Question

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

Answer 1

Given: `sqrt(3)` is irrational.

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

Proof [Step-wise]:

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

2. Then there exist integers a, b (b ≠ 0) such that `2 + 5sqrt(3) = a/b`, so `5sqrt(3) = a/b - 2`.

3. Hence `sqrt(3) = (1/5)(a/b - 2) = (a/(5b)) - (2/5)`.

4. The right-hand side `(a/(5b)) − (2/5)` is a difference of rational numbers, therefore rational.

5. So `sqrt(3)` would be rational, which contradicts the given that `sqrt(3)` is irrational.

6. Therefore the assumption in step 1 is false.