Question

Prove that `4 - 3sqrt(5)` is an irrational number.

Answer 1

Given: We are given the number `4 - 3sqrt(5)`.

To Prove: Prove that `4 - 3sqrt(5)` is an irrational number.

Proof [Step wise]:

1. Recall that `sqrt(5)` is an irrational number can be proven by contradiction assuming `sqrt(5) = a/b` where a, b are integers with no common factors, leading to contradiction.

2. Assume, for the sake of contradiction, that `4 - 3sqrt(5)` is rational.

Let `4 - 3sqrt(5) = r` where r is rational.

3. Manipulate the equation to isolate `sqrt(5)`:

`4 - r = 3sqrt(5)`

⇒ `sqrt(5) = (4 - r)/3`

4. Since r is rational and 4 and 3 are rational numbers, the right side `(4 - r)/3` is rational.

5. This implies `sqrt(5)` is rational, which contradicts the fact that `sqrt(5)` is irrational.

6. Hence, the assumption that `(4 - 3sqrt(5))` is rational must be false.

Therefore, `4 - 3sqrt(5)` is irrational.