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A Deadly Proof

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1 1 √2 Pythagoras: a² + b² = c²

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Can √2 be represented concretely?

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Pythagoreans whole numbers

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1² = 1 < 2 2² = 4 > 2 => √2 lies in between 1 and 2

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Pythagoreans fractions

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What next?

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Assume: √2 can be represented as a fraction... √2 = p/q

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...where p and q do not share any factors: 18/12 = 9/6 = 3/2

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√2 = p/q

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√2 = p/q => 2 = p²/q²

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√2 = p/q => 2 = p²/q² => 2q² = p²

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√2 = p/q => 2 = p²/q² => 2q² = p² => p² is even

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√2 = p/q => 2 = p²/q² => 2q² = p² => p² is even => p is even. Write p = 2r.

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2q² = p² and p = 2r

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2q² = p² and p = 2r => 2q² = 4r²

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2q² = p² and p = 2r => 2q² = 4r² => q² = 2r²

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2q² = p² and p = 2r => 2q² = 4r² => q² = 2r² => q² is even

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2q² = p² and p = 2r => 2q² = 4r² => q² = 2r² => q² is even => q is even. Write q = 2s.

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p = 2r and q = 2s

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p = 2r and q = 2s => p/q = (2r)/(2s)

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p = 2r and q = 2s => p/q = (2r)/(2s) => p/q reduces to r/s.

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"Proof by contradiction"

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Conclusion: √2 cannot be represented as a fraction.

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Pythagoreans:

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RIP Hippasus of Metapontum (maybe)