Question

Prove that `(2 + 3sqrt(5))` is an irrational number, given that `sqrt(5)` is an irrational number.

Answer 1

Given: `sqrt(5)` is an irrational number.

To Prove: `(2 + 3sqrt(5))` is an irrational number.

Proof [Step-wise]:

1. Assume, for contradiction, that `(2 + 3sqrt(5))` is rational.

2. Then there exist integers a and b (b ≠ 0) with `2 + 3sqrt(5) = a/b`.

3. Subtract 2 from both sides:

`3sqrt(5) = a/b - 2`

= `(a - 2b)/b`

4. Divide both sides by 3:

`sqrt(5) = (a - 2b)/(3b)`

5. The right-hand side `(a - 2b)/(3b)` is a rational number (ratio of integers), so this shows `sqrt(5)` is rational.

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

The assumption is false, therefore `(2 + 3sqrt(5))` is irrational.