Matpic
mathematics, in pictures
Ladder

Unique factorisation — the ladder

2 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    rung 1 · number
  2. The divisors of 60. Every divisor as a lattice point, one axis per prime, joined when one divides the other by a single prime.

    The shape of a number's divisors

    Lay a number's divisors out as a lattice with one axis per prime, and two of the most useful facts in arithmetic stop being formulas and become the width and the corner of a rectangle.

    rung 2 · number

All ladders