Question

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

Answer 1

Given: `sqrt(2)` is an irrational number.

To Prove: `5sqrt(2)` is an irrational number.

Proof [Step-wise]:

1. Assume, for contradiction, that `5sqrt(2)` is rational.

2. Then there exist integers a and b (b ≠ 0) with gcd(a, b) = 1 such that `5sqrt(2) = a/b`.

3. Divide both sides by 5 to get `sqrt(2) = a/(5b)`. The right-hand side `a/(5b)` is a rational number (ratio of integers).

4. This implies `sqrt(2)` is rational, which contradicts the given that `sqrt(2)` is irrational.

5. Therefore, the assumption in step 1 is false, so `5sqrt(2)` is not rational (i.e., it is irrational).

Hence, given `sqrt(2)` is irrational, `5sqrt(2)` is also irrational.