Proof that √2 is irrational


 * 1) Assume that √2 is a rational number, meaning that there exists an integer a and an integer b such that a / b = √2.
 * 2) Then √2 can be written as an irreducible fraction (the fraction is reduced as much as possible) a / b such that a and b are coprime integers and (a / b)2 = 2.
 * 3) It follows that a2 / b2 = 2 and a2 = 2 b2.
 * 4) Therefore a2 is even because it is equal to 2 b2 which is obviously even.
 * 5) It follows that a must be even. (Odd numbers have odd squares and even numbers have even squares.)
 * 6) Because a is even, there exists an integer k that fulfills: a = 2k.
 * 7) We insert the last equation of (3) in (6): 2b2 = (2k)2 is equivalent to 2b2 = 4k2 is equivalent to b2 = 2k2.
 * 8) Because 2k2 is even it follows that b2 is also even which means that b is even because only even numbers have even squares.
 * 9) By (5) and (8) a and b are both even, which contradicts that a / b is irreducible as stated in (2).