Wikipedia
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 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. ((a / b)n = an / bn)
4. Therefore a2 is even because it is equal to 2 b2. (2 b2 is necessarily even because it is 2 times another whole number; that is what “even” means.)
5. It follows that a must be even as (squares of odd integers are also odd, referring to b) or (only even numbers have even squares, referring to a).
6. Because a is even, there exists an integer k that fulfills: a = 2k.
7. Substituting 2k from (6) for a in the second equation of (3): 2b2 = (2k)2 is equivalent to 2b2 = 4k2 is equivalent to b2 = 2k2.
8. Because 2k2 is divisible by two and therefore even, and because 2k2 = b2, it follows that b2 is also even which means that b is even.
9. By (5) and (8) a and b are both even, which contradicts that a / b is irreducible as stated in (2).

Q.E.D.