MP Board · Class 10 · Mathematics · Real NumbersProve that √5 is an irrational number.

Let us assume to the contrary that √5 is a rational number. Then, we can find two coprime integers a and b (where b ≠ 0) such that √5 = a/b. Squaring both sides, we get 5 = a²/b², which implies a² = 5b². This means that 5 divides a², and therefore 5 divides a. We can write a = 5c for some integer c. Substituting this in a² = 5b², we get (5c)² = 5b², leading to 25c² = 5b², or b² = 5c². This shows 5 divides b², so 5 divides b. Thus, a and b have at least 5 as a common factor, which contradicts the fact that a and b are coprime. Hence, our assumption is false, and √5 is irrational.