Complex numbers
Named by 22 essays across 6 fields — each of them below, with the objects they name alongside it.
The shape in every picture of itself
One line of arithmetic, repeated, with a single complex number as its only input. Sort the numbers by whether the result stays bounded and the boundary between the two answers is the most complicated object anyone draws from a rule this short.
One c, one picture
The same iteration, with the parameter held still and the starting point varied instead. Every complex number gets its own picture, and moving the parameter a hair can shatter it into dust.
Where Newton's method goes instead
An algorithm designed to find roots, run from every starting point at once. Three roots, three basins, and a boundary at which all three are arbitrarily close — so a rule with no randomness in it has starting points whose answer cannot be predicted.
A loop that cannot miss the middle
Feed a circle into a polynomial and a closed loop comes out. A small circle gives a loop that does not enclose the origin; a large one gives a loop that goes round it as many times as the degree. Something has to happen in between, and that something is a root.
What the coefficients already know
Finding the roots of a polynomial is hard and often impossible in closed form. Reading off their sum, their product and how many of them are real is none of those things — those numbers are sitting in the coefficients, and no root-finding is required to get at them.
The polygon an equation forces
The n solutions of z to the n equals one are the corners of a regular polygon, and nearly everything about them — that they form a group, that they sum to zero, that the equation factors into pieces with whole-number coefficients — is that picture read carefully.
A multiplication that remembers the order
Four characters — i² = j² = k² = ijk = −1 — define a multiplication in which ab and ba are different numbers. Everything follows from them, including the fact that a rotation of a solid body has two names, and that two full turns are needed to get one of them home.
The number four points agree on
One inversion is a reflection and reverses orientation. Two of them compose to a motion, and what that motion leaves alone is a single number computed from any four points.
One sum, squared two ways
Add the p-th roots of unity, each taken with a plus or a minus according to whether its index is a square. The walk that results closes on a point at distance √p from the origin — and squaring that one number, evaluated two different ways, is the reciprocity law.
The identity that multiplies sums of squares
A sum of two squares times a sum of two squares is a sum of two squares, and the same is true of four and of eight. It is true of three of nothing: 3 and 5 are each a sum of three squares, 15 is not, and one small pair of numbers rules the case out forever.
The sphere that complex numbers live on
Add one point to the complex plane and it becomes a sphere. The rotations of that sphere are exactly the maps written as one linear expression divided by another, so a fact about turning a ball is a fact about dividing polynomials.
The centre is a choice
A Taylor series is nearly always written about zero, and nothing about the construction prefers zero. Moving the centre moves the interval the series works on, and moving it repeatedly walks the function into places its first series could never reach.
A denominator that reaches past the radius
The Taylor series of ln(1 + x) is useless beyond x = 1 however many terms it is given. The same coefficients spent on a numerator and a denominator converge at x = 3, and at x = 100, because a polynomial cannot imitate a singularity and a quotient of two polynomials can.
Covering rather than avoiding
Two arguments say no starting guess is safe: the boundary is fractal and some regions are permanently trapped. The repair is not a better guess. It is a fixed list of starting points, computed from the degree alone, from which every root of every polynomial of that degree is found.
One number under every bell
The area under e^(−x²) has no formula in terms of the usual functions, and yet the area under e raised to any downward quadratic is known exactly. Completing the square in the exponent moves and squeezes every such curve into the same one, so a single number — √π — pays for all of them.
Every power sum, from the coefficients alone
Raise the roots of a polynomial to the k-th power and add them. However the roots turn, the total is a whole number when the coefficients are, and Newton's identities produce it from the coefficients one step at a time — no root is ever found. Run the rule on x³ − x − 1 and out comes Perrin's sequence, whose terms know which numbers are prime, nearly.
The roots of the slope stay inside
Mark the roots of a polynomial in the complex plane and stretch a band around them. However the roots are arranged, the roots of the derivative land inside the band — never outside, never on a new frontier. The reason is a balance of pushes, the same reason makes the derivative of a cubic mark the foci of an ellipse nobody asked for, and a question about how far inside the roots must sit has been open since 1958.
Discs that fence in the eigenvalues
Draw one disc for each row of a square matrix, centred on the diagonal entry, with a radius equal to the sum of the sizes of everything else in that row. Every eigenvalue lies inside one of the discs, and a group of discs set apart from the rest holds exactly as many eigenvalues as it has discs. Nothing is solved to find them.
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.
Random roots crowd onto the circle
The roots of a random polynomial are not scattered across the plane. They gather within about 1/n of the unit circle, and their directions spread evenly round it. Paul Erdős and Pál Turán proved in 1950 that this is not a fact about randomness at all: any polynomial whose coefficients are all of roughly one size has roots whose directions are nearly even, and the inequality says exactly how nearly, in terms of nothing but the coefficients.
The sine of a figure eight
Walk round a circle and read the height against the distance walked, and the reading is the sine. Do the same on Bernoulli's figure eight and the reading is a new function, with an arc integral that has t⁴ where the circle's has t², a length Gauss found inside an average, two periods instead of one — and exactly the same list of equal divisions that compass and straightedge can draw.
Three real roots and no real radicals
The cubic x³ − 3x + 1 has three real roots, and Cardano's formula reaches every one of them by way of the cube roots of a complex number. That detour cannot be removed: Hölder proved in 1891 that no expression built from real radicals gives a root of an irreducible cubic whose roots are all real. The proof is three lines of the Galois correspondence, and its general form says that real radicals reach all-real roots only when square roots alone would.
Named alongside it
The objects these essays reach for when they reach for this one.
PolynomialRootsConvergenceIterationRoots of unityBasin of attractionFractalRoot-findingAnalytic continuationCoefficientConnectednessContinuity