Prove that if x2 is irrational then x is irrational using co

Solution

A proof by contraposition: If x is rational, then x = p/q for some integers p and q with q ?= 0. Then x^2 = p^2/q^2 , and we have expressed x^2 as the quotient of two integers, the second of which is not zero. This by definition means that x^2 is rational, and that completes the proof of the contrapositive of the original statement.