Let’s prove that √2 is irrational. The easiest way to do this is through proof by contradiction. So let’s assume that √2 is rational. By the definition of a rational number, this means that √2 can be written as some fraction a/b.

√2 = a/b

2 = (a/b)²

a² = 2b² such that a,b ϵ Z where a and b have no common factors.

We also know that the square of an even number is even, and the square of an odd number is odd. This means a can be written as a = 2c such that c ϵ Z. Now we can substitute a = 2c into the last equation.

(2c)² = 2b²

4c² = 2b²

b² = 2c²

This shows that b is also an even number, so b can be written as b = 2d such that d ϵ Z.

At the beginning, we assumed that a and b have no common factors, yet we have proven that if √2 is rational, then a and b would be even, and thus, have a common factor of 2. This contradict our original statement, and shows that √2 is rational is FALSE.