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Concept

Remainder

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.

Named by 7 essays across 4 fields — each of them below, with the objects they name alongside it.

Partial sums of sin x. sin x with its Taylor partial sums of degree 1, 3, 5, 9 about zero. Each extra term buys agreement over a wider interval and none of them is right everywhere.

One point's worth of information

A Taylor series claims that everything a function does, everywhere, is encoded in its behaviour at a single point. That claim is extraordinary, it is often true, and the cases where it fails are the interesting ones.

analysis · Taylor series Arithmetic on a dial of 12. A dial with 12 positions. Starting at 8 and stepping forward 9 places lands on 5, because the walk passes the top 1 time on the way.

Numbers that wrap

A clock does arithmetic. It has finitely many numbers, addition never leaves it, and multiplication behaves entirely differently depending on one property of the size of the dial.

discrete · Modular arithmetic One number, two dials: 3 and 5. A grid of remainder pairs, each cell holding the smallest number that leaves those two remainders.

Two dials at once

Watch one number on two clocks with different faces. If the faces share no factor, every pair of readings occurs exactly once — so two remainders name a number, and a hard calculation can be split into two easy ones.

number · Modular arithmetic Hamilton's method on 27 seats and 5 regions. A worksheet of populations, exact quotas, floors, remainders and the seats Hamilton's method awards to 5 regions.

The seat that vanishes when the house grows

Twenty-seven whole seats have to be divided between five regions whose exact shares are 15.417, 7.209, 1.755, 1.431 and 1.188. Every rule for rounding those five numbers breaks something, and the instance drawn here breaks all three of the classical ways at once.

applied · Apportionment What the degree-5 sum costs, and what the bound claims. The error of the degree-5 Taylor polynomial of sin x against x, on a logarithmic scale, with Lagrange's bound drawn above it. The bound exceeds the error by a factor of 1.8 at the right-hand end.

An error with an unknown in it

Taylor's theorem does not say a partial sum is close to anything. It says the error is one more derivative evaluated somewhere nobody can name, and everything the theorem is worth comes from what happens when that somewhere is replaced by the worst case.

analysis · Taylor series A series that gets better, then worse. The error of Euler's series against the number of terms kept, at x = 0.05 and 0.1 and 0.2. At 0.05 the error falls to 1.1e-8 at 20 terms and then climbs without limit. At 0.1 the error falls to 1.8e-4 at 10 terms and then climbs without limit. At 0.2 the error falls to 1.8e-2 at 5 terms and then climbs without limit.

A series that converges nowhere

Expand Euler's integral in powers of x and the coefficients are the factorials, so the series converges at no x but nought. Stopped at its smallest term it still computes the integral to within about e^(−1/x) — and every term added after that makes the answer worse.

analysis · Taylor series Class numbers of x² + ny², with Euler's idoneal numbers marked. A scatter of class number against n up to 2000, on a logarithmic scale, with the 65 idoneal numbers marked on the power-of-two levels.

Euler's sixty-five convenient numbers

For some n, whether a prime can be written as x² + ny² is settled by its remainder on division by 4n alone, the way a prime's remainder on division by four settles whether it is a sum of two squares. Euler found sixty-five such n, from 1 to 1,848, called them convenient, and used the largest to prove that 18,518,809 is prime. Every one of them is a number whose class group has no element of order more than two — and whether the list is complete is still not known.

number · Unique factorisation

Named alongside it

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

ConvergenceTaylor seriesApproximationBoundCyclic groupDerivativeGreatest common divisorModular arithmeticModulusPeriodicityPolynomial approximationRadius of convergence

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