Question details: Give me proof that (square root of 5) is not rational Solution If v5 is rational, then it can be expressed by some number a/b (in lowest terms). This would mean: (a/b)² = 5. Squaring, a² / b² = 5. Multiplying by b², a² = 5b². If a and b are in lowest terms (as supposed), their squares would each have an even number of prime factors. 5b² has one more prime factor than b², meaning it would have an odd number of prime factors. Every composite has a unique prime factorization and can\'t have both an even and odd number of prime factors. This contradiction forces the supposition wrong, so v5 cannot be rational. It is, therefore, irrational..