Try to prove the irrationality of using the approach of proof by contradiction. Will the same approach work for , , or ?

Question

KnowledgeBoat · Accepted Answer

Proof of irrationality of by contradiction : Assume is rational. Then , where p and q are integers with q ≠ 0 and gcd(p, q) = 1. Squaring both sides : So, p2 is divisible by 3, which means p is also divisible by 3. Let p = 3k for some integer k. Then : So, q2 is divisible by 3, which means q is also divisible by 3. Both p and q being divisible by 3 contradicts gcd(p, q) = 1. Hence, is irrational. Yes, the same approach works for , and , since 5, 7 and 10 are not perfect squares. The proof relies on the fact that if a prime number p divides n2, then p divides n. This works for any non-perfect square integer. Hence, is irrational, and the same approach proves , and are also irrational.

Mathematics

Try to prove the irrationality of using the approach of proof by contradiction. Will the same approach work for , , or ?

Whole Numbers

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