Matpic
mathematics, in pictures
Concept

Modular arithmetic — where it appears

5 essays name this object, across 2 fields. What follows is each of them, and the objects they name alongside it.
012345678910118 + 9= 17= 5 (mod 12)1 lap of the dial,then the remainder

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 12 lattice points sit on the circle — and 4 × (3 − 0) = 12divisors of 25: 1, 5, 25 are 1 mod 4, none are 3 mod 4

Two squares, and a lattice

Whether a prime is the sum of two squares is decided entirely by its remainder on division by four. A fact about circles is settled by a fact about remainders, and neither statement contains any hint of the other.

number · sums of two squares 1 string1 string5 strings5 strings5 strings5 strings5 strings5 strings32 strings fall into 8 necklaces32 strings in all: 2 constant ones, and 6 rings of 5so 32 − 2 = 5 × 6, and p divides a^p − a with nothing left over

Necklaces that prove a theorem

Thread five beads in two colours, thirty-two ways. Two of them are all one colour; the other thirty fall into rings of five. That count, and nothing else, is Fermat's little theorem.

number · fermats little theorem 0612391017134511281401234012mod 3 ↓mod 5 →every one of the 15 pairs is reached, exactly onceso a remainder mod 3 and a remainder mod 5 together name one number mod 15

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 slope 7/117 points below the line, 8 above, and 7 + 8 = 5 × 3 = 15(7 | 11) = −1 from the count below; (11 | 7) = +1 from the count above

Counting one rectangle, twice

Whether seven is a square modulo eleven, and whether eleven is a square modulo seven, are two unrelated-looking questions. Their answers are linked, and the link is a rectangle of dots counted along its rows and then along its columns.

number · quadratic reciprocity

Named alongside it

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

Counting two waysPrimesCyclic groupLatticeFermats little theoremGreatest common divisorOrderParityPeriodicityRemainderBijectionChinese remainder theorem

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