Question

Prove that `2/sqrt(7)` is irrational, given that `sqrt(7)` is irrational.

Answer 1

Given: `sqrt(7)` is irrational.

To Prove: `2/sqrt(7)` is irrational.

Proof [Step-wise]:

1. Write `2/sqrt(7)` in a form that factors out `sqrt(7)`:

`2/sqrt(7) = 2/sqrt(7) xx sqrt(7)/sqrt(7)` 

= `2/7 xx sqrt(7)`

2. Note that `2/7` is a nonzero rational number.

3. Suppose, for contradiction, that `2/sqrt(7)` is rational.

Then `2/7 xx sqrt(7)` is rational.

4. If a nonzero rational `2/7` times a number is rational, then that number must be rational.

Multiply the rational result by the rational `7/2` to recover `sqrt(7)`. 

Hence, `sqrt(7)` would be rational.

5. This contradicts the given that `sqrt(7)` is irrational. 

Therefore, the assumption in step 3 is false.