Question

Prove that `5 sqrt2 - 3` is an irrational number, given that `sqrt2` is an irrational number.

Answer 1

Let us assume that `5 sqrt2 - 3` is an irrational number.

Then we can find integers a and b(b ≠ 0) such that

`5 sqrt2 - 3 = a/b`

`5 sqrt2 = a/b + 3`

`5 sqrt2 = (a + 3b)/b`

`sqrt2 = (a + 3b)/(5b)`

This is a contradiction because the right-hand side is a rational number while given that `sqrt2` is an irrational. So, our assumption that `5 sqrt2 - 3` is rational is wrong. Hence, `5 sqrt2 - 3` is an irrational number.