The square root of 2 is not rational
Proof by contradiction
Let's assume that there exists a rational number q equal to √2. Therefore, by definition,
∃a∈N,b∈N∖{0},√2=q=abWe can also write it as an irreducible fraction, so that q=c/d, such that d does not divide c. From the properties of the square root, we have
q2=c2d2=2We can multiply both sides by b2 to obtain
c2=2d2