Modulus
Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.
Multiplying is turning
Complex numbers are introduced as an algebraic dodge for square roots of negatives. They are better understood as the arithmetic of rotation, at which point every rule stops needing to be remembered.
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.
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.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
Cyclic groupModular arithmeticCompositeCounting two waysFermats little theoremGreatest common divisorOrbitOrderPeriodicityPrimesRemainderArgument