Number Theory · Proof · Well-Ordering Principle

Most proofs that √2 is irrational require knowing something about even numbers. This one requires knowing only that every nonempty set of natural numbers has a smallest element — and then it finds a contradiction in that smallest element itself.

A proof that √2 is irrational — using nothing but a smallest natural number and the contradiction waiting inside it.

Step by step

We assume √2 is rational and look for a contradiction. If √2 = p/q for some integers p, q, then multiplying both sides by q gives q√2 = p, an integer. So the set S = { N ∈ ℕ : N√2 ∈ ℕ } is nonempty — q is in it. By the well-ordering principle, S has a smallest element k.

Now consider the number k(√2 − 1). Two things are true about it simultaneously, and together they are fatal.

Why this proof is different

The classical proof — the one in Euclid — writes √2 = p/q in lowest terms, squares both sides, and derives that both p and q must be even, contradicting the lowest-terms assumption. It works, but it requires the machinery of common factors and reduced fractions.

This proof needs none of that. It never writes √2 as a fraction. It never mentions even numbers. The only arithmetic it uses is that 1 < √2 < 2 — which follows from 1² = 1 and 2² = 4 — and that the difference of two integers is an integer. Everything else is the well-ordering principle doing its job.

A proof that proceeds by structure rather than by calculation tends to be one you understand rather than merely remember — the distinction the Feynman Technique is built around. Once you have seen why the descent is unavoidable, the result is obvious. Before you see it, no amount of arithmetic makes it feel necessary.

The Pythagoreans discovered that √2 is irrational — that the diagonal of the unit square cannot be expressed as a ratio of whole numbers — and reportedly found it disturbing enough to suppress. They were not the last to want mathematics to behave differently: the Indiana Pi Bill of 1897 attempted to settle the value of π by legislative vote. The proof above answers its own question with rather more finality than any bill could. If such a ratio existed, there would have to be a first natural number witnessing it. There isn't.

If the irrationality of √2 made you curious about π — another number that refuses to behave — six books approach it from equally oblique angles: history, category theory, and at least one very well-timed pun.

Keep wandering

A few more pieces in the same spirit — math, design, and slow attention.

[
Paper that stands up
Henry Billingsley’s 1570 English Euclid — paper solids that still beat a headset
Open
](https://abakcus.com/articles/billingsley-euclid)
[
The Feynman Technique: A Complete Guide
The four-step learning method developed by Nobel Prize-winning physicist Richard Feynman. How it works, why it works, and how to apply it to anything.
Open
](https://abakcus.com/articles/feynman-technique)
[
Seven questions, June 7, 1869
MIT's first algebra entrance exam. The questions haven't changed. Everything else — tuition, class size, acceptance rate — is unrecognisable.
Open
](https://abakcus.com/articles/mit-1869)
[
The legislature that tried to vote on π
1897 — Indiana’s House passed a bill that implied π = 3.2. Then a mathematician walked in.
Open
](https://abakcus.com/articles/indiana-pi-bill)
[
The Mathematics of Sudoku
On counting all valid 9×9 grids (6.7 sextillion), solving algorithms that work on paper, and why Sudoku is fundamentally a graph coloring problem.
Open
](https://abakcus.com/articles/sudoku-mathematics)
[
The world's hardest Sudoku — 23 clues, one solution
Arto Inkala's 2012 puzzle: 4.5% trivialization rate, 153 human attempts, three months to design — and what “hardest” actually measures.
Open
](https://abakcus.com/articles/inkala-sudoku)

A Number That Cannot Be the Smallest

Step by step

Why this proof is different