Logarithm
Named by 30 essays across 6 fields — each of them below, with the objects they name alongside it.
The curve that is its own slope
There is exactly one shape of exponential curve whose steepness at every point equals its height at that point. The number that produces it is 2.71828…, and it was not chosen for elegance.
A sum whose terms vanish and whose total does not
Add a half, a third, a quarter, and keep going. The terms shrink to nothing and the total passes every number there is — but so slowly that no computation will ever watch it happen.
The same terms, in a different order, adding to whatever is asked
Flip alternate signs in the harmonic series and it converges. Reorder the terms — add nothing, remove nothing — and it converges to any number chosen in advance. Addition stops being commutative, and the picture shows where it goes.
How fast two orbits part
The word "sensitive" is an adjective. Averaging the logarithm of one derivative along an orbit turns it into a number — one that says how many steps of prediction the map allows, and whose sign says whether it allows any.
The question nobody can answer
Halve it if it is even, triple it and add one if it is odd. Every number anyone has tried comes down to one. Nobody can prove they all do, and the reason is not that the problem is hard to state.
Counting what has no formula
There is no expression that gives the nth prime, and yet the number of primes below a bound is predictable to within a fraction of a per cent — by a function that is not a formula for the primes but an integral of the wrong-looking quantity.
The slope of the mirror image
Undoing a function is reflecting its graph in the diagonal, and a reflection turns a slope into its reciprocal. That single observation supplies the derivative of every inverse — the logarithm, the roots, the inverse trigonometric functions — without differentiating any of them.
The area that names the number
The number e can be defined without mentioning slopes at all. Slide right along the curve 1/x until the area underneath reaches exactly one, and stop. That is where e is, and the reason logarithms turn multiplication into addition is visible in the same picture.
A closer start buys only time
Two trajectories from almost the same place separate exponentially, and the rate does not depend on how close they began. Halving the initial error buys one fixed interval of extra agreement, and no amount of precision buys more than a fixed number of those.
The heuristic that cannot be a proof
There is a two-line argument that the Collatz conjecture is true, it is convincing, and everybody who works on the problem believes it. It also cannot be turned into a proof, and understanding exactly where it fails is more instructive than the argument itself.
![Partial sums of x − x²/2 + x³/3 − … on [0, 1]. Partial sums of the power series x − x²/2 + x³/3 − … drawn on the interval from 0 to 1 with the function the series sums to inside the interval, and the values at x = 1 marked.](/f/partial-sums-of-x-x2-2-x3-3-on-0-1-e402dfbd.svg)
A sum read from inside
The series 1 − 1/2 + 1/3 − 1/4 + … adds to log 2, and the reason is not in the series. Its power series equals log(1 + x) inside the interval, and the value at the edge is read off by continuity. Abel's theorem says when that reading is honest — and the series 1 − 1 + 1 − …, which has no sum, is read the same way as a half.
An endless region with a finite area
A region that runs off to infinity can still have a finite area, and for the curves 1/xᵖ the exponent that makes the far end finite is exactly the one that makes the end at zero infinite. 1/x fails at both, no power succeeds at both, and a horn can hold less than π while needing infinite paint.
A rectangle cut by a curve
Integration by parts is taught as the product rule run backwards. It is also a picture: an increasing curve cuts a rectangle into two pieces, one of them the area under the curve and the other the area beside it, and the formula says only that the pieces fill the rectangle. Run repeatedly, the same cut produces the factorials and Wallis's product for π.
Two families of solutions, and a box that holds both
Replace the 1 in Pell's equation by 7 and x² − 2y² = 7 still has infinitely many solutions — but they fall into exactly two families, each one an orbit of the same multiplication, and every family has a member inside a box whose size is fixed in advance. How many families there are is then a count of factors, and 3 has none.
Sixty needs two digits and sixty-one needs ten
The smallest solution of x² − 60y² = 1 is x = 31. For x² − 61y² = 1 it is x = 1,766,319,049. The jump is not an accident of 61: the solution is exactly the product of one period's complete quotients, so its size is set by how long the continued fraction takes to come home — and for the cattle problem Archimedes is said to have posed, that product has 103,273 digits.
Units that form a lattice
In the whole numbers only 1 and −1 have whole-number reciprocals. In the integers of a bigger field there can be infinitely many such units, and they are not scattered: take logarithms of their sizes under each way of placing the field in the real or complex numbers, and the units land exactly on a lattice. How many dimensions that lattice has is a count of those placements, and the area of its cell is a number no formula gives.
The matrix that has no logarithm
Every square matrix has an exponential, and a matrix exponential is always invertible. The converse fails, and it fails in a way that can be read straight off the eigenvalues — a real matrix with eigenvalues −1 and −2 is no exponential at all, while minus the identity is the exponential of a whole continuum of matrices that do not even commute with each other.
No integrand sits on the border
The area under 1/x out to infinity is infinite and the area under 1/x² is finite, so it is natural to look for the dividing line. There is none. Divide any divergent integrand by its own running integral and it still diverges, more slowly; divide any convergent one by the square root of its tail and it still converges, more slowly. Paul du Bois-Reymond proved in 1873 that no single integrand can separate the two families — and the slow ones are slow beyond imagining.
Almost none of the roots are real
Pick the coefficients of a polynomial of degree a thousand at random, each one an independent draw from the bell curve, and ask how many of its thousand roots are real. The answer is about five. Mark Kac found in 1943 that the average grows only like (2/π) ln n, and the reason can be read off the real line itself: the real roots crowd towards +1 and −1 and spread evenly on a logarithmic scale of distance from them.
Five where four were promised
In one unknown, a polynomial with three terms has at most two positive roots, whatever its degree. The natural guess for two equations in two unknowns, each with three terms, was two times two: four. It is wrong. Bertrand Haas found two such equations in 2002 whose curves cross five times in the positive quadrant, all five crowded within a tenth of one corner, and five turned out to be the most there can ever be.
When a table of moves came from steady rates
A process that jumps between states at constant rates, watched once a year, produces a table of yearly moves that is the exponential of its rates. Most tables that anyone could write down are not — a random three-state table is only about one time in twenty-three — and the few that are can have two different sets of rates behind them, but only once the process has forgotten where it started.
Multiplying makes the digit one common
Multiply a few random numbers together and the product starts with 1 about 30 per cent of the time and with 9 under 5 per cent — Benford's law, which no factor contains. The logarithm of a product is a sum, the central limit theorem spreads that sum across many powers of ten, and once it is spread its fractional part is uniform. The approach is geometric, at a rate fixed by a single number for each kind of factor; sums never get there, and the powers of two get there with no randomness at all.
A share that depends on the average
What share of the whole numbers begin with the digit 1? Counted up to N, the answer swings between one ninth and five ninths for ever, and averaging the swing over N, even three times over, narrows it without removing it. Weight each number n by 1/n instead and the share settles at log₁₀ 2 — Benford's value — and so does every other sensible weighting that counts each decade alike. The primes behave the same way. Benford's law for the counting numbers is not a fact about them; it is a fact about how they are averaged.
What a missing string of digits leaves behind
Strike from the harmonic series every term whose denominator contains 42 and the rest adds up to 228.45. Strike out 99 instead and it adds up to 253.28. The difference is not about which numbers are lost but about waiting: 99 overlaps itself, so it takes 110 random digits on average to turn up where 42 takes 100, and the sum is almost exactly ln 10 times that wait.
The length round an ellipse
An ellipse has an area anyone can write down, πab, and a perimeter nobody can: it is an integral with no expression in the functions of school mathematics. Kepler guessed it, Euler guessed better, and in 1914 Ramanujan wrote down a formula whose error falls like the fifth power of how far the ellipse is from a circle — off by four hundredths of a per cent even for a segment. Gauss had already found the way to compute it exactly, doubling the correct digits at every step.
The corners of a random hull
Drop ten thousand points at random into a triangle and stretch a band round them: about twenty points touch it. Drop them into a disc and about seventy-three do. The count grows like a logarithm in one shape and like a cube root in the other, and the reason is where the boundary bends.
The best average is the worst bet
A coin lands heads six times in ten and pays even money. Stake everything on every toss and the expected wealth grows by a fifth a toss — and the bettor is ruined at the first tail, which is certain to come. Stake a fifth of wealth each time instead and the average grows far more slowly, while the typical bettor grows faster than with any other fraction. The expectation is maximised by the strategy almost nobody should use.
What not knowing the coin costs
Kelly's rule stakes the edge, and the edge is the one thing a bettor never has. Estimate it from twenty tosses and stake the estimate, and a lucky start can turn a favourable coin into certain ruin. Learn it toss by toss instead, with Laplace's rule, and the cost of ignorance shrinks to a factor of about the square root of the number of tosses — because a bettor who learns is exactly a bettor who split the money among every fixed fraction at once.
The route round that rates cannot avoid
A table of yearly moves between states came from steady rates only if it is the exponential of a rate matrix. With two states that is true of half of all tables, with three of one in twenty-four, with four of about one in two thousand, and of none of 400,000 tables drawn at random with five. Even a table that is nine parts staying put fails more and more often as states are added, and always for one reason: rates that carry 1 to 2 and 2 to 3 also carry 1 to 3, and a table whose direct move is rarer than that route round cannot have come from them.
A rule that counts lets the sum escape
Strike out every number containing a 9 and the harmonic series converges, to 22.92. Keep only the numbers with as many 7s as 3s and it diverges — slowly, like the square root of the number of digits. Ask for two such balances and it still diverges; ask for three and it converges. The dividing line is Pólya's theorem that a random walk comes home in one and two dimensions and not in three.
Named alongside it
The objects these essays reach for when they reach for this one.
Harmonic seriesConvergenceIntegralLimitExpectationConvergence rateDerivativeDeterminantDivergenceEigenvalueApproximationContinuity