Suppose that the diagonal of a square with a side length of 1 can be written as the ratio of integers to integers (P: Q) and PQ has no common divisor (when Q= 1, P: Q is an integer.

Pythagorean theorem: (p/q) 2 = 1 2+ 1 2

That is, p 2 = 2q 2.

Because 2Q^2 is even, that is, P 2 is even, so P is even (the square of any odd number is also odd).

Because PQ has no common divisor, Q is odd and P is even. Let p = 2a and p 2 = 4a 2 = 2p 2.

Q^2=2a^2

That is, q 2 is even, even and odd, so it can't be expressed by integer and integer ratio, so it is irrational.

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