Question

Prove that `(4sqrt(2) + 5/3)` is an irrational number given that `sqrt(2)` is an irrational number.

Answer 1

Given: `sqrt(2)` is irrational.

To Prove: `(4sqrt(2) + 5/3)` is irrational.

Proof [Step-wise]:

1. Suppose, for contradiction, that `4sqrt(2) + 5/3` is rational. Then there exist integers a, b (b ≠ 0) such that `4sqrt(2) + 5/3 = a/b`.

2. Subtract `5/3` from both sides: `4sqrt(2) = a/b - 5/3`.

3. Divide both sides by 4: `sqrt(2) = (1/4)(a/b - 5/3) = a/(4b) - 5/12`.

4. The right-hand side `a/(4b) - 5/12` is a difference of rational numbers, hence rational. Thus `sqrt(2)` would be rational.

5. This contradicts the given that `sqrt(2)` is irrational.

Therefore the assumption in step 1 is false.