Prove that is an irrational number.

Question

KnowledgeBoat · Accepted Answer

We will use the proof by contradiction. Assume is a rational number. Then it can be written as , where p and q are integers with q ≠ 0 and gcd(p, q) = 1 (lowest form). ⇒ Squaring both sides : So, p2 is divisible by 5, which means p is also divisible by 5. Let p = 5k for some integer k. Substituting : So, q2 is divisible by 5, which means q is also divisible by 5. But this means both p and q are divisible by 5, contradicting our assumption that gcd(p, q) = 1. Hence, our initial assumption is wrong. Hence, is an irrational number.

Mathematics

Prove that is an irrational number.

Whole Numbers

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