Question

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

Answer 1

Given: `sqrt(3)` is irrational.

To Prove: `(4 - sqrt(3))` is irrational.

Proof [Step-wise]:

1. Suppose, for contradiction, that `(4 - sqrt(3))` is rational.

2. Then there exist integers a and b (b ≠ 0) with gcd(a, b) = 1 such that `4 - sqrt(3) = a/b`.

3. Rearranging gives `sqrt(3) = 4 - a/b = (4b - a)/b`, which is a rational number (quotient of integers).

4. This contradicts the given fact that `sqrt(3)` is irrational.

5. Therefore, the assumption in step 1 is false, so `(4 - sqrt(3))` is not rational.