Question

Given that `sqrt(2)` is an irrational number, prove that `5 - 2sqrt(2)` is also an irrational number.

Answer 1

Given: `sqrt(2)`

To prove: `5 - 2sqrt(2)`

Proof: 

1. Assume the opposite: Assume that `5 - 2sqrt(2)` is a rational number.

2. Definition of rationality: If  `5 - 2sqrt(2)` is rational, it can be written as `5 - 2sqrt(2) = p/q`, where p and q are integers and q ≠ 0.

3. Rearrange the equation:

`5 - p/q = 2sqrt(2)`

`(5q - p)/q = 2sqrt(2)`

`(5q - p)/(2q) = sqrt(2)`

4. Analyze the components: Since p and q are integers, 5q – p is an integer and 2q is an integer. Therefore, the left-hand side, `(5q - p)/(2q)`, must be a rational number.

5. Identify the contradiction: The equation implies that a rational number equals `sqrt(2)`. However, it is given that `sqrt(2)` is an irrational number.

6. Conclusion: The assumption that `5 - 2sqrt(2)` is rational leads to a contradiction. Therefore, `5 - 2sqrt(2)` must be an irrational number.