PROVE SQUARE ROOT OF 2 IS IRRATIONAL NUMBERS.

PROVE SQUARE ROOT OF 2 IS IRRATIONAL NUMBERS.
To prove that the length of the diagonal of a square of unit side cannot be represented by rational number it suffices to show that square root of 2 is irrational. To this end we first observe that, for a positive integer s, s2 is even if and only if s is even. Now suppose for the purpose of argument, that square root of 2 is rational, that is, square root of 2 =a/b, where a and b are relatively prime integers. Then
a = square root of 2b
or
a2 = 2b2
since a2 is twice an integer, we see that a2, and hence a, must be even. Put a =2c. Then the last equation becomes
4c2 = 2 b2
Or
2c2 = b2
From which we conclude that b2, and hence b, must be even. But this is impossible since a and b were assumed to be relative prime. Then the assumption square root of 2 is rational has led us to this imposible situation, and must be for clear in mathematic. So square root of 2 is irrational number.