Question

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

Answer 1

Given: `sqrt(2)` is irrational.

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

Proof [Step-wise]:

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

2. Then there exist integers p and q (q ≠ 0) with (p, q) = 1 such that `3 + 5sqrt(2) = p/q`.

3. Rearranging, `5sqrt(2) = p/q - 3 = (p - 3q)/q`, so `sqrt(2) = (p - 3q)/(5q)`.

4. The right-hand side `(p - 3q)/(5q)` is a rational number (ratio of integers), so this expresses `sqrt(2)` as a rational number.

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

The assumption is false; therefore `3 + 5sqrt(2)` is irrational.