as the sample grows. The answer is exponential in the sample size, and the exponent is computed from the summand before any sample size is chosen.
• Largest prime factor 2 essays — The biggest prime that divides a whole number. For a number chosen at random up to x it is typically about x^0.62, its distribution is given by the Dickman function, and it decides how hard the number is to factor by trial division.
• Latin rectangle 1 essay
• Latin square 16 essays — A square array in which every symbol appears exactly once in each row and each column. It is the multiplication table of a quasigroup, and the question of when two can be superimposed with all symbol pairs distinct is two centuries old.
• Lattice 43 essays — The regular array of points reached by taking whole-number steps along a fixed set of directions. It is where number theory and geometry meet: counting its points inside a region turns arithmetic questions into questions about area.
• Lattice paths 7 essays — Paths on a grid built from steps in a fixed set, usually one to the right or one upward. Counting them is a matter of choosing which steps go which way, so their numbers are binomial coefficients, and families of them that never touch are counted by a determinant.
• Lattice point 4 essays — A point of the plane or of space whose coordinates are all whole numbers. Counting the lattice points inside a region estimates its area or volume, and the error in the count is where much number theory lives.
• Lattice walk 1 essay
• Laurent series 1 essay
• Law of cosines 2 essays — The relation giving the third side of any triangle from the other two and the angle between them, whose extra term vanishes at a right angle. It is the Pythagorean theorem with a correction term, and the correction vanishes exactly when the angle is a right angle.
• Law of large numbers 11 essays — The statement that the average of independent repetitions of a measurement settles on the underlying mean as their number grows. It is what makes an average an estimate at all, and it says nothing about how fast — that is a separate and finer question.
• Law of sines 2 essays — The identity that in any triangle each side divided by the sine of the angle opposite it gives the same value, twice the circumradius. It converts a statement about angles into one about lengths, and is the usual way a trigonometric proof of a geometric theorem gets started.
• Lcm 1 essay
• Leaf 2 essays — A point of a tree joined to exactly one other point. Every tree with two or more points has at least two leaves, and removing a leaf leaves a smaller tree, which is how most proofs about trees proceed.
• Least common multiple 1 essay
• Least squares 4 essays — The rule that fits a model to data by minimising the total of the squared discrepancies. Squaring is what makes the answer a projection: the fitted values are the point of the subspace the model can reach that is nearest the data, and the leftover meets that subspace at a right angle.
• Lebesgue constant 2 essays — The largest factor by which interpolation at a given set of points can magnify an error in the data, taken over the whole interval. It grows exponentially for equally spaced points and only like the logarithm of their number for Chebyshev's.
• Lebesgue integral 1 essay
• Legendre symbol 8 essays — A quantity that is plus one when a is a non-zero square modulo the prime p, minus one when it is not, and nought when p divides a. It multiplies over its top argument, which reduces any such question to the primes and the two supplements.
• Legendre transform 2 essays — The construction sending a convex function to the record of its own tangent lines, indexed by slope. It converts minimisation into evaluation, addition into an infimal convolution, and undoes itself — which is why duality appears wherever it does.
• Lexicographic order 2 essays — The order in which words appear in a dictionary: compare the first letters, and where they agree compare the next. It is used to pick a canonical representative from a collection of equivalent objects, because it costs nothing to state and two parties can arrive at the same one independently.
• Lifting 5 essays — Following a path or a map upstairs through a covering, so that it projects back to the original. A path lifts uniquely once a starting point above is chosen, and whether a loop lifts to a loop is what decides which subgroup a cover names.
• Lifting map 2 essays — Sending a planar point to the point above it on the paraboloid, so that its height is the square of its distance from the origin. It turns every circle in the plane into a plane in space, and every inside-the-circle question into an inequality.
• Light cone 1 essay
• Likelihood 6 essays — The probability of the observed evidence under a given hypothesis, read as a measure of how well that hypothesis explains it. In Bayes' theorem the ratio of two likelihoods is the factor by which the evidence multiplies the odds between the hypotheses.
• Limit 44 essays — The value a sequence or a function approaches as closely as anybody likes, whether or not it ever arrives. Its existence is separate from its value, and separate again from the rate at which the approach happens.
• Limit function 1 essay
• Limit ordinal 3 essays — An ordinal with things below it and nothing immediately below: the least thing above an increasing collection rather than one step past a predecessor. Induction along the ordinals needs a case of its own for these, and that case is where arithmetic's own strength runs out.
• Limit shape 6 essays — The curve or region that a large random or combinatorial object settles onto once it is scaled down by its size. A ball in a lattice becomes a polygon, a random partition's diagram becomes a fixed curve, and the fluctuations about the limit are usually where the harder questions are.
• Line at infinity 1 essay
• Line integral 1 essay
• Linear algebra 1 essay
• Linear code 1 essay
• Linear convergence 1 essay
• Linear dependence 2 essays — A relation among vectors in which a non-trivial combination of them adds to zero. Any list longer than the dimension of the space is dependent, which is how a counting of dimensions proves that some polynomial equation must exist.
• Linear extension 1 essay
• Linear independence 4 essays — The property of a set of vectors that none of them is a combination of the others, so that each contributes a direction the rest cannot reach. It is what makes coordinates unique, and Gram-Schmidt tests for it as a side effect by measuring how much of each vector survives the removal of its shadows.
• Linear inequality 1 essay
• Linear map 2 essays — A function between vector spaces that respects addition and scaling, sending lines through the origin to lines or to the origin. Once bases are chosen it is a matrix, and its kernel and image measure what it loses and what it reaches.
• Linear program 16 essays — The problem of making a weighted sum as large or as small as possible subject to finitely many straight-line inequalities. Its optimum is always at a corner of the feasible region, which is what makes searching corners a complete method.
• Linear programming 14 essays — Maximising a linear expression subject to linear inequalities. Its optimum is always attained at a corner of the region the inequalities cut out, which is what makes it solvable and what makes the corners' arithmetic decide everything about it.
• Linear recurrence 5 essays — A rule that gives each term of a sequence as a fixed weighted sum of the terms just before it. Its solutions are sums of powers of the roots of one polynomial, which is why such sequences grow geometrically and can be computed quickly by repeated squaring of a matrix.
• Linear system 11 essays — A collection of linear equations in several unknowns, solved by elimination. A great many quantities in probability that look like limits — expected times, absorption chances, hitting times — are exact solutions of one instead.
• Linearisation 1 essay
• Linearity 12 essays — The property that a quantity applied to a sum is the sum of the quantities, and scales with a multiple. It is what makes superposition work, and it is the property every approximation in this collection tries to arrange locally.
• Link 1 essay
• Linking number 11 essays — A whole number attached to two disjoint closed loops in space, counting with signs how often one passes through a surface the other bounds. No deformation keeping the loops apart can change it, so a non-zero value proves they cannot be separated.
• Liouville function 1 essay
• Liouville number 3 essays — A number approachable by fractions faster than any fixed power of the denominator allows. Every one of them is transcendental, since an algebraic number of degree d cannot be approached better than the dth power permits, and one is built by spacing decimal digits factorially.
• Lipschitz function 2 essays — A function that changes by at most a fixed multiple of the change in its input, so that |f(x) − f(y)| ≤ L|x − y| for some constant L. When L is below one the function is a contraction and has a unique fixed point.
• List decoding 1 essay
• Listing 1 essay
• Literal 3 essays — A variable, or the negation of a variable — the smallest piece from which a clause is built. Working in literals rather than in variables is what allows a single inference rule to cancel one variable between two clauses.
• Littlewood polynomial 1 essay
• Local global principle 1 essay
• Local minimum 5 essays — A point at which a function is no larger than at every nearby point. For a convex function there is only one such value and it is the global minimum; without convexity a local minimum can be arbitrarily worse than the best, and nothing local tells them apart.
• Local optimum 1 essay
• Local search 3 essays — A method that improves a candidate solution by small changes, keeping a change whenever it helps. It is fast and often good, but it can stop at a solution no single change improves that is far from the best.
• Locality 9 essays — The restriction that each part of a system is affected only by what is immediately next to it. It is what makes a cellular automaton's global behaviour surprising, since every rule in it sees only its immediate surroundings.
• Locus 4 essays — The set of all points satisfying a stated condition, taken as a shape in its own right. Describing a curve as one turns a geometric condition into an equation, which is the whole method of coordinate geometry.
• Log-concavity 4 essays — The property of a sequence whose every term squared is at least the product of its two neighbours, so that the logarithms form a concave sequence. It forces the sequence to rise to a peak and fall without dips, and it follows whenever the generating polynomial has only real roots.
• Logarithm 30 essays — The exponent a base must be raised to in order to give a number, which turns multiplication into addition. Turning products into sums is exactly why it appears in the prime counting theorem and in every entropy.
• Logarithmic spiral 1 essay
• Logic 2 essays — The study of which conclusions follow from which assumptions, made precise enough to be checked mechanically. It supplies the notion of consistency that every aggregation problem is measured against.
• Logical connective 1 essay
• Logistic map 15 essays — The one-parameter rule that sends a number to r times itself times one minus itself, on the unit interval. As the parameter rises it goes from a fixed point through period doubling into chaos, all with one quadratic rule.
• Longest common subsequence 1 essay
• Longest increasing subsequence 2 essays — The longest run of terms of a sequence, not necessarily adjacent, that rises from left to right. For a random permutation of n numbers its length is close to twice the square root of n, and the same question can be asked of points in a square or a cube.
• Loop 4 essays — A path in a space that ends where it began. Which loops can be shrunk to a point is the first question topology asks of a space, and the answer already distinguishes a disc from a ring.
• Lorenz system 4 essays — Three differential equations for a crude model of convection, written down by Edward Lorenz in 1963. Their solutions never settle and never repeat, winding round the butterfly-shaped attractor that made sensitive dependence famous.
• Lottery 1 essay
• Lower bound 33 essays — A proof that no method can do better than some amount, as opposed to a method that achieves it. Lower bounds are usually harder than upper bounds because they must defeat every possible strategy, including ones nobody has thought of.
• Lowest terms 5 essays — A fraction written so that its numerator and denominator share no factor above one. Every fraction has exactly one such form, and nearly every argument about rational numbers begins by taking it, since it is where unique factorisation gets its grip.
• Lucas' theorem 4 essays — The statement that a binomial coefficient is odd exactly when the binary digits of the lower index sit under those of the upper. It is the reason shading the odd entries of Pascal's triangle produces a fractal, and it generalises to any prime with that prime's digits in place of binary.
• Lyapunov exponent 13 essays — The average rate at which two nearby orbits of a map separate, measured as the logarithm of a factor per step. A positive value is what chaos means quantitatively, and its reciprocal is how many steps a computed orbit tracks the true one before the rounding has been amplified to full size.
• Lyapunov function 1 essay
• LYM inequality 1 essay
• Lyndon word 1 essay

• Magic square 1 essay
• Mahler measure 1 essay
• Majority 5 essays — The rule that accepts a proposition when more than half of those voting do. It is the only anonymous rule treating acceptance and rejection alike, and on an agenda whose propositions imply one another it can produce a collectively inconsistent verdict.
• Majority game 1 essay
• Majority rule 12 essays — The rule that settles a yes-or-no question by giving it to the side with more votes. It is the only rule that treats voters alike, treats the two answers alike and never punishes a candidate for gaining support — May's theorem — and it can still produce a set of verdicts that contradict one another when several questions are decided at once.
• Mandelbrot set 8 essays — The parameters for which the orbit of zero under squaring-and-adding stays bounded. It is connected, its boundary has dimension two, and which parameters lie in it is decided by an escape-time test.
• Manifold 1 essay
• Manipulation 1 essay
• Mantel theorem 1 essay
• Many valued logic 4 essays — A logic whose sentences can take more than the two values true and false, such as a third value or a whole interval of degrees. Its laws are those that hold under every assignment of values, and which classical laws survive depends on the tables chosen.
• Mapping class group 1 essay
• Marginal contribution 10 essays — What a player adds to a coalition on joining it — the value of the group with them, less its value without. Averaging it over every order in which the players could have arrived is the Shapley value, and the four conditions that pick out that average are conditions on the averaging rather than on the contributions.
• Marked straightedge 3 essays — A ruler carrying two scratches a fixed distance apart, slid until the marked segment's ends lie on two stated curves. The slide solves an equation of degree three or more, which is why it trisects any angle where compass and straightedge cannot.
• Market clearing 4 essays — A state of prices at which every buyer is content with what they get and nothing is left over that someone would pay more for. In the assignment problem such prices exist, and they are exactly the dual certificate that proves an assignment is the best.
• Market design 1 essay
• Markov chain 21 essays — A process moving between states with fixed probabilities that depend only on the state it is in, never on how it got there. Its long-run behaviour is read off the powers of its transition matrix, and the stationary distribution is the direction that matrix leaves alone.
• Markov chain monte carlo 1 essay
• Markov partition 2 essays — A division of a space into pieces that a map carries exactly across whole pieces, never part of the way into one. Orbits are then coded by walks on a finite graph, so counting periodic points and measuring entropy become computations with a matrix.
• Martingale 10 essays — A sequence of fortunes in a fair game: whatever has happened so far, the expected next value equals the current one. Stopped at a sensible moment it still averages its starting value, which turns waiting times and winning chances into bookkeeping about fair bets.
• Mass distribution principle 1 essay
• Mass partition 1 essay
• Matching 22 essays — A set of pairings in which no element appears twice. Whether one covering everything exists is decided by a condition on every set of elements at once, and a largest one is found by improving a partial one along alternating paths.
• Matrix 34 essays — A rectangular grid of numbers, read as the instructions for a linear map. Multiplying two of them composes the maps, which is why the chain rule in several variables is a matrix product.
• Matrix exponential 5 essays — The power series 1 + A + A²/2 + A³/6 + …, applied to a square matrix A, which always converges. It solves the linear differential equation with matrix A, and unlike the logarithm it can be applied to every square matrix.
• Maximal element 2 essays — An element of a partially ordered set with nothing strictly above it, though other elements may be incomparable to it. Zorn's lemma guarantees one whenever every chain has an upper bound, which is how spanning trees, bases and maximal ideals are shown to exist without being built.
• Maximin 1 essay
• Mean field 1 essay
• Mean value theorem 4 essays — The statement that a differentiable function's change across an interval equals its slope somewhere inside, times the interval's length. It converts a bound on the derivative into a bound on the function, which is how nearly every error estimate in analysis is obtained.
• Measure 24 essays — An assignment of size to the parts of a set, adding up whenever disjoint parts are put together. It is what makes it possible to say that a set has length zero while still having uncountably many points.
• Measure zero 12 essays — The property of a set that can be covered by intervals of total length as small as desired. Every countable set has it, and so do uncountable ones such as the Cantor set — so it is a notion of smallness quite separate from how many members a set has.
• Measurement 3 essays — In quantum mechanics, a complete set of yes-or-no questions, one for each direction of an orthonormal basis, of which exactly one is answered yes. The state fixes the probabilities of the answers, and the Kochen–Specker theorem says the answers themselves cannot all be fixed in advance.
• Medial axis 2 essays — The set of points inside a shape that have two or more nearest points on its outline: the Voronoi diagram of the boundary itself. With the distance to the boundary recorded at each point it rebuilds the shape exactly, and it is where fires lit all round the edge would meet.
• Median 6 essays — The middle value of a list once it is sorted, with as many values at or below it as at or above. Unlike the average it ignores how far the extremes lie, and in a distributive lattice the same idea, taken coordinate by coordinate, is the majority vote that picks out a middle element.
• Median of means 1 essay
• Median voter 3 essays — The voter whose ideal position on a single left-to-right scale has as many voters on each side. When preferences fall away from each voter's ideal, the median's position beats every other in a straight vote, which is why competing parties are pulled towards it.
• Mediant 8 essays — The fraction formed by adding two fractions' numerators and adding their denominators. It is not addition and not an average, it lies strictly between its two parents, and repeating it from the right starting pair produces every positive rational exactly once and already in lowest terms.
• Membership 1 essay
• Membership pattern 1 essay
• Mersenne prime 2 essays — A prime of the form 2ⁿ − 1, such as 3, 7, 31 and 127; n must itself be prime for it to have a chance. Each one gives an even perfect number, 2ⁿ⁻¹(2ⁿ − 1), and every even perfect number arises this way — so the two lists are the same length, and nobody knows whether either is infinite.
• Mertens constant 3 essays — The constant 0.2615 by which the sum of the reciprocals of the primes up to x exceeds log log x. Franz Mertens proved in 1874 that the difference converges, and the constant fixes the average number of distinct prime factors of a whole number.
• Mertens function 1 essay
• Method of exhaustion 7 essays — Pinning a quantity down by bounding it above and below with pieces that can be made as small as needed. It is what Archimedes used for the circle and it is the ancestor of the limit, arriving two thousand years before one was defined.
• Metric 1 essay
• Metropolis algorithm 1 essay
• Midpoint 3 essays — The point halfway along a segment, equidistant from both ends. It is the piece of metric information a straightedge cannot produce, and having one is equivalent to being able to draw parallels.
• Miller rabin 1 essay
• Milnor invariant 1 essay
• Min-max theorem 5 essays — A theorem saying that the largest value of one quantity equals the smallest value of another that bounds it from above. Max-flow min-cut, linear-programming duality and the minimax theorem are the standard cases, and each turns a claim of optimality into a checkable certificate.
• Minimal counterexample 1 essay
• Minimal polynomial 10 essays — The smallest-degree polynomial with whole-number coefficients that a given number satisfies. Its degree is the degree of the extension the number generates, which is what settles constructibility questions.
• Minimality 1 essay
• Minimax 7 essays — The most a cautious chooser can guarantee, which in a zero-sum game is the same number the opponent can hold them to. For a zero-sum game the two players' guarantees coincide, which is what the minimax theorem establishes.
• Minimum cut 1 essay
• Minimum distance 3 essays — The smallest number of places in which two different codewords of a code disagree. A code corrects up to half of it in errors, which makes it the single number describing what a code can do.
• Mixed strategy 11 essays — A weighting over the options open to a chooser, judged by the payoff it returns on average. Allowing them is what makes an equilibrium always exist, since a game with no equilibrium in pure choices may have one in weights.
• Mixing 1 essay
• Mixing time 3 essays — The number of steps a Markov chain needs, from its worst starting state, to come within a fixed distance — usually a quarter — of its stationary distribution. It is what decides how long a chain must run before its output can be used.
• Möbius band 7 essays — A strip joined end to end after a half twist, leaving one side and one edge. It is the standard example of non-orientability, and cutting it down the middle gives one longer two-sided band rather than two.
• Mobius function 3 essays — The function on whole numbers that is 0 at any number with a repeated prime factor, and otherwise +1 or −1 as the number of prime factors is even or odd. It inverts summing over divisors, and its values sum to the counts that inclusion and exclusion produce.
• Mobius transformation 5 essays — A map of the plane, extended by one point at infinity, given by a quotient of two linear expressions in a complex variable. Such maps carry circles and lines to circles and lines, preserve angles, and are exactly the compositions of an even number of inversions in circles.
• Modal logic 7 essays — The logic of an operator meaning *necessarily*, evaluated over diagrams of worlds and arrows. Its axioms are not a matter of taste: each holds on exactly the diagrams whose arrows have a stated property.
• Model 10 essays — A structure giving each symbol of a formal language a meaning, so that every sentence of it becomes true or false. Exhibiting one is how a statement is shown unprovable from a set of axioms, since a derivation would have to hold in every model.
• Model misspecification 2 essays — The situation in which the statistical model used to analyse data does not contain the process that actually produced them. Inference still converges, but to the member of the model closest to the truth, and its stated uncertainty can be badly overconfident.
• Modular arithmetic 63 essays — Arithmetic on a finite dial, in which numbers differing by a multiple of the dial's size count as the same number. Addition and multiplication are well defined on the classes, which is what makes the dial a ring and not merely a labelling.
• Modular group 5 essays — The group of transformations z ↦ (az + b)/(cz + d) with whole-number entries and ad − bc = 1, acting on the upper half of the complex plane. It carries any pair of neighbouring fractions to any other, and its motions tile the half-plane with congruent curved triangles.
• Modulus 4 essays — The size of the dial that arithmetic is being done on — the number that counts as zero. Whether the dial's arithmetic has division depends entirely on it: every non-zero element is invertible exactly when it is prime.
• Moment generating function 3 essays — The average of e raised to the power λX, for a random quantity X, taken as a function of λ. Its derivatives at nought give the moments, independent sums multiply it, and every exponential tail bound of the Chernoff kind is read off it by choosing λ well.
• Monodromy 5 essays — The permutation of a covering's sheets obtained by following a loop in the base and noting which sheet each lift ends on. It records the whole covering as an action of the loops, and a branch point is a point round which it is not the identity.
• Monoid 1 essay
• Monotone 1 essay
• Monotone function 1 essay
• Monotone property 2 essays — A property of graphs that cannot be destroyed by adding edges, such as being connected or containing a triangle. Every such property has a threshold in random graphs, and whether the threshold is sharp is decided by locality.
• Monotone subsequence 1 essay
• Monotonicity 6 essays — The property that a rule's output moves in one direction when its input moves in one direction — more of something in never gives less of something out. It is the property most often quietly assumed and most often violated by rules that look reasonable.
• Monsky theorem 1 essay
• Monte Carlo 14 essays — Computing a deterministic quantity by averaging a random process designed to have that quantity as its mean. Its error falls like one over the square root of the sample size whatever the dimension, which makes it slow everywhere and the only workable method in high dimensions.
• Monty hall 1 essay
• Moore graph 1 essay
• Morley theorem 4 essays — Morley's trisector theorem: the adjacent trisectors of any triangle's angles meet in three points that form an equilateral triangle. Discovered in 1899, it holds for every triangle whatever, and its proofs run backwards from the equilateral triangle rather than forwards from the given one.
• Morley triangle 1 essay
• Morse theory 1 essay
• Moving knife 2 essays — A division procedure in which a knife sweeps continuously across a good and a participant calls stop when the piece passed over is worth a set share. It gives exact proportional divisions, but it needs continuous attention rather than a finite sequence of questions.
• Multilinearity 2 essays — The property of being linear in each argument separately, with the others held fixed. It is what lets a function of several vectors be expanded over a basis one argument at a time, and it is the first of the three conditions that leave the determinant as the only possibility.
• Multinomial 1 essay
• Multiplication 1 essay
• Multiplicative function 10 essays — A function on the whole numbers whose value at a product of two coprime numbers is the product of its values at each. Its values are determined by what it does at prime powers, which is what turns a sum over the integers into a product over the primes.
• Multiplicative order 2 essays — The smallest power of a number that leaves remainder 1 when divided by a given modulus. It is the period of the remainders of the powers, and it is the length of the repeating block in the expansion of a fraction.
• Multiplier 1 essay
• Myerson value 1 essay

• Napoleons theorem 1 essay
• Nash equilibrium 16 essays — A choice by every participant such that none of them can do better by changing their own while the rest stay fixed. Its existence for finite games follows from a fixed-point theorem, and finding one is a different matter from knowing it is there.
• Nash welfare 1 essay
• Natural deduction 3 essays — A proof system whose rules come in pairs, one saying how to make a formula with a given connective and one saying how to use one. Its distinguishing feature is the discharge, which withdraws an assumption once its consequence has been derived and so turns a conditional derivation into a proof of an implication.
• Natural frequencies 1 essay
• Natural logarithm 5 essays — The power to which e must be raised to reach a given number. Its derivative is one over the input, which is what makes it appear whenever a sum over the whole numbers is estimated.
• Nearest-neighbour 1 essay
• Necessity 1 essay
• Necklace 6 essays — An arrangement of coloured beads in a ring, counted as one object when arrangements that differ only by a rotation are the same. Counting necklaces is the standard application of Burnside's lemma, and the count of prime length gives a proof of Fermat's little theorem.
• Negation 2 essays — The logical operation that turns a statement into its opposite, true exactly when the original is false. Applied to a quantified sentence it swaps every for some and some for every, and pushes itself inward until it lands on the property at the end.
• Negative correlation 1 essay
• Neighbourhood 1 essay
• Nested intervals 1 essay
• Network 1 essay
• Neusis 3 essays — A construction step that slides a segment of fixed length until both ends meet stated curves. It reaches numbers of degree three, five and six over the rationals, so it trisects an angle and builds the eleven-sided polygon, and its exact reach is not characterised.
• Newtons method 15 essays — The iteration that replaces a guess at a root by where the tangent there crosses the axis. Its derivative at a simple root is exactly zero, so the error is squared at each step and the number of correct digits doubles — which is why it converges in a handful of steps and why it fails badly at a multiple root.
• Next-bit test 1 essay
• Non-crossing partition 1 essay
• Non-measurable set 6 essays — A set of points to which no length, area or volume can be assigned consistently with translation and countable addition. Such sets exist because of the axiom of choice, and a ball can be cut into five of them and reassembled into two balls.
• Non-orientable 8 essays — A surface on which a consistent sense of clockwise cannot be chosen, because some route reverses it. A shape carried round the offending route comes back mirrored, and no consistent choice of normal exists.
• Nonconstructive 14 essays — Said of a proof that establishes something exists while giving no way to find it. Such a proof is worth having and worth flagging, since it leaves the question of finding the object entirely open.
• Norm 16 essays — A measure of the size of a vector or a number, and the notion of distance that follows from it. Which one is chosen decides what nearest means, and only the ordinary one is unchanged by rotation.
• Normal distribution 21 essays — The bell-shaped distribution that sums of many small independent contributions settle into. It is the fixed point of convolution followed by rescaling, which is why it and no other shape is what sums arrive at.
• Normal form 10 essays — A standard way of writing an expression, chosen so that equivalent expressions written that way are identical. Having one turns the question of whether two expressions are equal into the question of whether two strings are identical.
• Normal number 4 essays — A number whose digits in a given base contain every block of digits as often as a random sequence would. Almost every number is normal, but no familiar constant such as the square root of two or pi is known to be.
• Normal order 2 essays — A function has normal order g(n) when f(n) is close to g(n) for almost every n, though not necessarily on average. The number of distinct prime factors has normal order log log n, which is the content of the Hardy–Ramanujan theorem.
• Normal subgroup 5 essays — A subgroup carried into itself by conjugation with every element of the group. It is exactly the condition under which the cosets can be multiplied, so it is what makes a quotient group exist, and a chain of them with abelian steps is what solvability by radicals requires.
• Normal vector 2 essays — A vector perpendicular to a surface or a plane at a point, used to say which way the surface faces there. Under a linear map it is carried by the inverse transpose rather than by the map itself, which is why the normal of a stretched surface is not the stretched normal.
• Notation 1 essay
• Nowhere dense 2 essays — A set is nowhere dense when its closure contains no interval — however small a window, the set misses some sub-window entirely. A countable union of such sets is meagre, and Baire's theorem says the whole line is never meagre, which is what makes nowhere-dense sets a usable notion of small.
• Nowhere differentiable 1 essay
• NP-hard 5 essays — The property of a problem that every problem whose solutions can be checked quickly reduces to it in polynomial time. A fast algorithm for any one such problem would give fast algorithms for all of them, and none is known.
• Nucleolus 2 essays — The division of a group's winnings whose loudest complaint is quietest, with remaining ties broken by the second loudest and so on. It exists and is unique for every game, agrees with the stable divisions whenever any exist, and reproduces a division table from the Mishnah.
• Null-player 1 essay
• Nullity 2 essays — The dimension of the subspace a linear map sends to nothing. It plus the rank is the dimension of the map's source, so it measures exactly what was lost; on a graph's incidence matrix it counts the independent cycles.
• Numerical artefact 1 essay
• Numerical integration 5 essays — Approximating an integral by a weighted average of the integrand's values at chosen points. How the points are chosen — on a grid, at random, evenly spread, or on a lattice — decides how fast the error falls as more points are used.

• Object language 1 essay
• Odd cycle 1 essay
• Odd perfect number 1 essay
• Oligarchy 1 essay
• Online algorithm 1 essay
• Open problem 23 essays — A question that has been stated precisely, attacked seriously, and not settled. The useful ones come with an identified obstruction — a named reason the available methods stop short — rather than merely an absence of answers.
• Open set 3 essays — A set containing a neighbourhood of each of its points, so that no point of it sits on its own edge. The open sets of any space form a model of the constructive logic, in which negation takes the inside of the complement and loses the boundary.
• Operation set 18 essays — The fixed collection of moves a construction is allowed, which decides what can and cannot be reached. Changing it changes the answer completely: trisection is impossible with straightedge and compass and easy with a marked ruler.
• Optimal stopping 10 essays — Choosing when to accept an offer that arrives in a sequence, with no going back. What the answer looks like depends entirely on what the chooser is told and what counts as success, and changing either produces a different constant from the same arrival process.
• Optimality 10 essays — The property of being best under a stated objective, together with a proof that nothing else does better. The proof is the substance: a value with no argument that nothing beats it is a candidate rather than an optimum.
• Optimisation 20 essays — Finding the value of a variable that makes a quantity as small or as large as possible, subject to constraints. Whether it is tractable turns almost entirely on convexity, which decides not how fast an answer is found but whether a found answer can be recognised.
• Orbit 31 essays — The set of places a point is carried to by repeating an operation until it returns. It may be finite, may return exactly, or may never repeat, and which of the three happens is the first question asked of any iteration.
• Order 5 essays — The number of times an operation has to be repeated before it returns to where it started. It divides the size of the group, which is Lagrange's theorem applied to the subgroup that one element generates.
• Order isomorphism 1 essay
• Order lattice 6 essays — An ordering in which every two elements have a greatest lower bound and a least upper bound, whatever else is incomparable. The power set of any collection is one, with intersection and union playing the two roles.
• Order of an element 6 essays — The smallest number of times an element must be combined with itself to give the identity. It divides the size of the group — Lagrange's theorem — and knowing an element's order exactly is how one certifies that a group is as large as it is claimed to be.
• Order statistics 1 essay
• Order type 5 essays — What is left of an ordered set when the identity of its members is forgotten and only the arrangement is kept. Two sets share an order type when some relabelling matches them in order, and the order types of well-ordered sets are the ordinals.
• Ordering 1 essay
• Ordinal 7 essays — The order type of a well-ordered set: a way of counting that continues past the finite while recording arrangement rather than size. Ordinals can be compared, added and multiplied, and a process that strictly decreases one at every step must terminate.
• Ordinal arithmetic 1 essay
• Ordinary line 6 essays — A line through exactly two points of a finite set in the plane. The Sylvester–Gallai theorem says that any finite set of points not all on one line has at least one, and later work bounds how few there can be.
• Oresme 1 essay
• Orientability 7 essays — The property of a surface that a consistent sense of turning can be chosen at every point at once. It fails exactly when carrying a frame round some loop returns it mirrored, and it is what a one-sided surface lacks, stated without reference to any surrounding space.
• Orientation 21 essays — A consistent choice of which way round a surface or a path is traversed, on the surfaces where such a choice can be made at all. It exists on a sphere or a torus and not on a Mobius band, and its absence is what makes a surface non-orientable.
• Orientation double cover 2 essays — The two-sided surface that lies two to one over a surface, each of its points being a point below together with a choice of sense of turning there. It is connected exactly when the surface below is one-sided, and it doubles the Euler characteristic.
• Orthogonal latin squares 6 essays — Two Latin squares of the same order such that superimposing them produces every ordered pair of symbols exactly once. Families in which every pair is orthogonal exist at every prime power order, are the same objects as finite planes, and are what Euler's officer problem asks for.
• Orthogonal matrix 1 essay
• Orthogonality 23 essays — The relation between two directions at right angles, so that neither has any component along the other. It is what makes coefficients computable one at a time, which is why a Fourier coefficient is a projection and not a fitted parameter.
• Orthonormal basis 2 essays — A basis whose vectors are mutually perpendicular and each of length one. Its value is that a coordinate becomes a single inner product rather than the solution of a system, so coefficients can be computed independently of each other and a truncated expansion is the best approximation available from the directions kept.
• Oscillation 3 essays — The oscillation of a function at a point is how far its values still spread in arbitrarily small neighbourhoods of that point. In the limit it is the gap between the largest and smallest nearby values, and it is zero exactly where the function is continuous.
• Outer billiards 1 essay
• Outside option 1 essay

• P adic numbers 6 essays — Numbers built by measuring size through divisibility by a prime p, so that high powers of p count as small. In them 1 + 2 + 4 + … converges to −1, and equations can be solved digit by digit by lifting a solution modulo p.
• P-value 3 essays — The chance, if a stated null hypothesis is true, of a result at least as extreme as the one observed. A small value counts against the null, but it measures surprise under that one hypothesis and says nothing about how much likelier any alternative makes the data.
• Pade approximant 2 essays — A quotient of two polynomials whose own series agrees with a function's series for as many terms as its coefficients allow. Because its denominator can vanish it can imitate poles and cuts, and it often converges where the Taylor series built from the same coefficients diverges.
• Pairing function 1 essay
• Pairs 1 essay
• Pairwise majority 7 essays — The relation formed by taking candidates two at a time and letting the larger side of the electorate settle each pair. It need not be transitive, and the cycles it can produce are why a majority verdict may fail to assemble into a ranking.
• Paley graph 2 essays — The graph on the residues modulo a prime of the form 4k + 1, with two residues joined when their difference is a square. It is half-dense, highly symmetric and behaves in many ways like a random graph.
• Pants decomposition 1 essay
• Parabola 5 essays — The conic section at the boundary between closed and unbounded, whose points are as far from a focus as from a line. It is the cut parallel to the cone's own slope, and every ray arriving parallel to its axis reflects through the focus.
• Paraboloid 1 essay
• Paradox 2 essays — An argument from acceptable premises by acceptable steps to a conclusion that seems unacceptable. Resolving one means finding which premise or step fails, and several famous paradoxes turned out to mark the boundary of a theory rather than an error in it.
• Parallel 1 essay
• Parallel lines 1 essay
• Parallel postulate 2 essays — Euclid's fifth assumption, that through a point beside a line there is exactly one line that never meets it. Denying it gives consistent geometries, which is what the disc model exhibits and what settled two thousand years of attempts to prove it.
• Parameter 2 essays — A quantity held fixed while a system runs and varied between runs, so that a family of behaviours is indexed by it. Plotting a measured quantity against one is how a bifurcation, a plateau or a threshold becomes visible at all.
• Parametrisation 4 essays — A map from an interval to a space, whose image is a curve and which carries more information than the image does. Length, speed and direction belong to the map rather than to the set of points, so one curve traced three times over is three times as long.
• Pareto efficiency 1 essay
• Parity 74 essays — Whether a whole number is even or odd, which is all that a surprising number of arguments turn on. It is the simplest invariant there is, and it settles puzzles that no amount of searching would: a move that preserves it cannot change it.
• Parity check 1 essay
• Parrondos paradox 1 essay
• Parseval identity 1 essay
• Partial latin square 1 essay
• Partial order 7 essays — A way of comparing elements that need not compare every pair, being reflexive, transitive, and never putting two distinct elements each below the other. Subsets under inclusion and whole numbers under divisibility are the standard examples, and the pairs it leaves uncompared are what its antichains count.
• Partial sum 2 essays — The sum of the first n terms of a series, whose limit as n grows is the series' value if it has one. How fast the partial sums approach that limit decides how useful the series is for computation, and acceleration methods transform them to approach it faster.
• Partition 12 essays — A way of writing a whole number as a sum of whole numbers with the order disregarded. The counts have no simple closed form and are handled instead through a product whose factors decide how many copies of each part are used.
• Partition function 1 essay
• Partitions 1 essay
• Pascals triangle 5 essays — The triangle of binomial coefficients, each entry the sum of the two above it. Row n lists the ways to choose 0, 1, … n things from n, and the triangle's patterns — symmetry, row sums of 2ⁿ, divisibility — are identities about those choices.
• Path-connected 1 essay
• Pell equation 10 essays — The equation x² − Dy² = 1 in whole numbers, for a D that is not a perfect square. It always has infinitely many solutions, and every one of them is a power of the smallest under a multiplication that is really multiplication of numbers of the form a + b√D.
• Penrose tiling 1 essay
• Pentagon 1 essay
• Pentagonal number 1 essay
• Pentagonal number theorem 1 essay
• Percolation 5 essays — A random model in which each edge or site of a grid is kept with some probability, and the question is whether the kept parts connect over long distances. Above a critical probability they do, and below it they do not.
• Perfect code 3 essays — A code whose correction balls fill the whole space of words exactly once, with nothing left over. They are rare — the repetition codes, the Hamming codes and the Golay code are the complete binary list — and they are the codes that meet the sphere-packing bound.
• Perfect matching 4 essays — A set of edges in a graph covering every vertex exactly once, pairing the vertices off. Counting perfect matchings gives the permanent of a bipartite graph's adjacency matrix, and for planar graphs a determinant computes it.
• Perfect number 10 essays — A number equal to the sum of its own proper divisors, as six is one plus two plus three. Every even one comes from a Mersenne prime by Euclid's rule, and whether any odd one exists is unknown.
• Perfect square 1 essay
• Perimeter 2 essays — The length of the boundary of a plane region. Compared with the area it encloses it measures roundness, since the isoperimetric inequality makes the disc the shape of least perimeter for its area.
• Period 6 essays — How many steps a repeating sequence takes before it comes back to where it started. For a generator it bounds how much output is usable, and for a linear recurrence over a finite field it is decided by whether a polynomial is primitive.
• Period-doubling 10 essays — A bifurcation at which a cycle loses stability and is replaced by one of twice its length. The parameter gaps between successive doublings shrink by a universal ratio, which is the same for a whole class of maps.
• Periodic orbit 43 essays — An orbit that returns exactly to a value it has already taken, and then repeats forever. Its length and stability are what a bifurcation diagram records, and a period-three orbit forces orbits of every other period.
• Periodicity 20 essays — The property of a pattern that repeats itself exactly after a fixed step. It is what a Fourier series assumes, and imposing it on a function that lacks it is what produces the artefacts at the ends.
• Permanent 5 essays — The sum over all permutations of the same products a determinant takes, with the minus signs deleted. It counts perfect matchings and the ways of extending a Latin rectangle, and having no cancellation in it, it cannot be evaluated by elimination.
• Permutation 60 essays — A rearrangement of a collection that sends each member to one place and leaves nothing doubled up. The parity of one is invariant under the way it is written as swaps, which is why some puzzle positions are unreachable.
• Permutation parity 1 essay
• Perpendicular bisector 3 essays — The line of points equally far from two given ones, at right angles to the segment joining them. The three of a triangle meet at the centre of its circumscribed circle, which is why that circle exists at all.
• Persistence 1 essay
• Petersen graph 3 essays — A graph of ten vertices and fifteen edges, every vertex with three neighbours and no cycle shorter than five. It is the counterexample to many plausible statements about graphs and the best-known member of several small extremal families.
• Phase portrait 1 essay
• Phase transition 10 essays — A qualitative change in a system's behaviour produced by a small change in a parameter. In a random structure it is a property whose probability moves from near zero to near one over a vanishing range, and the sharpness is a fact about the limit rather than about any finite case.
• Phyllotaxis 2 essays — The arrangement of leaves or seeds around a stem, set by a fixed turn between successive ones. The turn is close to the golden angle, which is the arrangement that avoids lining up for the longest.
• Pi 33 essays — The ratio of a circle's circumference to its diameter, and a constant that turns up in a great many places with no circle in them. It is the ratio a circle fixes, and its appearances in counts of lattice points and in Buffon's needle come through that ratio rather than by coincidence.
• Pick theorem 3 essays — The rule giving a lattice polygon's area as the interior dot count plus half the boundary count minus one. It is exact, it has no analogue in three dimensions, and its exactness comes from the boundary being made of segments between lattice points.
• Piecewise isometry 1 essay
• Piecewise-linear 2 essays — Built from finitely many straight pieces joined end to end, as a function whose graph is a broken line or a surface made of flat faces. Such objects are simple enough to compute with exactly and flexible enough to approximate any continuous one.
• Pigeonhole 2 essays — The observation that distributing more things than there are places forces some place to take two. It proves existence without producing anything, and its power comes entirely from choosing what to count as a place.
• Pigeonhole principle 25 essays — The observation that putting more things than boxes into the boxes forces two of the things to share one. It is the shortest existence argument in mathematics, and the skill in using it is entirely in the choice of boxes.
• Pivotal player 1 essay
• Place value 2 essays — A way of writing numbers in which each digit is multiplied by a power of the base according to its position, so 307 means three hundreds, no tens and seven ones. Reading digits in a prime base turns divisibility questions about binomial coefficients into questions about carries.
• Planar graph 20 essays — A graph that can be drawn in the plane with no two of its edges crossing. Its edge count is bounded by three times its vertices, which is what rules out the complete graph on five points.
• Planarity 8 essays — The property of a graph that it can be drawn in the plane with no two edges crossing. Kuratowski's theorem says exactly two graphs obstruct it, so failing is always failing for one of two reasons.
• Platonic solids 12 essays — The five convex solids whose faces are congruent regular polygons meeting alike at every corner. There are five because the angles at a corner must fall short of a full turn, which is a counting argument rather than a search.
• Poincare section 1 essay
• Point at infinity 1 essay
• Point set 8 essays — A finite collection of points in the plane or in space, studied for the patterns its points must form. Questions about point sets ask how many lines, distances or empty shapes are forced, whatever the arrangement.
• Pointwise convergence 4 essays — A sequence of functions converges pointwise when, at each point separately, the values converge to the limit's value there. It is weaker than uniform convergence, which asks one rate to work everywhere at once, and the gap between them is where continuity, integrals and limits stop being preserved.
• Poisson approximation 8 essays — The approximation that the number of occurrences of many rare, nearly independent events behaves like a Poisson count with the same mean. The chance of none is then about e to the minus that mean, which turns an awkward exact calculation about coincidences into a one-line estimate.
• Polar 1 essay
• Pollard rho 1 essay
• Polya urn 2 essays — An urn from which a ball is drawn at random and returned with another of the same colour, so every draw makes its own colour likelier. Its share of red settles on a random value, and the number of red draws is exactly uniform when it starts with one ball of each colour.
• Polygon 8 essays — A closed curve made of finitely many straight segments joined end to end. Almost every statement about curves is easy for polygons and hard in general, because a polygon has finitely many corners, a length, and a triangulation, and a curve need have none of them.
• Polygon clipping 1 essay
• Polyhedron 16 essays — A solid bounded by flat faces meeting along straight edges. Its corners, edges and faces satisfy Euler's relation, which is what limits how many regular ones can exist — and the answer to that is five.
• Polynomial 35 essays — An expression built from a variable by adding and multiplying only, with whole-number powers. Its degree bounds its number of roots, and over the complex numbers it has exactly that many when they are counted properly.
• Polynomial approximation 7 essays — Replacing a function by a polynomial that agrees with it in some stated sense. The sense may be agreement at a point to high order, exact agreement at several points, or the smallest possible worst error across an interval, and a polynomial optimised for one of those loses badly under another.
• Polynomial method 3 essays — Proving that a set must be large by building a polynomial of low degree that would vanish on it if it were small. A polynomial of low degree cannot vanish everywhere it is forced to, and the contradiction gives the bound.
• Polynomial roots 1 essay
• Polytope 13 essays — The generalisation of a polygon and a polyhedron to any number of dimensions, bounded by flat pieces of one dimension lower. Regular ones are infinite in number in two dimensions, five in three and six in four, after which exactly three exist in every dimension.
• Poncelet porism 1 essay
• Poset 4 essays — A set with a rule saying that some pairs of its elements come before others, consistently and without cycles, leaving the rest incomparable. It is the mathematics of partial information: what is known about an order, before everything has been compared.
• Positive definite 6 essays — The requirement that a rule for multiplying two vectors give a strictly positive answer whenever a vector is multiplied by itself, unless the vector is zero. It is what turns such a rule into a geometry, since it makes the self-product a squared length, and it is the property whose loss reverses the Cauchy-Schwarz inequality.
• Post classes 1 essay
• Posterior 2 essays — The probability distribution over hypotheses after the data are taken into account, proportional to the prior times the likelihood. It is what Bayes' theorem produces, and it is only as trustworthy as the list of hypotheses and the model behind the likelihood.
• Potential function 2 essays — A single number attached to each state of a game that falls by exactly the amount a mover gains, at every unilateral move. Nobody is trying to change it and it is nobody's payoff; its whole use is that improving moves walk downhill on it, so they cannot cycle and must reach a state from which nothing improves.
• Potential game 1 essay
• Power diagram 1 essay
• Power law 4 essays — A relation in which one quantity varies as a fixed power of another, so that it appears as a straight line on logarithmic axes. Its exponent is often the only thing that survives when the details of a system change.
• Power of a point 2 essays — The product of the two distances from a point to a circle along any line through the point, which is the same for every such line. It is positive outside the circle, negative inside and zero on it, and equals the squared tangent length outside.
• Power series 11 essays — An infinite sum of powers of a variable, standing for a function wherever the sum settles. Its radius of convergence is set by the nearest point where the function misbehaves, including points off the real line.
• Power set 2 essays — The collection of all sub-collections of a given set, including the empty one and the whole. It is always strictly larger than the set itself, by the diagonal argument, which is why the sizes of infinite sets never stop increasing.
• Predicate logic 2 essays — The language of statements built from properties and relations of objects using and, or, not, and the quantifiers for every and there exists. Its sentences say things about whole structures, and the order in which quantifiers appear changes what they mean.
• Prediction 3 essays — A statement about a system's future derived from its present state. In a chaotic system its useful lifetime grows only as the logarithm of the precision of the starting measurement, which is why better instruments buy so little.
• Preference profile 14 essays — The whole input to a voting rule: one complete ranking of the candidates, submitted by every voter. Every impossibility theorem in voting is a statement about what no rule can do across all of them.
• Presburger arithmetic 2 essays — The first-order theory of the whole numbers with addition but no multiplication. Every sentence of it can be decided true or false by an algorithm, because the sets it defines are unions of arithmetic progressions, which a finite automaton reading digits can recognise.
• Price of anarchy 2 essays — The ratio between the worst cost of a stable outcome and the cost of the best outcome a planner could impose. Stable means that no one can gain by changing course alone, and the ratio measures how much a system loses by leaving choices to the individuals in it.
• Primality test 9 essays — A computation certifying that a number is composite without producing a factor. Raising a base to one less than the number is the cheapest such test, and the composites it cannot catch are exactly those whose unit orders all divide one less than the number.
• Prime 10 essays — A whole number above one whose only divisors are one and itself. Every whole number above one is a product of primes in exactly one way, which is why they are the objects arithmetic is built from rather than a curiosity about divisibility.
• Prime constellation 1 essay
• Prime counting function 1 essay
• Prime factorisation 8 essays — The expression of a whole number greater than one as a product of primes, which is unique apart from the order of the factors. Finding it is easy to state and slow to do for large numbers, and its uniqueness is what lets a number be read off from its primes.
• Prime gaps 3 essays — The distances between consecutive primes. Near a large number x they average about the logarithm of x, and the longest gap below x appears to grow like the square of that logarithm, though far less than that is proved.
• Prime implicant 1 essay
• Prime number theorem 11 essays — The statement that the number of primes up to x is about x divided by its natural logarithm, with the ratio tending to one. Proved in 1896 by Hadamard and de la Vallée Poussin, it says the primes thin out like 1/log x.
• Prime power 7 essays — A prime raised to a whole-number exponent, which is exactly the sizes at which a finite field exists. A finite field exists at exactly these sizes and no others, which is why there is no field with six elements.
• Primes 49 essays — The whole numbers above one divisible by nothing but themselves and one, out of which every other whole number is built. How many lie below a bound is estimated by the logarithmic integral, and how they are arranged locally remains largely unknown.
• Primitive element 6 essays — A member of a group whose powers run through the whole group, or of a field extension that generates all of it. Its existence turns a structure into a list of powers, so that multiplication becomes addition of exponents.
• Primitive root 5 essays — A number whose powers, taken modulo a prime, run through every non-zero remainder before repeating. Every prime has one, and choosing one turns multiplication of remainders into addition of exponents, the way logarithms do for real numbers.
• Primitive triple 2 essays — A Pythagorean triple whose three numbers share no common factor, so that it is not a whole multiple of a smaller one. Every triple is a multiple of exactly one primitive one, and the primitive ones are what a parametrisation or a tree is asked to produce.
• Primorial 2 essays — The product of all the primes up to a given prime, written p#, as 2 × 3 × 5 = 30 for p = 5. Euclid's proof adds one to it to make a number no listed prime divides, and its logarithm is close to p by the prime number theorem.
• Private information 2 essays — Knowledge that one party to an exchange has and the others lack, such as a seller's own cost or a buyer's own value. Much of the theory of auctions and bargaining is about what can be achieved when that knowledge cannot simply be asked for.
• Probabilistic algorithm 1 essay
• Probabilistic method 6 essays — Proving that an object with some property exists by showing that a randomly chosen one has it with positive probability. The argument produces nothing anyone can look at, and in several settings the objects it promises have never been constructed.
• Probability 16 essays — A number between nought and one measuring how much of a space of outcomes an event takes up, adding over disjoint events. It is a measure normalised to total one, and everything about it follows from that.
• Probability density 8 essays — A non-negative function whose integral over a region gives the chance of landing in it, and whose integral over everything is one. It is the continuous counterpart of a list of probabilities, and its value at a point is a rate rather than a chance.
• Probability distribution 1 essay
• Probability model 2 essays — A precise statement of which outcomes are possible and how likely each is, chosen so that a question about chance becomes a calculation. Different models of one situation can give different answers, so the model is part of the claim rather than a detail of it.
• Product 1 essay
• Product formula 1 essay
• Projection 28 essays — A map that drops a dimension, casting a shape onto a line, a plane or a surface along a chosen family of rays. It is what loses information deliberately, and what survives it is exactly what the projection was chosen to preserve.
• Projective geometry 2 essays — The geometry of incidence, in which parallel lines meet at a point at infinity and only properties preserved by projection from a point are kept. Cross-ratios survive and lengths do not, so its theorems are about lines meeting and points lining up.
• Projective map 1 essay
• Projective plane 24 essays — A geometry in which any two points lie on one line and any two lines meet in one point, so there are no parallels. Adding a line at infinity to the ordinary plane produces one, and it is what makes the conics a single family rather than four.
• Projective transformation 1 essay
• Proof 1 essay
• Proof by contradiction 11 essays — An argument that assumes what it means to refute, derives an impossibility from it, and concludes the assumption was false. It establishes existence without producing anything, which is why Brouwer and other intuitionists rejected it for that purpose.
• Proof complexity 1 essay
• Proof system 14 essays — A fixed stock of axioms and inference rules together with a definition of what counts as a derivation. Its value is measured twice: whether everything true is derivable in it, and how long the derivations are forced to be.
• Proportionality 6 essays — The property of a division among n people that each of them values their own share at a full nth of the whole or more. It is the weaker of the two standard fairness conditions, and unlike envy-freeness it can be achieved by a simple moving-knife procedure.
• Proposition 3 essays — A statement that is either true or false, the basic object a logic assigns values to. In judgement aggregation the propositions are the questions put to a group, and how they are logically connected decides whether majority voting on each can stay consistent.
• Propositional formula 1 essay
• Provability 2 essays — The property of having a proof in a stated formal system, which is itself expressible inside the system when the system is strong enough. Read as a modal operator it obeys a complete and surprising set of laws, of which Löb's is the strongest.
• Prufer code 2 essays — A list of n − 2 labels that records a tree on n labelled points, by repeatedly removing the smallest leaf and writing down its neighbour. Every list is the code of exactly one tree, which is why there are n^(n−2) labelled trees.
• Pseudoprime 1 essay
• Pseudorandomness 16 essays — The property of a sequence produced by a rule that nevertheless passes the statistical tests one would apply to a random one. The tests are always passed by some rules with visible structure, so passing them is evidence about which tests were run rather than about the absence of pattern.
• Ptolemy inequality 1 essay
• Ptolemys theorem 1 essay
• Pure strategy 1 essay
• Pythagorean theorem 5 essays — The statement that the squares on the two shorter sides of a right triangle together hold the area of the square on the longest. Read as a definition rather than a fact it says how distance is computed, and changing the exponent in it gives a whole family of distances.
• Pythagorean triples 6 essays — Triples of whole numbers that are the sides of a right triangle, so that the first two squares add to the third. They are parametrised by two coprime integers of opposite parity, and a tree of three matrices produces every primitive one exactly once.

• Quadratic convergence 2 essays — The behaviour of an iteration whose error is roughly squared at each step, so the number of correct digits doubles. Newton's method has it near a simple root, and it is what lets a correction outrun losses that would defeat a slower method.
• Quadratic extension 1 essay
• Quadratic field 1 essay
• Quadratic form 21 essays — A homogeneous quadratic expression in several variables, written as a symmetric matrix sandwiched between a vector and itself. Its level sets are the conics and quadrics, and the signs of its eigenvalues decide which - a classification unchanged by any invertible change of coordinates.
• Quadratic irrational 7 essays — A number of the form (a + b√D)/c with whole numbers a, b, c and D not a perfect square. These are the irrational roots of quadratic equations with whole coefficients, and they are exactly the numbers whose continued fractions eventually repeat.
• Quadratic polynomials 6 essays — Expressions whose highest power is a square, and the curves those expressions describe. Completing the square solves them all, and the discriminant decides how many real roots there are before any solving is done.
• Quadratic reciprocity 7 essays — The law relating whether p is a square modulo q to whether q is a square modulo p, for two odd primes. The two answers agree unless both primes are 3 modulo 4, and the theorem has upwards of two hundred proofs because each generalises differently.
• Quadratic residue 19 essays — A number that is the square of something else in modular arithmetic. Exactly half the non-zero residues modulo an odd prime are squares, and joining two points when their difference is one gives colourings with unusually few large single-coloured sets.
• Quadrature 1 essay
• Quantifier 19 essays — A word settling how many things a statement is about — every one of them, or at least one. Which one is used, and in which order, is where most of the content of a mathematical statement lives.
• Quantifier depth 3 essays — How deeply the quantifiers in a sentence are nested, which bounds how many objects it can refer to at once. It is the resource an Ehrenfeucht-Fraisse game measures, and a fixed depth is always defeated by structures that are large enough.
• Quantifier elimination 2 essays — Replacing every formula that says some value exists by an equivalent formula with no quantifier in it at all. In a theory that allows it every question reduces to checking quantifier-free facts, which is how the arithmetic of addition alone was shown to be decidable.
• Quantifier order 5 essays — The difference between one thing serving every case and each case having its own, which swapping two quantifiers changes. It is the difference between continuity and uniform continuity, and between a bound existing and a single bound working everywhere.
• Quantisation 1 essay
• Quasi isometry 2 essays — A map between spaces that distorts every distance by at most a fixed factor and a fixed amount, so that the two look alike from far enough away. Changing a group's generators is one, and what survives it — growth, ends, amenability — is what the geometric theory of groups studies.
• Quasi-monte carlo 3 essays — Numerical integration by averaging over points placed deliberately evenly rather than at random. On smooth integrands its error falls far faster than random sampling's, at the cost of the error bar random sampling provides, which randomising the points restores.
• Quasi-random sequence 1 essay
• Quasigroup 1 essay
• Quaternion 10 essays — A number of the form a + bi + cj + dk, multiplied by Hamilton's rule, in which the order of a product matters. Unit quaternions perform rotations of three-space by conjugation and of four-space in pairs, and their length is multiplicative.
• Quaternions 1 essay
• Quota 8 essays — A region's exact share of the seats being divided — its population times the house size, over the total — which is almost never a whole number. A rule respects quota when every region ends with the floor or the ceiling of its own, and the classical divisor methods do not.
• Quota rule 4 essays — An aggregation rule accepting a proposition when at least a stated number of voters do. Majority and unanimity are the two extremes, and raising the quota buys consistency by allowing the body to take no view at all.
• Quotation 1 essay
• Quotient 3 essays — The object obtained by declaring some elements of a structure to be the same and keeping only what survives that identification. Its points are the classes of the relation, and its structure is whatever the original structure induces on them.
• Quotient group 1 essay

• Radian 2 essays — The unit of angle equal to the arc length it cuts from a circle of radius one, so that a full turn is 2π. In radians the derivative of the sine is the cosine with no stray factor, which is why calculus uses them.
• Radical axis 1 essay
• Radical extension 3 essays — A field reached from the rationals by adjoining, one at a time, roots of numbers already present — square roots, cube roots and so on. A number is expressible by radicals exactly when it lies in such a field, and Galois theory decides which polynomials have their roots there.
• Radius of convergence 9 essays — The distance from a power series' centre out to where it stops converging. It equals the distance to the nearest point where the function is not analytic, measured in the complex plane whether or not the function was ever meant to leave the real line, and it can also be read off the growth of the coefficients alone.
• Rado graph 2 essays — The countable graph that almost surely results from joining each pair of vertices with a fair coin. It is unique up to isomorphism, characterised by its extension property, and its first-order theory is what large random graphs obey.
• Radon theorem 1 essay
• Ramsey number 7 essays — The smallest number of points at which every two-colouring of the pairs contains a given number all agreeing. Only a handful are known exactly; the upper bounds come from counting arguments and the lower ones from constructions or from proofs that constructions exist.
• Ramsey theory 1 essay
• Random 1 essay
• Random code 2 essays — A code whose codewords are drawn at random rather than designed. Shannon's proof that reliable communication is possible up to the channel capacity works by showing that the average random code is good, which establishes that a good code exists without exhibiting one.
• Random field 2 essays — A random function of position, such as a landscape whose height at each point is a random variable correlated with the heights nearby. Gaussian random fields are fixed by their covariance, and averages of local features, like the Euler characteristic of a level set, can be computed exactly.
• Random function 2 essays — A function from a finite set to itself chosen uniformly from all possible functions. Iterating it from any start must eventually cycle, and the typical tail and cycle lengths grow like the square root of the set's size.
• Random graph 19 essays — A graph on a fixed set of points in which each possible edge is present independently with the same probability. Almost every property of one holds either with probability near zero or near one, with the change happening over a narrow range of that probability.
• Random group 1 essay
• Random mapping 2 essays — A function from a finite set to itself chosen uniformly among all such functions. Its structure — cycles with trees hanging from them — has exact laws, and it models hashing, pseudo-random generators and factoring methods.
• Random matrix 4 essays — A matrix whose entries are drawn at random, studied through the statistics of its eigenvalues. Large ones obey universal laws, such as the semicircle and repulsion between neighbouring eigenvalues, that do not depend on the entries' distribution.
• Random matrix product 1 essay
• Random model 1 essay
• Random permutation 7 essays — An ordering of a list chosen so that every one of the possible orderings is equally likely. Shuffling a pack of cards thoroughly produces one, and many guarantees about algorithms hold on average over such an ordering rather than for every order.
• Random walk 50 essays — A path built by taking each step in a direction chosen at random, independently of the ones before. Its position after n steps has variance exactly n, which is why space scales as the square root of time in the limit.
• Randomisation 1 essay
• Randomised algorithm 2 essays — A method that makes random choices as it runs, analysed by the chance that it succeeds or by its expected running time. Repeating it independently drives the chance of failure down as far as is wanted.
• Randomness 16 essays — The property of a sequence that no shorter description of it exists, or that no efficient procedure predicts it — two definitions that do not coincide. What a generator provides is neither; it is a short description arranged so that finding it is hard or unnecessary.
• Rank 19 essays — The number of independent directions a linear map's output can reach, which is the dimension of its image. It is the dimension of the image, and it plus the nullity is the dimension of the space the map started from.
• Rate 1 essay
• Rate function 2 essays — The exponent in a large-deviation estimate: the number for which an unlikely event's chance falls like e to the minus n times it. It is zero at the typical value and positive elsewhere, and its smallest value over a set is that set's own rate.
• Rational approximation 9 essays — A fraction standing in for a number that is not one, and the question of how much accuracy a given denominator buys. How well a number can be approximated is measured against the denominator, and the golden ratio is the worst case.
• Rational distance 1 essay
• Rational number 6 essays — A number that is one whole number divided by another, such as 3/4 or −7. The rationals are dense — one lies between any two numbers — yet they can be listed one after another, and that combination is what lets a function behave differently at every fraction than it does everywhere else.
• Rational points 3 essays — The points of a curve whose coordinates are both fractions. A circle through one such point has infinitely many, found by lines of rational slope, and whether a curve has any at all is a question about primes and remainders.
• Rational root theorem 5 essays — The rule that a rational root of a whole-number polynomial has a numerator dividing the constant term and a denominator dividing the leading one. It reduces the search for rational roots to a finite list, and its failure to produce one is how irrationality is often established.
• Rayleigh quotient 1 essay
• Real numbers 2 essays — The whole continuum of magnitudes, filling every gap the fractions leave between them. Their completeness is what every limit argument uses, and it is exactly what the rationals lack.
• Real rooted polynomial 4 essays — A polynomial with real coefficients all of whose roots are real numbers. Its coefficients are then log-concave and unimodal, and families such as matching polynomials and many graph polynomials are real-rooted for structural reasons.
• Rearrangement 2 essays — Moving the pieces of a shape without changing any of them, so that whatever the pieces measure is unchanged. It is what a dissection proof does, and the whole force of such a proof is that area survives it while shape does not.
• Recurrence 22 essays — The return of a system arbitrarily close to a state it has already been in. For a fair walk on the line it happens with certainty, and in three dimensions it does not, which is a fact about dimension rather than about walks.
• Recurrence relation 5 essays — A rule giving each term of a sequence from the terms before it, with enough starting values to begin. Solving one turns a step-by-step rule into a formula, and the solutions of a linear one are powers of the roots of its characteristic equation.
• Recursion 25 essays — A definition that refers to itself on smaller inputs, with a base case to stop it. It is what makes a short definition describe an unbounded object, and the base case is what stops the description from being empty.
• Recursion theorem 1 essay
• Reduction 5 essays — A procedure turning any solution of one problem into a solution of another, so that the second is at least as hard as the first. It is how hardness is transferred, and every claim that a construction is secure is a reduction to something believed hard.
• Redundancy 3 essays — The part of a coded message that carries no new information, added so that errors in transmission can be detected or corrected. A code with more redundancy has fewer codewords for its length but can survive more damage, and the art of coding is getting the most protection for the least of it.
• Reed solomon 3 essays — A code whose messages are the coefficients of a polynomial and whose words are its values at fixed points. Any k of the values recover the message, so it meets the Singleton bound exactly and loses nothing to erasures.
• Reed solomon code 2 essays — An error-correcting code whose codewords are the values of low-degree polynomials at distinct points of a finite field. Any k of its n symbols recover a message of k symbols, so it survives the loss of any n minus k of them.
• Reflection 13 essays — A rigid motion of the plane that flips it about a line, leaving that line fixed and swapping the two sides. It preserves distances and tangency and reverses orientation, which is why reflecting a graph in the diagonal gives the inverse function.
• Reflection group 1 essay
• Reflection principle 3 essays — The correspondence folding a path about the first moment it touches a level, which turns a question about a path's history into one about its endpoint. It counts paths that touch a barrier by counting all paths from a mirrored start, and it degrades as soon as there are two barriers.
• Reflexive 1 essay
• Refutation 9 essays — A derivation ending in a contradiction, showing that the assumptions it started from cannot all hold at once. It is a certificate of impossibility, checkable line by line without repeating the search that produced it.
• Region count 2 essays — How many pieces a drawing cuts its surface into, counted including the unbounded one. It enters Euler's relation as the face count, which is why the unbounded region has to be included.
• Register machine 1 essay
• Regular language 1 essay
• Regular polygon 15 essays — A closed figure whose sides are all the same length and whose angles are all the same size. Which ones are constructible is decided by the Fermat primes, which is Gauss's theorem and the reason a seventeen-sided one can be drawn.
• Reidemeister moves 6 essays — The three local changes to a knot diagram that generate every deformation of the knot itself. Any two diagrams of the same knot are joined by a sequence of them, which is what makes checking an invariant a finite task.
• Relation 3 essays — An equation a presentation imposes on words in its generators, saying that some product is the identity. Adding one is what distinguishes a quotient from a free object, and in topology it is what gluing a disc along a loop does to a fundamental group.
• Relative consistency 1 essay
• Relative entropy 3 essays — A measure of how far one probability distribution is from another: the average, under the first, of the logarithm of the ratio of their probabilities. It is never negative, is nought only when they agree, and is the exponential rate at which a large sample mistakes one for the other.
• Relaxation 6 essays — A weaker problem obtained by dropping a constraint, whose answer bounds the original's. When its corners happen to satisfy the dropped constraint the two problems coincide, and when they do not the difference is a measurable gap.
• Remainder 7 essays — What is left of one number once as many whole copies of another as will fit have been taken away. It is what the Euclidean algorithm iterates on, and it is the whole content of arithmetic on a dial.
• Renormalisation 6 essays — Rescaling a small part of a system so that it looks like the whole, then studying the rescaled system instead. In dynamics it explains why small copies of a picture reappear inside it, and why some constants are the same across many different systems.
• Rent division 1 essay
• Replicator dynamics 1 essay
• Residual 3 essays — What a fit fails to account for - the difference between the data and the values the model produces. Under least squares it is perpendicular to everything the model could have used, so a visible pattern in it is evidence about a missing column rather than about the measurements.
• Residual entropy 1 essay
• Residue 1 essay
• Residue class 2 essays — All the whole numbers that leave the same remainder when divided by a fixed number, such as the numbers ending in 7 in base ten. Questions about primes are often asked class by class, since a class can hold infinitely many primes only if its remainder shares no factor with the divisor.
• Resolution 9 essays — A proof rule that combines two clauses disagreeing about one variable into a single clause that leaves that variable out. Repeated until the empty clause appears, it proves that a set of clauses cannot all be true, and it is the reasoning inside modern satisfiability solvers.
• Resultant 2 essays — A determinant built from the coefficients of two polynomials that vanishes exactly when they share a root. It turns two equations in two unknowns into one equation in one, which is how elimination works without finding any root.
• Retraction 1 essay
• Return map 3 essays — A return map records where a flow next crosses a chosen surface, as a function of where it crossed last. When the flow contracts strongly, its crossings fall on a curve and the map is a function of one variable, so the theory of iterated maps of an interval applies to the flow.
• Reuleaux triangle 6 essays — The constant-width shape bounded by three circular arcs, each centred at the opposite corner. It is the constant-width shape of least area, and it rolls under a plank as smoothly as a circle while having no centre to turn about.
• Revenue equivalence 2 essays — The theorem that many auction rules raise the same expected revenue when bidders' values are independent and they bid in their own interest. It says the format matters less than the allocation and the lowest type's payoff, and it fails when its assumptions do.
• Reversibility 4 essays — The property of a random process that as much probability flows from one state to another as flows back. It is a condition on each pair of states separately, and it is sufficient for a given distribution to be the one the process settles into.
• Riemann hypothesis 5 essays — The conjecture, from 1859, that every non-trivial zero of the Riemann zeta function has real part one half. It is equivalent to the primes being as evenly spread as randomness allows, and it is the most famous open problem in mathematics.
• Riemann sphere 2 essays — The complex numbers with a single point at infinity adjoined, which stereographic projection turns into an ordinary sphere. On it a rational function is a map with a degree and no exceptions, and division by zero becomes an ordinary value.
• Riemann sum 6 essays — An approximation to an area built from rectangles, and the thing an integral is the limit of. Its limit as the rectangles narrow is the integral, and whether that limit exists is what integrability means.
• Riemann zeta function 2 essays — The function ζ(s) given by the sum of 1/n^s over the positive integers, and its continuation to the complex plane. It encodes the primes through Euler's product, and its values at whole numbers, such as ζ(2) = π²/6, appear far from number theory.
• Right angle 2 essays — A quarter turn, the angle between two perpendicular lines, and the case in which two squares add to the third. It is where the law of cosines loses its extra term, and where two squares add to a third rather than falling short or overshooting.
• Right triangle 3 essays — A triangle with one angle of ninety degrees. Its three sides satisfy the Pythagorean relation, and the whole-numbered instances of that relation are the Pythagorean triples, of which there are infinitely many.
• Rigid motion 3 essays — A transformation of the plane or space preserving every distance: a translation, a rotation, a reflection, or a composition of those. Which of them a construction is allowed decides what it can reach, and forbidding rotation gives dissection a second invariant beside area.
• Rigidity 1 essay
• Risk aversion 2 essays — A preference for a certain amount over a gamble with the same average, measured by how much average one would give up to remove the risk. Maximising the expected logarithm of wealth is one form of it.
• Robinson schensted 1 essay
• Robustness 2 essays — The property of a method or a count that a small or rare error in its input changes its output only a little. A robust estimate of the centre of data tolerates a few wild values that would drag an average anywhere.
• Rogers ramanujan identities 1 essay
• Rolles theorem 1 essay
• Rook polynomial 3 essays — The polynomial whose coefficient of x to the k counts the ways to place k non-attacking rooks on a board of cells. Its coefficients turn inclusion–exclusion into a count of the permutations that avoid the board, and it multiplies over blocks that share no row or column.
• Root 1 essay
• Root system 1 essay
• Root-finding 6 essays — The computation of where a function is zero, as opposed to the argument that a zero exists. The methods are iterations, and the fast ones are contractions near the root whose behaviour elsewhere can be arbitrarily wild.
• Roots 12 essays — The values at which a polynomial takes the value zero. Their number is bounded by the degree, and over the complex numbers there are exactly that many when multiplicity is counted, which is the fundamental theorem of algebra.
• Roots of unity 19 essays — The complex numbers whose nth power is one, sitting at the corners of a regular polygon on the unit circle. They form a cyclic group under multiplication, they add to zero, and their orthogonality is what makes discrete Fourier analysis work.
• Rotation 13 essays — A motion of the plane or of space that fixes a point or a line and preserves every distance and orientation. Composing two gives another, so the rotations leaving an object where it was form a group, whose size measures how symmetric the object is.
• Rotation number 13 essays — The average amount by which a map advances a point around a circle, per step. It exists for every such map, never falls as the parameter rises, and sticks at each rational over a whole interval because a periodic orbit there attracts.
• Rounding 9 essays — Replacing a quantity by a nearby whole number or simpler value, and taking on whatever that difference costs. The error it introduces is bounded by half a unit and accumulates, which is what makes a long computation's error analysis necessary.
• Rounding error 1 essay
• Rule 110 1 essay
• Rule 30 1 essay
• Rule 90 1 essay
• Russell paradox 2 essays — The contradiction produced by the set of all sets that are not members of themselves, which belongs to itself exactly when it does not. It showed that not every describable property forms a set, and it forced set theory onto axioms that say which collections exist.

• Saddle point 4 essays — A saddle point is a critical point that is a maximum along one direction and a minimum along another, like a mountain pass. On a surface it is where two pieces of a region above a height join or a loop closes, and it counts minus one towards the Euler characteristic.
• Sample space 8 essays — The collection of every outcome a random process could produce, taken as the ground the probabilities sit on. Fixing it is the first step in any probability question, and most paradoxes in the subject come from two people using different ones.
• Sampling 16 essays — Drawing a limited number of cases from a large or infinite collection in order to say something about the whole. What makes a sample informative is how it was drawn rather than how large it is, since a systematic omission is not repaired by more of the same.
• Satisfiability 18 essays — The property of a formula for which some assignment of values makes it true. Deciding it for a formula in many variables is the model hard search problem, since the assignments to try double with each variable.
• Saturation 1 essay
• Scale 1 essay
• Scaling 27 essays — Enlarging or shrinking everything by one factor, so that shapes are preserved and sizes are not. Which exponent to scale by is often the whole question: only one power leaves a limit behind, and finding it is most of the work.
• Schlafli symbol 1 essay
• Schur theorem 1 essay
• Scissors congruence 1 essay
• Scrambling 2 essays — Randomising an evenly spread point set by flipping the binary digits of its coordinates at random, each flip chosen afresh within every block. The points stay one to every elementary box, so they keep their evenness, but each is now uniformly random, so repeated scramblings give an unbiased estimate and an error bar.
• Seating 1 essay
• Secant 3 essays — A secant is the straight line through two points of a curve, and its slope is the change in height divided by the change in position between them. Letting the two points merge gives the tangent, which is how the derivative is defined.
• Second derivative 1 essay
• Second moment 2 essays — The average of the square of a random quantity, which with the mean gives its variance. Showing the second moment is close to the square of the mean proves a count is rarely zero, the step that locates thresholds.
• Second-order logic 3 essays — Logic whose quantifiers may range over sets and relations of objects, not only over the objects themselves. Its existential part describes exactly the problems whose solutions can be checked quickly, and on infinite structures it can pin down the whole numbers up to isomorphism.
• Self affinity 4 essays — Being built from copies of itself contracted by different factors in different directions. It is the weakening of self-similarity that separates the two definitions of dimension: the pieces are rectangles rather than squares, so a grid of squares can never match their shape at any scale.
• Self-dual function 1 essay
• Self-intersection 1 essay
• Self-reference 6 essays — A statement or object that describes itself, usually by way of a numbering that lets it be named inside its own system. It is what makes a sentence able to talk about its own provability, and it is the engine of the incompleteness and halting arguments alike.
• Self-reproduction 1 essay
• Self-similarity 34 essays — The property of a shape that contains smaller copies of the whole of itself. It can be exact, as in a construction, or statistical, as in a random walk seen at three magnifications.
• Selfridge conway 1 essay
• Semicircle law 3 essays — Wigner's law that the eigenvalues of a large random symmetric matrix, suitably scaled, are spread with a semicircular density. It holds whatever the distribution of the entries, provided they are independent with equal variance.
• Sensitive dependence 16 essays — The property that two starting points however close eventually lead to orbits that are far apart. It makes long-range prediction impossible without making the rule any less determined, which is the distinction chaos rests on.
• Sensitivity 1 essay
• Sensor network 2 essays — A field of many simple detectors spread over a region, each reporting a small local reading such as how many targets it can sense. The question is what can be recovered from the readings together, like the number of targets, without knowing which target each sensor sees.
• Separating hyperplane 1 essay
• Series 1 essay
• Series acceleration 1 essay
• Series tail 1 essay
• Set operations 1 essay
• Set theory 1 essay
• Shadow price 4 essays — The rate at which the best value of an optimisation problem improves when one constraint is relaxed by a unit. It is the dual variable of that constraint, and it is zero for any constraint that is not binding at the optimum.
• Shadowing 2 essays — The property that a computed sequence, each step correct to within a small amount, is the true orbit of some nearby starting point. Where it holds, every statistical claim drawn from a computed chaotic orbit is a claim about a genuine orbit, which is what makes such computations trustworthy.
• Shapley shubik index 1 essay
• Shapley value 12 essays — The rule that gives each player the average of what they add to the group already present, taken over every order in which the players could arrive. Four conditions — efficiency, symmetry, a player who adds nothing getting nothing, and additivity — leave it as the only possibility.
• Shear 5 essays — A map that slides each line parallel to a fixed direction by an amount depending on its distance, leaving area unchanged. It is what Euclid's proof of the Pythagorean theorem uses, since sliding a parallelogram along its base leaves its area alone.
• Sheffer stroke 1 essay
• Shift map 4 essays — The map that deletes the first symbol of an infinite string and moves the rest one place along. Many maps of an interval are this map in disguise once each point is replaced by its itinerary, which is what makes symbolic dynamics work.
• Shift register 1 essay
• Shortest path 1 essay
• Shortest vector 1 essay
• Shrinkage 1 essay
• Sidon set 1 essay
• Sierpiński triangle 1 essay
• Sieve 10 essays — A method of counting or finding numbers by removing, in turn, those divisible by each prime in a list. Counting what survives is inclusion–exclusion, and stopping that count early is what lets sieves bound quantities the exact formula cannot reach.
• Sieve of eratosthenes 1 essay
• Sign 5 essays — The single bit a permutation carries: whether it is built from an even or an odd number of swaps. It is the only label on permutations that multiplies correctly into a commutative group, and it is what makes a determinant possible.
• Signature 1 essay
• Similar triangles 9 essays — Triangles with the same angles, so that one is an enlargement of the other and their sides are in fixed ratio. The fixed ratio of their sides is what makes trigonometry possible, since the ratios depend on the angles and not on the size.
• Similarity 2 essays — The relation between two square matrices that are the same linear map written in different bases, one equal to P⁻¹AP for the other. Similar matrices share eigenvalues, trace and determinant, because those belong to the map and not to the coordinates used to write it down.
• Similarity dimension 1 essay
• Simple closed curve 2 essays — A loop drawn on a surface that never crosses or touches itself. On a torus the loops of this kind that cannot be shrunk away are exactly the classes (p, q) whose two counts share no common factor.
• Simple game 2 essays — A cooperative game in which every coalition is worth either one or nothing, so the structure is entirely a matter of which coalitions win. A weighted vote is one, and a member's power in it is the fraction of arrival orders in which they are the one who carries a coalition over the quota.
• Simple group 3 essays — A group with no way of being broken into a smaller group and a quotient, other than trivially. The smallest non-abelian one is the rotation group of the icosahedron, and its indivisibility is why the general fifth-degree equation has no formula in radicals.
• Simple harmonic motion 1 essay
• Simplex 6 essays — The set of lists of non-negative numbers adding to a fixed total, which for three numbers is a triangle. Distributions and divisions both live in one, so a claim about every distribution becomes a claim about a region of it.
• Simplex method 2 essays — An algorithm for linear programming that walks from corner to corner of the feasible polytope, always along an edge that improves the objective, until no edge does. It is fast in practice, though some programs force it through exponentially many corners.
• Simply connected 1 essay
• Simulation 28 essays — Estimating a probability or an average by generating many random instances and counting what happens. It converges at a rate of one over the square root of the number of runs, which is slow, and its value is as a check on an exact computation rather than a substitute for one.
• Sinc 1 essay
• Sine 10 essays — The vertical coordinate of a point travelling round the unit circle, as a function of the angle turned through. It is the derivative of nothing simpler than cosine, and it is what a circle looks like from the side as it turns.
• Single-peaked preferences 1 essay
• Singleton bound 2 essays — The limit that a code of length n over q symbols, with any two words differing in at least d places, holds at most q to the power n − d + 1 words. Deleting d − 1 positions leaves every word distinct, which is the whole proof, and codes that meet it exactly are called maximum distance separable.
• Singular matrix 3 essays — A matrix that collapses space onto something smaller, so that it cannot be undone and its determinant is zero. It has a non-trivial kernel, so the equation it defines either has no solution or a whole family of them.
• Singular value 1 essay
• Singular value decomposition 1 essay
• Singularity 8 essays — A point where a function fails to be analytic - a pole, a branch point, or something worse. Its importance is that it is never local news: the distance from an expansion point to the nearest singularity is exactly the radius on which that expansion converges, however far away and however invisible it is.
• Skeleton 2 essays — A thin, branching set of curves inside a shape that records its structure: where its parts are and how they join. The medial axis is the standard skeleton, and its branches follow the shape's protrusions, which is why it describes a shape well and is so sensitive to noise.
• Skolem sequence 1 essay
• Slope field 1 essay
• Small divisors 1 essay
• Smallest prime factor 1 essay
• Smooth number 3 essays — A whole number all of whose prime factors are small, below some bound. Smooth numbers are rarer than they first seem, and their frequency governs how fast many factoring and discrete-logarithm methods run.
• Smoothing 1 essay
• Snark 1 essay
• Sobol sequence 2 essays — A sequence of points in the unit cube built from binary digits so that its points spread evenly into boxes made by halving the sides. Taken a power of two at a time, it puts the same number of points in every such box of equal volume, and it is the most used quasi-random sequence for integration.
• Solid of revolution 1 essay
• Solvability 1 essay
• Solvable group 2 essays — A finite group that can be broken down, one normal subgroup at a time, into pieces that are all abelian. A polynomial can be solved by radicals exactly when its Galois group is solvable, which is why the general quintic cannot be.
• Sorting 1 essay
• Soundness 6 essays — The property of a proof system that everything derivable in it is true. It is what makes a derivation evidence at all, and it is usually the easy half to establish, since it needs only that each rule preserves truth.
• Space average 1 essay
• Space-filling curve 1 essay
• Spanning surface 1 essay
• Spanning tree 20 essays — A set of a graph's edges that reaches every vertex and closes no loop, using one fewer edge than there are vertices. Every connected graph has one, and counting them is Cayley's formula when the graph is complete.
• Spectral gap 3 essays — One minus the second-largest eigenvalue of a Markov chain's transition matrix. It sets the rate at which the chain forgets its start, so that its reciprocal, the relaxation time, measures how slowly the chain mixes.
• Spectral radius 2 essays — The largest absolute value among a matrix's eigenvalues. Powers of the matrix grow or shrink at that rate, so it decides whether a series of powers converges, and for a walk on a graph it sets how fast the chance of returning home decays.
• Spectral test 2 essays — A measure of how badly a linear congruential generator's outputs, taken in pairs or triples, line up on parallel planes. It finds the widest gap between those planes exactly, by a shortest-vector computation, rather than estimating it from a sample.
• Spectral theorem 3 essays — The theorem that a map equal to its own adjoint has real eigenvalues and eigenvectors perpendicular to one another. It holds under whichever inner product the map is self-adjoint for, and it is why symmetric matrices, reversible chains and Fourier series all split into independent directions.
• Spectrum 8 essays — The list of how much of each frequency a signal holds, which describes it as completely as its shape over time does. Reading it is how a wave is decomposed, and the fact that convolution multiplies spectra is why it settles so many problems.
• Sperner lemma 1 essay
• Sperner theorem 3 essays — The statement that the largest collection of subsets of a set with no two comparable is a whole layer — all the subsets of one size, the middle one. It is the founding result of extremal set theory, and its proofs generalise far better than the statement does.
• Sphere 14 essays — The set of points at a fixed distance from a centre, in any number of dimensions. Its defining property in topology is that it separates: a sphere in n-dimensional space cuts the complement into exactly two pieces, and that is true in every dimension whether or not the sphere sits tamely.
• Sphere minus a point 1 essay
• Sphere-packing 2 essays — An arrangement of equal balls that touch but do not overlap, judged by the share of space they fill. In the plane the best is the hexagonal pattern of coins, in space the stall of oranges; in most higher dimensions the densest packing is still unknown.
• Sphere-packing bound 2 essays — The limit on how many codewords a code can hold, being the number of words divided by the size of one correction ball. A code meeting it exactly is called perfect, and the Hamming codes are the standard family that does.
• Spherical excess 1 essay
• Spherical geometry 3 essays — The geometry of the surface of a sphere, where the straight lines are great circles and every two of them meet. A triangle's angles add to more than two right angles by an amount equal to its area, so triangles of different sizes are never similar.
• Splitting field 1 essay
• Square numbers 5 essays — The counts that fill a square array — 1, 4, 9, 16 and onward. They are the partial sums of the odd numbers, which is the gnomon picture and the discrete ancestor of the product rule.
• Square root 8 essays — A number whose square is a given number, in whatever arithmetic is in use. Modulo a prime one may or may not exist, and when it does it can be lifted digit by digit to every power of the prime; among ordinary numbers taking one halves an exponent.
• Square root of one 1 essay
• Square root rule 1 essay
• Squarefree 1 essay
• Squaring the circle 3 essays — The problem of constructing, with straightedge and compasses alone, a square whose area equals a given circle's. It is impossible, because it would require constructing the square root of π, and π is transcendental; curves drawn by other means, such as a spiral, can do it.
• Squeeze 2 essays — An argument that traps a quantity strictly between two bounds where nothing of its kind can lie. The classical cases put a number that would have to be whole strictly between nought and one, which is how both the irrationality of e and that of pi are established.
• Stabiliser 3 essays — The set of symmetries in a group that leave a chosen object exactly where it is. It is a subgroup, and its size multiplied by the size of the object's orbit gives the size of the whole group — the orbit–stabiliser theorem, which is how most counts of symmetric configurations are made.
• Stability 18 essays — Whether a small disturbance to a state dies away or grows, decided by the rule's slope there. It is decided by whether the map's slope at the fixed point is under one in size, and it is what separates an attracting point from a repelling one.
• Stable matching 11 essays — A pairing of two sides in which no two people on opposite sides would both rather have each other than what they hold. One always exists and is found by deferred acceptance, and which of several is reached depends on which side proposes.
• Stable set 1 essay
• Standard deviation 1 essay
• Standard error 1 essay
• Star 1 essay
• Star polygon 1 essay
• State space 12 essays — The set of all configurations a system can be in, with one point for each. Drawing the rule as a motion on it is what turns a question about a sequence into a question about a picture.
• Stationary distribution 9 essays — A distribution over a Markov chain's states that the chain leaves unchanged: as much flows into each state as out. On a finite chain that can reach everywhere it exists, is unique, and is what a single long run's time averages settle on.
• Stationary measure 1 essay
• Statistic 1 essay
• Statistical description 1 essay
• Statistical mechanics 1 essay
• Steiner system 1 essay
• Steiner tree 4 essays — The shortest network joining a set of points when extra junction points may be added anywhere. Every junction meets exactly three roads at 120°, there are at most two fewer junctions than points, and finding the best arrangement is NP-hard.
• Steiner triple system 2 essays — A set of points together with triples of them, arranged so that every pair of points lies in exactly one triple. One exists precisely when the number of points leaves remainder one or three on division by six, which is what two divisions coming out whole requires and is also, unusually, enough.
• Stereographic projection 7 essays — The correspondence between a sphere with one point removed and a plane, made by shining rays from that point. It takes every circle to a circle or a line and preserves every angle exactly, which is why it is the projection navigators and crystallographers use.
• Stern brocot tree 9 essays — The binary tree of fractions built by repeatedly taking the mediant of two neighbours. Every positive rational appears in it exactly once and already in lowest terms, because the construction preserves primitivity rather than restoring it by cancelling.
• Stick number 1 essay
• Stirling approximation 4 essays — The estimate n! ≈ √(2πn)(n/e)ⁿ, accurate to within a fraction of a per cent from n = 10. It turns products of many factors into exponentials, which is how most asymptotic counts in combinatorics are made.
• Stirling formula 1 essay
• Stopping time 3 essays — A rule for when to stop a random process that decides using only what has happened so far, never what is still to come. Fair games stopped at such times stay fair on average, which turns many waiting-time questions into short calculations.
• Straightedge 6 essays — An instrument that draws the line through two given points and nothing else. Its operations are preserved by every projective transformation, which is exactly why it cannot find a midpoint on its own.
• Straightedge and compass 8 essays — The two classical drawing operations — the line through two points and the circle centred at one through another — and nothing else. What they can reach is exactly the numbers built from the four operations and square roots, which is what settles the classical impossibilities.
• Strange attractor 10 essays — A bounded set that nearby trajectories are drawn onto and on which they never settle. It has zero volume because the flow contracts volumes, and a dimension that is not a whole number because it is a sheet times a Cantor set.
• Strategic manipulation 1 essay
• Strategic voting 1 essay
• Strategy 6 essays — A rule telling a player what to do at every position that can arise, rather than a single good move. A winning strategy is one that leads to a win against every opposing play, so exhibiting one settles a game outright.
• Strategy-proofness 4 essays — The property that submitting one's true ranking is never beaten by submitting any other, whatever everybody else does. No reasonable voting rule over three or more candidates has it, which is the Gibbard-Satterthwaite theorem.
• Sturmian word 2 essays — A two-letter word that never repeats yet has only n + 1 different blocks of each length n, the fewest possible. Every one is the record of a straight line of irrational slope crossing a square grid, and each is balanced: equal-length stretches differ by at most one letter.
• Subadditivity 2 essays — The property that the value for a combined piece is at most the sum of the values for its parts, as with a shortest route. Fekete's lemma then makes the value per unit of size converge, which is how limit shapes are proved to exist.
• Subdivision 2 essays — A graph obtained from another by replacing edges with paths through new points of degree two. A graph contains a subdivision of K5 or K3,3 exactly when it cannot be drawn in the plane without crossings, which is Kuratowski's theorem.
• Subformula property 3 essays — The property of a proof that every formula appearing anywhere in it is part of the formula it concludes. Proofs without cuts or detours have it, which is what makes them searchable and lets an interpolant be read off them.
• Subgame perfect equilibrium 1 essay
• Subgroup 13 essays — A subset of a group that is a group in its own right under the same operation, closed under combining and under inverses. A group's structure is read off its subgroups, and for a finite group each of their sizes divides its own.
• Subplane 1 essay
• Subset 1 essay
• Subset-sum 2 essays — The problem of deciding whether some selection from a list of numbers adds up exactly to a given target. It is hard in general, and the best known methods split the list and look for a collision between the halves' sums.
• Subspace 5 essays — A subset of a vector space that is itself a vector space - closed under addition and under scaling, and so containing the origin. The set of everything a matrix can produce is one, which is why a system with no solution is a question about whether a point lies in a subspace rather than about arithmetic.
• Substitution 3 essays — A rule that replaces each symbol of a word, or each tile of a tiling, by a fixed pattern of symbols or smaller tiles. Repeating it grows a whole word or tiling from one seed, and the matrix counting what each piece becomes fixes how often each piece appears in the limit.
• Successor 1 essay
• Sudoku 1 essay
• Sum of cubes 2 essays — A whole number written as a total of cubes of whole numbers. With positive cubes the question is how many are needed for every number, which is Waring's problem; with negative cubes allowed it becomes a hard search with numbers of many digits.
• Sum of squares 5 essays — A number or polynomial written as a total of squares of others, as 13 is 4 plus 9. Which whole numbers are sums of two, three or four squares has exact answers, and a polynomial written this way is visibly never negative, which is how many inequalities are certified.
• Sums of two squares 16 essays — The question of which whole numbers are a² + b², settled for the primes by their residue modulo four. The count of representations is a divisor sum, and the whole subject is the arithmetic of the Gaussian integers in disguise.
• Sumset 1 essay
• Superattracting 1 essay
• Superpermutation 1 essay
• Superposition 1 essay
• Support function 4 essays — For a convex shape, the distance from a chosen origin to the supporting line with a given outward normal. It describes the shape by the lines that touch it rather than by its boundary, and it turns adding shapes into adding functions.
• Supporting line 4 essays — A line touching a curve at a point and lying below it everywhere. A convex function has one at every point, which is what turns a statement about a derivative at one place into a bound holding across the whole domain.
• Supremum 6 essays — The least thing lying above a whole collection, whether or not the collection contains it. A limit ordinal is the supremum of everything below it, and the difference between having a supremum and containing one is what separates a limit from a successor.
• Surface 4 essays — A space that looks like a piece of the plane near every one of its points. The closed ones are classified completely by orientability and a single whole number, which makes them the largest family of spaces anybody can list.
• Swing 1 essay
• Syllogism 3 essays — An argument from two premises about classes to a conclusion about classes, such as all A are B and all B are C, so all A are C. Of the 256 forms Aristotle's scheme allows, only 24 are valid, and a diagram of three circles decides which.
• Symbolic dynamics 12 essays — The description of an orbit by the sequence of regions it visits, turning a map into a shift on strings of symbols. It converts questions about orbits into questions about words, and for a map that doubles it makes the orbit literally the digits of the starting point.
• Symmedian 1 essay
• Symmetric chain decomposition 1 essay
• Symmetric function 7 essays — A polynomial in several variables that is unchanged when the variables are permuted among themselves. The sum and the product of a polynomial's roots are the basic examples, and every symmetric function of the roots is a polynomial in the coefficients.
• Symmetric matrix 6 essays — A square matrix equal to its own reflection across the leading diagonal. Its eigenvalues are always real and its eigenvectors always perpendicular, which is false of matrices in general and is what makes quadratic forms and covariances tractable.
• Symmetrisation 1 essay
• Symmetry 34 essays — A transformation that leaves an object looking exactly as it did before. The symmetries of an object form a group, which is what makes symmetry a thing to compute with rather than merely to notice.
• Symmetry group 13 essays — The set of all transformations leaving an object where it was found, taken together with composition. Its size measures how symmetric the object is, and its structure distinguishes objects that a count of symmetries alone cannot tell apart.
• Synchronisation 5 essays — The falling into step of coupled oscillators or chaotic systems, so that their phases or states come to agree. It switches on at a threshold of coupling set by how strongly the units differ or how fast chaos separates them.
• Syndrome 1 essay
• System of equations 1 essay

• Tableau 4 essays — A proof by decomposition: assume the opposite of what is wanted and take it apart. Every branch either contradicts itself or describes a way for the assumption to hold, so every branch closing is a proof, and a branch staying open hands over a counterexample rather than merely failing.
• Tail bound 8 essays — A statement of how much probability can lie beyond a given distance from the centre of a distribution. Its strength follows from its hypothesis: a variance gives a bound falling like one over the square of the distance, an exponential moment gives one falling exponentially.
• Tangency 15 essays — The meeting of a curve and a line or another curve that touch without crossing at the point of contact. It is preserved by reflection, which is why the slope of an inverse function is the reciprocal of the original's.
• Tangent 7 essays — A line that touches a curve at a point without crossing it there, pointing in the curve's own direction. For a circle it is perpendicular to the radius, and the square of its length from an outside point is that point's power.
• Tautology 1 essay
• Taylor series 10 essays — The infinite sum built from a function's derivatives at a single point. Its partial sums are the polynomials matching the function to successively higher order there, so it claims that everything the function does everywhere is encoded in its behaviour at one place - which is often true and sometimes spectacularly false.
• Telescoping 1 essay
• Tent map 3 essays — A map of an interval to itself made of two straight pieces meeting at a peak, both steeper than the diagonal. It is the simplest chaotic map, and the return map of the Lorenz system is close to one.
• Termination 12 essays — The property of a procedure guaranteed to stop, usually because some whole-number quantity strictly decreases at every step. Proving it usually means exhibiting the decreasing quantity, which is the same move as an argument by descent.
• Ternary expansion 2 essays — The writing of a number between 0 and 1 in base three, as a sequence of digits 0, 1 and 2. The Cantor set is exactly the numbers with an expansion using no 1, which is why base three is the natural language for its arithmetic.
• Thales theorem 1 essay
• The core 1 essay
• The Euler–Mascheroni constant 3 essays — The constant gap between the harmonic partial sums and the natural logarithm, near enough 0.5772. It is what the naive sieve heuristic for the primes gets wrong by, and whether it is irrational is not known.
• The integral 1 essay
• Three distance theorem 1 essay
• Threshold 14 essays — The value of a parameter at which a property changes from almost never holding to almost always holding. For a monotone property of a random graph one always exists, and locating it is usually a matter of counting what the property requires.
• Threshold function 1 essay
• Threshold rule 8 essays — A stopping rule of the form: pass over everything until some condition is met, then take the first thing that qualifies. The condition may be a position, a value or a relative rank, and which of those it is decides how much the rule can achieve.
• Tiling 15 essays — A covering of a region by pieces meeting edge to edge with no gap and no overlap. Whether a set of pieces can tile the plane is undecidable in general, though any particular attempt either closes up or does not.
• Time average 2 essays — The average of a quantity along one trajectory of a system, taken over longer and longer stretches of time. Ergodic theory asks when it equals the space average over all states, which is when one long run can stand in for the whole system.
• Top trading cycles 1 essay
• Topological entropy 7 essays — The rate at which the number of distinguishable orbit segments of a map multiplies with each step, measured as a logarithm. For a coded system it is the growth rate of the allowed words, and for a map of an interval it is the growth rate of the monotone pieces of its iterates.
• Topological invariant 20 essays — A quantity left unchanged by bending and stretching, and therefore able to prove that two shapes are not the same shape. It can prove two shapes different and never that they are the same, which is the asymmetry every invariant argument carries.
• Topology of logic 3 essays — The reading under which formulas are open sets of a space, conjunction is intersection and negation is the inside of the complement. It makes the failure of excluded middle a geometric fact about boundaries, and makes every finite model of the constructive logic a finite order.
• Torus 17 essays — The surface of a doughnut — a sphere with one handle, on which a closed curve need not separate the surface in two. Its characteristic is zero, and a map drawn on it may need seven colours where a map in the plane needs four.
• Total variation 3 essays — A distance between two probability distributions: the largest difference in the chance they give to any one event. It equals half the sum of the differences state by state, and a small distance means no question can tell the two apart well.
• Totient 9 essays — The count of numbers below a given one sharing no factor with it. It counts the invertible elements on a dial of that size, and its product formula over the prime factors is what makes it computable.
• Tournament 5 essays — A contest in which every pair of players meets once and one of the two wins, drawn as a complete graph with every edge given a direction. A tournament can contain cycles in which each player beats the next, so winning more games does not by itself decide who is strongest.
• Trace 8 essays — The sum of the entries down the leading diagonal of a square matrix. It is unchanged by any change of basis and equals the sum of the eigenvalues, so it is a property of the map rather than of the array that happens to describe it.
• Transcendence 8 essays — The property of a number that no polynomial with whole-number coefficients has it as a root. Establishing it for a particular number is hard, and it is what puts a length beyond the reach of straightedge and compasses.
• Transcendental number 1 essay
• Transfer matrix 2 essays — A matrix that counts how the allowed configurations of one slice of a system can follow those of the previous slice. Its powers count configurations of a long strip, and its largest eigenvalue gives their growth rate per slice.
• Transfer operator 1 essay
• Transfinite induction 1 essay
• Transfinite recursion 1 essay
• Transience 1 essay
• Transient 3 essays — The early part of an orbit, before it has settled onto whatever it settles onto. Discarding it is what makes a bifurcation diagram legible, and how long it lasts is a separate question from where it ends up.
• Transition map 1 essay
• Transition matrix 3 essays — A square table whose row for each state lists the chances of being found in each state one step later, so every entry is non-negative and every row sums to one. Multiplying it by itself gives the chances over several steps, and its eigenvalues say how fast the starting state is forgotten.
• Transitive 1 essay
• Transitivity 2 essays — The property that if A beats B and B beats C, then A beats C. Rankings by a single number always have it; majority votes, dice and races between coin patterns can lack it, and then there may be no best option at all.
• Translation 1 essay
• Translation surface 1 essay
• Transport 2 essays — Transport is the carrying of a vector, frame or other structure continuously along a path, one small step at a time, without any sudden change. Comparing what returns at the end of a closed loop with what set out reveals twisting — a flipped orientation, a turned frame — that no single point can see.
• Transposition 3 essays — A permutation that exchanges two things and leaves everything else where it was. Every permutation is a product of transpositions, and whether the number needed is even or odd is always the same for a given permutation — its sign.
• Transversal 7 essays — A choice of one member from each of several sets, with all the choices distinct. Whether one exists is decided by comparing the size of every subfamily with the size of the union of its sets.
• Trapping region 1 essay
• Travelling salesman 4 essays — The problem of finding the shortest closed tour through a set of points, visiting each once. It is hard to solve exactly for many points, so it is studied through lower bounds from linear programs and through approximations with guarantees.
• Tree 3 essays — A connected graph with no cycles, so that between any two of its points there is exactly one path. A tree on n points has n minus 1 edges, and trees are what spanning, searching and labelling problems on graphs are built from.
• Trefoil 1 essay
• Triangle 1 essay
• Triangle centres 3 essays — Points defined from a triangle so that they do not depend on how it is placed or labelled, such as the centroid, the incentre and the circumcentre. Thousands are catalogued, and the relations between them are a large part of triangle geometry.
• Triangle inequality 2 essays — The rule that going directly between two points is never longer than going through a third. It holds for straight-line distance and defines a metric, and it is what lets a tour shortcut repeated visits without growing longer.
• Triangle-free 3 essays — The property of a graph that no three of its points are joined in pairs. A triangle-free graph on n points has at most a quarter of n squared edges, reached only by two equal halves joined completely, and every denser graph is forced to contain a triangle.
• Triangle-free graph 1 essay
• Triangular numbers 5 essays — The counts of dots in a triangular arrangement - 1, 3, 6, 10 and onward - each the running total of the whole numbers up to its own index. They are the third diagonal of Pascal's triangle, because a diagonal there is the running total of the one before it, and twice one of them is a rectangle.
• Triangulation 21 essays — A division of a shape into triangles meeting edge to edge, with no gap and no overlap. Refining one is the standard way to turn a continuous question into a finite one, which is how Sperner's lemma reaches Brouwer's theorem.
• Tricolourability 2 essays — The property of a knot diagram that its arcs can be given three colours with the three at every crossing all alike or all different. It is a knot invariant because the three Reidemeister moves preserve it, which is what makes it able to tell the trefoil from the unknot.
• Trigonometry 2 essays — The relations between the angles and side lengths of triangles, built on the sine, cosine and tangent. It turns a geometric configuration into an equation that can be manipulated, which is how a statement about a picture becomes a computation and, sometimes, a proof.
• Trimmed mean 1 essay
• Trisection 6 essays — The division of an angle into three equal parts, which straightedge and compass cannot perform in general. Its impossibility follows from the cube roots involved, and it is one of three classical constructions ruled out by the same argument.
• Truncation 1 essay
• Truth 1 essay
• Truth function 9 essays — A rule computing the truth value of a compound statement from the truth values of its parts and nothing else. There are sixteen of them on two inputs, and a single one can generate all the rest.
• Truth table 6 essays — A table listing every assignment of true and false to a formula's variables and the value the formula takes on each. With n variables it has 2^n rows, which are the corners of an n-dimensional cube, and it decides any question about a formula by exhaustion.
• Tseitin formula 1 essay
• Turan graph 3 essays — The complete multipartite graph whose vertices are split into a given number of parts as equal in size as possible. Turán's theorem says it has the most edges of any graph on those vertices without a clique one larger than the number of parts.
• Turing machine 4 essays — An idealised computer that reads and writes symbols on an unbounded tape, one cell at a time, under a finite table of rules. The table is indexed by the machine's state and the symbol read, and whatever can be computed at all can be computed by such a machine.
• Turning number 2 essays — The number of full turns the tangent direction of a closed curve makes as the curve is traced once. It does not change when the curve is deformed without kinks, and it is the invariant that sorts closed curves in the plane up to such deformation.
• Tverberg theorem 1 essay
• Twin primes 5 essays — Pairs of primes differing by two. Nobody knows whether there are infinitely many, and the reciprocals of the ones there are add to a finite number — which rules out the argument that settles the primes themselves.
• Twist map 1 essay
• Type theory 2 essays — A system of rules that gives every term in a calculation a type and allows only combinations whose types fit. It began as a guard against paradox and now underlies programming languages and proof assistants, where a program's type is the statement it proves.

• Ulam spiral 1 essay
• Unbiased estimator 4 essays — An estimator whose average over all possible samples equals the quantity it estimates. Unbiasedness says nothing about the spread, and an unbiased estimate can vary so wildly from sample to sample that it is useless in any single one.
• Uncertainty principle 1 essay
• Uncountability 1 essay
• Undecidability 10 essays — The property of a question for which no algorithm gives the right answer on every input. It is a statement about every possible algorithm, and it is established by self-reference rather than by exhausting the possibilities.
• Undecidable sentence 3 essays — A statement that a given list of axioms neither proves nor refutes. Its existence is a theorem about the axioms rather than a report of unfinished work, and it is established by exhibiting two structures obeying the axioms that disagree about the statement.
• Unfolding 6 essays — Reflecting the table instead of the ball, so that a bouncing path becomes one straight line through a tiling of reflected copies. It removes the corners from a trajectory without changing it, and turns a question about a billiard into a question about a surface.
• Uniform acceleration 1 essay
• Uniform boundedness 1 essay
• Uniform convergence 10 essays — Convergence of a sequence of functions in which the largest gap to the limit, taken over the whole domain at once, goes to zero. It is strictly stronger than every point settling separately, and it is what a limit needs in order to inherit continuity from its members.
• Uniform distribution 7 essays — The distribution that gives every outcome in a set the same probability, or every interval of the same length the same chance. It is the model of a fair choice, and checking that a procedure really produces it is often harder than it looks.
• Uniformity 4 essays — Convergence at a rate that works everywhere at once, rather than separately at each point. It is the difference a quantifier order makes, and it is what allows a limit to be exchanged with an integral or a sum.
• Unimodular 3 essays — Having determinant one, for a matrix of whole numbers. Such a matrix has a whole-number inverse and takes the integer lattice exactly onto itself, and the condition is what keeps a mediant in lowest terms and what makes two Ford circles touch.
• Union bound 2 essays — The inequality that the chance of at least one of several events is at most the sum of their separate chances. It needs nothing about how the events overlap, which is why it is the first estimate reached for when many different things could go wrong.
• Unique factorisation 19 essays — The fact that every whole number above one is a product of primes in exactly one way, order aside. Written as an identity between a sum and a product it becomes Euler's product, which is where the analytic study of the primes begins.
• Uniqueness 8 essays — The property of a specification that exactly one object satisfies it. A characterisation needs two halves — that the conditions leave only one object, and that each condition is needed, which is shown by exhibiting something satisfying all the others and failing that one.
• Unit 5 essays — An element of a number system with a multiplicative inverse inside that same system, as 1 and −1 are among the integers. Among numbers a + b√D the units are the solutions of x² − Dy² = ±1, generated by −1 and one fundamental unit.
• Unit circle 6 essays — The circle of radius one centred at the origin, on which the trigonometric functions are read off directly. Every point on it satisfies the Pythagorean relation, and its rational points are exactly the Pythagorean triples in disguise.
• Unit distance graph 2 essays — A graph whose vertices are points of the plane, with two joined exactly when they are one unit apart. How many edges it can have, and how many colours it needs, are two of the oldest open questions in discrete geometry.
• Unit fraction 1 essay
• Unit group 1 essay
• Unit propagation 2 essays — A step in solving logical formulas: when a clause has only one undecided literal left, set that literal to make the clause true, and repeat. It is the cheap deduction every modern satisfiability solver runs between its guesses.
• Unitary divisor 1 essay
• Universal cycle 2 essays — A cyclic sequence in which every object of some family, such as every word of a given length, appears exactly once as a window of consecutive symbols. De Bruijn sequences are the case of words; subsets and permutations are harder.
• Universality 12 essays — The property of a system able to simulate any computation whatever, and therefore as powerful as any computer. It arrives at astonishingly small rule sets, and the price of it is that nothing general can be decided about what such a system will do.
• Unknotting number 1 essay
• Upper sum 1 essay

• Validity 2 essays — The property of an argument whose conclusion is true in every situation in which all of its premises are true, whatever its terms are taken to mean. It depends on the form of the argument alone, and a single situation making the premises true and the conclusion false refutes it.
• Valuation 1 essay
• Valuation measure 5 essays — One person's stated worth of every part of a thing being divided, adding across parts and totalling the whole. The procedures assume only that it adds over pieces and totals the whole, which is what makes them work without any comparison between people.
• Van der waerden 1 essay
• Variance 34 essays — The average of the squared distances of a quantity's values from its mean, whose square root is the standard deviation. It adds over independent quantities, which is the single fact behind the square root in every diffusion and every limit theorem.
• Variogram 1 essay
• Vector bundle 3 essays — A vector bundle is a family of vector spaces, one over each point of a space, varying continuously from point to point. The lines across a Möbius band over its centre circle are the smallest twisted one, and whether nonzero choices can be made continuously measures the twist.
• Vector field 1 essay
• Venn diagram 6 essays — A family of overlapping closed curves arranged so that every pattern of membership occupies exactly one region. It exists for any number of sets, though past three the curves cannot all be circles.
• Verification 1 essay
• Vertex configuration 2 essays — The list of face sizes met in order when walking round a single corner of a polyhedron. It is what a solid with every corner alike is classified by, and the enumeration of which such lists can close up is how the Archimedean solids are counted.
• Vertex enumeration 2 essays — Listing the corners of a region carved out by linear constraints, by solving the systems of equations that hold with equality there. It is how a linear objective is maximised without calculus, since the best value is always taken at a corner.
• Vertex-cover 2 essays — A set of a graph's vertices that touches every edge. Finding the smallest is hard in general, and relaxing the choice of each vertex from yes-or-no to a fraction gives a linear programme whose answer is always within a factor of two.
• Vertex-transitive 1 essay
• Vietas formulas 1 essay
• Volume 11 essays — The amount of space a solid occupies, measured against a chosen unit cube. It is additive over pieces that do not overlap, and the factor by which a linear map multiplies it is that map's determinant.
• Volume form 1 essay
• Voronoi diagram 11 essays — The division of a space into one region per site, each holding the points closer to that site than to any other. Every boundary is a bisector, so the regions are convex polygons, and the diagram answers which source got here first.
• Voting power 1 essay
• Voting rule 11 essays — Any recipe turning a collection of submitted rankings into a verdict — a winner, or an ordering of the candidates. Every one of them fails at least one condition that looks indispensable, which is what the impossibility theorems establish.

• Warings problem 1 essay
• Wave equation 1 essay
• Weierstrass function 1 essay
• Weighted average 2 essays — A sum of values each multiplied by a non-negative weight, with the weights adding to one. The result always lies between the smallest and largest value, and in the plane any weighted average of points lies inside their convex hull.
• Weighted majority 1 essay
• Weighted voting 4 essays — A voting rule in which each member casts a fixed number of votes and a motion passes when the votes for it reach a quota. A member's power to decide outcomes need not follow its share of the votes, and can even be zero.
• Well defined 1 essay
• Well-ordering 7 essays — An order on a set in which every non-empty subset has a least element. The whole numbers carry one and the real numbers carry none that anybody has described, and the assertion that every set has one is equivalent to the axiom of choice.
• Winding number 23 essays — How many times a closed curve goes round a chosen point, counted with a sign for direction. It cannot change under continuous deformation, which is what turns it into a proof that a polynomial has a root inside a circle.
• Witness 1 essay
• Word complexity 1 essay
• Word metric 6 essays — The distance between two group elements, counted as the fewest generators needed to get from one to the other. It turns a group into a geometric object whose large-scale features — growth, the shape of balls, the size of their boundaries — do not depend on the generators chosen.
• Work 1 essay
• Worst case 1 essay
• Writhe 4 essays — The sum of the signs of a knot diagram's crossings, each counted plus or minus one according to how the two directed strands cross. Adding or removing a twist changes it by one, which makes it the correction that turns the Kauffman bracket into an invariant.

• Young tableau 1 essay

• Zero divisor 4 essays — A non-zero element that multiplies with another non-zero element to give zero, which no field can contain. Their absence is what makes cancellation legal, and a dial has none exactly when its size is prime.
• Zero-one law 1 essay
• Zero-sum game 9 essays — A game in which one party's gain is exactly the other's loss, so the whole situation is described by a single number per cell. Mixing forces the two cautious valuations to coincide, and that common value is what each side can guarantee whatever the other does.
• Zorns lemma 3 essays — The statement that a partially ordered set in which every chain has an upper bound contains a maximal element. It is equivalent to the axiom of choice, and it is the usual way that axiom is used to build bases, maximal ideals and other objects no construction can name.

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