Question

Prove that `1/sqrt(3)` is irrational, given that `sqrt(3)` is irrational.

Answer 1

Given: `sqrt(3)` is irrational.

To Prove: `1/sqrt(3)` is irrational.

Proof [Step-wise]:

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

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

3. Multiply both sides by `sqrt(3) : 1` 

= `(asqrt(3))/b`

So, `sqrt(3) = b/a`.

4. But `b/a` is rational, so this shows `sqrt(3)` is rational contradicting the given that `sqrt(3)` is irrational.

5. Therefore, the supposition that `1/sqrt(3)` is rational is false.

Hence, `1/sqrt(3)` is irrational.