Question

Show that the following numbers are irrational.

`3 - sqrt(5)`

Answer 1

Let us assume that `3-sqrt5` is rational. Then, there exist positive co-primes a and b such that

`3-sqrt3=a/b`

`sqrt5=3-a/b`

`sqrt5=(3b-a)/b`

Here we see that `sqrt5` is a rational number which is a contradiction as we know that `sqrt5` is an irrational number

Hence `3-sqrt5` is irrational.