Proof That √2 Is Irrational (Annotated)
The classic proof that √2 is irrational, with color-coded handwritten notes explaining each step and phrase.
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Cartoon by Ben Orlin, Math with Bad Drawings. https://cartoons.mathwithbaddrawings.com/2015-01-01-proof-that-2-is-irrational/ (CC BY-NC 4.0)Transcript
"Proof: √2 is irrational."
"Suppose √2 is rational. [we suppose this to show it must be false] Then √2 = a/b, in lowest terms [i.e., simplified; no common factors]. 2 = a²/b² [squaring both sides]. 2b² = a² [multiplying by b²]. Thus, a is even, since its square is even. So a = 2k [that's what it means to be even]. 2b² = 4k². b² = 2k². So b is also even. This is a contradiction."
[Margin notes: "the number that, when squared, gives 2"; "inexpressible as a ratio of integers"; "a ratio of integers"; "odd² is odd, so if a² is even then a is even"; "because it's not simplified"]