Question

Prove that `3/sqrt(5)` is irrational, given that `sqrt(5)` is irrational.

Answer 1

Given: `sqrt(5)` is irrational.

To Prove: `3/sqrt(5)` is irrational.

Proof [Step-wise]:

1. Assume, for contradiction, that `3/sqrt(5)` is rational.

2. Then there exist integers a and b (b ≠ 0) with gcd(a, b) = 1 such that `3/sqrt(5) = a/b`.

3. Rearranging gives `sqrt(5) = 3 xx b/a = (3b)/a`, which is a ratio of integers and therefore rational.

4. This contradicts the given fact that `sqrt(5)` is irrational.

Therefore, `3/sqrt(5)` is irrational.