Question

Given that `sqrt(2)` is irrational, prove that `(5 + 3sqrt(2))` is an irrational number.

Answer 1

Let us assume, to the contrary that `(5 + 3sqrt2)` is rational.

That is, we can find coprime a and b `(b != 0)` such that `5 + 3sqrt2 = a/b`

Therefore, `5 - a/b = 3 sqrt2`

`=> 5/3 - a/(3b) = sqrt2`

`=> (5b -a)/(3b) = sqrt2`

Since, a and b are integers, we get `(5b - a)/(3b)` is rational and so `sqrt2` is rational.

Which contradicts the fact that `sqrt2` is irrational.

So our assumption was wrong that `(5 + 3sqrt2)` is rational.

Hence we conclude that `5 + 3sqrt2` is irrational.