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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. first split 2 × 180first split 18 × 203602180229022245222315222335360182029210233225both end in 2 × 2 × 2 × 3 × 3 × 5the same primes, the same number of times, in a different order — and that is the theorem

    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. 124361251020153060× 2 →× 3 ↑2^2 × 3 × 5 — 3 × 2 × 2 = 12 divisors

    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