Matpic
mathematics, in pictures
Concept

Proof by contradiction

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.

Named by 11 essays across 4 fields — each of them below, with the objects they name alongside it.

Euclid's construction on 2, 3, 5, 7. The product of the listed primes plus one, divided by each of them in turn; every division leaves one over.

There is no last prime

Euclid's argument is often described as producing a new prime from any finite list. It does not, and the number it builds is frequently composite — which makes the proof more interesting rather than less.

number · Infinitude of primes Two factor trees of 360. The same number split two different ways, both ending in the same primes.

One way to factor, and no other

Every number breaks into primes in exactly one way. That is so familiar it is hard to see as a claim at all — until it is put beside an arithmetic where it is false, and where six has two different factorisations that cannot be reconciled.

number · Unique factorisation Two squares of side 12 inside one of side 17. Two overlapping squares laid into opposite corners of a larger one, with the overlap and the two uncovered corners marked.

The square that cannot shrink

The usual proof that the square root of two is irrational is about even and odd numbers. There is a proof about squares instead, in which a supposed solution is folded into a smaller one — and the folding is a drawing.

number · Irrationality The diagonal, and the row built to be off the list. A table of rows of ones and zeros with the diagonal marked, and beneath it the row obtained by flipping every diagonal entry.

The row that is not on the list

Write down a list of infinite sequences, any list at all, and there is a rule that builds a sequence missing from it. The rule reads one entry from each row, and it is the single most reused argument in this field.

logic · Diagonalisation Independence of irrelevant alternatives, broken by Borda. Two profiles that agree on every voter's ranking of two candidates and differ only in where the others sit, with the rule's verdict between the two reversed.

Four conditions, and no rule that has all of them

Five reasonable rules can return five different winners on one set of ballots, which invites the obvious question of which one is right. The answer is that the conditions anybody would write down cannot all hold at once — and here each named rule's own violation is found by search rather than quoted.

applied · Voting rules Between every number and its double. The interval from n to twice n, drawn for n up to 26, with the primes inside each marked. Every interval contains at least one.

Always one before the double

A density says what happens on average and permits long empty stretches. This says something a density cannot — that the stretch from any number to twice it contains a prime, at every scale, without exception.

number · Prime distribution The tail that would have to be a whole number. For each denominator, the value of q! times the tail of the series for e, plotted against the band between zero and one where no whole number lies, with the bound 1/q above it.

A tail too small to be a whole number

If e were a fraction with denominator q, then q! times e would be a whole number. It splits into a whole part and a tail, the tail is squeezed strictly between nothing and one, and there is no whole number there.

number · Irrationality The polynomial that squeezes π. On the left, xⁿ(π − x)ⁿ/n! drawn at several degrees, its largest value falling toward nothing; on the right, the derivatives of the same polynomial at zero, every one a whole number.

An integral that cannot be a whole number

Niven's proof that π is not a fraction is the same squeeze as the one for e, with a much harder multiplier. A polynomial supplies the whole number; its own smallness supplies the contradiction; and both halves are computable.

number · Irrationality The primes below 100,000, by remainder mod 4. A bar for each remainder on division by 4, showing how many primes below 100000 leave it. The 2 classes sharing no factor with 4 hold near-equal counts; the rest are empty or hold one prime.

Infinitely many of one kind

Euclid's argument produces a prime nobody had listed, and says nothing about what it looks like. Ask for infinitely many primes ending in 3, or leaving a remainder of 1 on division by 4, and the same construction has to be aimed — and for most targets nobody knows how to aim it.

number · Infinitude of primes The tangent built from a continued fraction. The curve tan x on (−1.55, 1.55) with 4 of Lambert's convergents: a straight line, then rational curves that bend ever closer to the tangent and follow it towards its poles.

The fraction Lambert built for the tangent

The first proof that π is not a fraction, from 1761, does not look at π at all. It writes the tangent as an endless continued fraction, shows that the fraction's value at any rational point other than zero cannot be rational — because its tails are trapped between nothing and one — and then notes that tan(π/4) = 1.

number · Irrationality The powers of 2 in 1 to 12: one number, 8, stands alone. The whole numbers from 1 to 12, each with a bar whose height is the power of 2 dividing it. A single number has the tallest bar, which is why the harmonic number H(12) has an even denominator and an odd numerator.

The sum that steps over every whole number

The harmonic sum 1 + 1/2 + 1/3 + … passes 2 at the fourth term, 3 at the eleventh, 4 at the thirty-first, and eventually every whole number there is. It never lands on one. The proof is a single number in the list 1, 2, …, n that carries more factors of two than any other — and the same arithmetic makes the numerators divisible by squares of primes they have no business knowing about.

analysis · Harmonic series

Named alongside it

The objects these essays reach for when they reach for this one.

DivisibilityIrrationalityPrimesUnique factorisationContinued fractionsCounting argumentCounting two waysDescentExistence proofFactorialHarmonic seriesIntegrality

All concepts