Question

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

Answer 1

Given: `sqrt(3)` is an irrational number.

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

Proof [Step-wise]:

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

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

3. Rearranging gives `sqrt(3) = a/b - 2 = (a - 2b)/b`.

4. The right-hand side `(a - 2b)/b` is a ratio of integers, hence rational.

5. Therefore, `sqrt(3)` would be rational, contradicting the given that `sqrt(3)` is irrational.

The assumption that `(2 + sqrt(3))` is rational leads to a contradiction; hence `(2 + sqrt(3))` is irrational.