Question

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

Answer 1

Let us assume `5 + 2sqrt(3)` is rational, then it must be in the form of `p/q` where p and q are co-prime integers and q ≠ 0

i.e. `5 + 2sqrt(3) = p/q`

So `sqrt(3) = (p - 5q)/(2q)` ......(i)

Since p, q, 5 and 2 are integers and q ≠ 0, RHS of equation (i) is rational. But LHS of (i) is `sqrt(3)` which is irrational. This is not possible.

This contradiction has arisen due to our wrong assumption that `5 + 2sqrt(3)` is rational.

So, `5 + 2sqrt(3)` is irrational.