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Concept

Irrationality

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.

Named by 10 essays across 3 fields — each of them below, with the objects they name alongside it.

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 Every candidate for a rational square root, tried. A column for each of 2, 3, 4, 5, 6, 7, 8, 9, listing the whole numbers that divide it with their squares, and the verdict the search returns.

Which roots refuse to be fractions

The square root of two is not a fraction, and neither is the square root of three, five, six or seven. The rule behind the list turns an infinite question into a search over the divisors of a single number — and the search finishes.

number · Irrationality 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 13 record approximations in 26 turns. The distance from π to each fraction the descent passes, against its denominator, on logarithmic axes. 13 of them beat every fraction with a smaller denominator.

The fractions that beat every smaller one

Walking down the tree towards a number produces a sequence of fractions closing in on it. Most of them are steps along the way; a few are the best approximations there are — closer than every fraction with a smaller denominator — and which few is decided by where the turns change direction.

number · Stern brocot An additive function that is nowhere a line. A square window with 1157 points of the graph of an additive function that sends √2 to 0, scattered across the window and meeting every part of it, beside a dashed diagonal marking the straight line y = x.

A function that adds and is nowhere a line

Every continuous function with f(x + y) = f(x) + f(y) is a straight line through the origin. Drop continuity and, given the axiom of choice, there are others — functions that add perfectly and whose graphs are scattered densely over the whole plane. A finite piece of the construction can be drawn exactly; the whole of it needs a basis of the real numbers that no one can write down.

logic · Axiom of choice 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 continued fraction of e. Bars for the first 30 continued-fraction terms of e: mostly ones, with every third bar rising in a straight staircase, 2, 1, 2, 1, 1, 4, 1, 1, 6, ….

The pattern in e's continued fraction

Written as a continued fraction, e is 2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8 — two ones, then the next even number, for ever. Euler found the pattern and proved it with a differential equation. A proof from 2006 needs only three integrals, each of which turns out to be exactly the error of one of e's own convergents.

number · Irrationality A pentagon, its pentagram, and the pentagon inside. A regular pentagon with its diagonals drawn as a pentagram, enclosing a smaller pentagon, repeated 3 times inward; every pentagon has diagonal-to-side ratio φ.

The diagonal no unit measures

Draw the five diagonals of a regular pentagon and they make a star with a smaller pentagon at its centre. Subtract the side from the diagonal and what is left is the smaller pentagon's diagonal; subtract that from the side and what is left is its side. The pentagon has handed back a smaller copy of itself, and it will do so for ever — which means no unit, however small, measures both the side and the diagonal an exact whole number of times.

geometry · Golden ratio Apéry's proof for ζ(3), as three numbers. log₁₀ of 2·lcm(1, …, n)³, of |a(n)ζ(3) − b(n)| and of their product for n up to 60: slopes 1.262, -1.546, -0.284 per step.

The race that makes ζ(3) irrational

Roger Apéry's 1978 proof that the sum of the reciprocal cubes is not a fraction comes down to a race between two numbers. A whole-number multiplier grows by a factor of ten every 0.79 steps; the gap it multiplies shrinks by a factor of ten every 0.65. The gap wins, by a margin of about seventeen per cent, and that margin is the whole proof.

number · Irrationality

Named alongside it

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

Continued fractionsProof by contradictionDiophantine approximationInfinite descentIntegralityRational approximationDescente, the numberFactorialIncommensurabilityLowest termsPi

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