Question

Prove that `((3 - 4sqrt(2)))/7` is an irrational number, given that `sqrt(2)` is an irrational number.

Answer 1

Given: `sqrt(2)` is irrational.

To Prove: `((3 - 4sqrt(2)))/7` is irrational.

Proof [Step-wise]:

1. Assume, for contradiction, that `((3 - 4sqrt(2)))/7` is rational.

2. Then there exist integers p and q (q ≠ 0) with `((3 - 4sqrt(2)))/7 = p/q`.

3. Multiply both sides by 7 to get `3 - 4sqrt(2) = (7p)/q`, which is rational (since `p/q` is rational).

4. Rearranging gives `-4sqrt(2) = (7p)/q - 3`, so `sqrt(2) = (3 - 7p)/(4q)`.

5. The right-hand side `(3 - 7p)/(4q)` is a rational number difference and quotient of rationals, so this implies `sqrt(2)` is rational.

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

The assumption leads to a contradiction, so `((3 - 4sqrt(2)))/7` must be irrational.