Why √2 is not rational
About this app
The indirect proof with tiles: 2q² = p² splits the p² square into two equal halves, odd squares always keep a middle tile, p = 2a quarters the square and q turns even, so the stamp “reduced” breaks and the same figure returns one step smaller. In Try mode, test your own fractions: even 99/70 misses by exactly one tile.
With explain mode: an AI tutor walks you through the app, points out what matters in the scene and answers your questions.
The proof
Suppose √2 were rational, √2 = p/q with integers p and q in lowest terms. Squaring gives p² = 2q². So p² is even, and since the square of an odd number is odd, p must be even: p = 2a. Substituting gives 4a² = 2q², so q² = 2a², and q is even as well. Then p/q could be reduced by 2, contradicting the assumption. Therefore √2 is irrational. This indirect proof is one of the best known in mathematics.
What the app shows
The app turns the equation p² = 2q² into tiles: a square of p² tiles would have to contain as many tiles as two squares of q² tiles each. Six steps follow the proof: assumption, p² even, p even (odd squares always keep a centre tile), q even, and the contradiction as the “reduced” stamp breaks. The last step shows infinite descent: the same figure returns one level smaller, p > q > a > b > …, which is impossible for positive integers.
Try it
In “Try” mode you choose p and q yourself or pick one of the near misses 3/2, 7/5, 17/12, 41/29, 99/70 and 239/169. The app pours the 2q² tiles into the p² pile and shows the difference. For these approximations exactly one tile is left over or missing, which is as close as it gets. Counting factors of 2 explains why: p² contains an even number of them, 2q² an odd number.
Frequently asked questions
- Why is the square root of 2 irrational?
- If √2 = p/q were a fraction in lowest terms, then p² = 2q². It follows that p is even and then q is even too, so the fraction could be reduced by 2 after all. This contradiction shows that √2 cannot be written as a ratio of integers.
- What is a proof by contradiction?
- In a proof by contradiction you assume the opposite of what you want to show and derive a logical contradiction from it. Since a true assumption cannot lead to a contradiction, the assumption must be false. In the proof for √2 the assumption is that √2 is a fraction in lowest terms.
- Which fractions come closest to √2?
- Good approximations are 3/2, 7/5, 17/12, 41/29, 99/70 and 239/169. For each of them p² − 2q² = ±1, so they solve Pell’s equation. 99/70 ≈ 1.414286 is already very close to √2 ≈ 1.414214 but never equal.
- What can I do in this app?
- In “Proof” mode you go through the proof by contradiction in six steps with tile pictures and formulas. In “Try” mode you test your own fractions p/q and see by how many tiles p² and 2q² differ.
Subject: Visual proofs
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