Question

Prove that `sqrt(2) + sqrt(3)` is irrational.

Answer 1

Given: Let `sqrt(2)` and `sqrt(3)` denote the positive square roots of 2 and 3 known to be irrational.

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

Proof [Step-wise]:

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

2. Then there exist co-prime integers a, b (b ≠ 0) such that `sqrt(2) + sqrt(3) = a/b`.

3. Rearranging gives `sqrt(3) = a/b - sqrt(2)`.

4. Square both sides: `3 = (a/b - sqrt(2))^2`

= `a^2/b^2 - ((2a)/b)sqrt(2) + 2`

5. Move terms to isolate `sqrt(2)`: `((2a)/b)sqrt(2) = a^2/b^2 - 1`.

6. Hence `sqrt(2) = (a^2 - b^2)/(2ab)`. The right-hand side is a rational number (ratio of integers).

7. This shows `sqrt(2)` is rational, contradicting the known fact that `sqrt(2)` is irrational.

Therefore `sqrt(2) + sqrt(3)` is irrational.