Question

Prove that the following number is irrational.

`sqrt(2) + sqrt(5)`

Answer 1

To prove:

`sqrt(2) + sqrt(5)` is irrational

Proof by Contradiction:

Let `sqrt(2) + sqrt(5)` be rational.

Then:

⇒ `sqrt(2) + sqrt(5) = r` (where r is rational)

Now subtract `sqrt(2)` from both sides:

⇒ `sqrt(5) = r - sqrt(2)`

Here:

• r is rational
• `sqrt(2)` is irrational
• ⇒ `r - sqrt(2)` is irrational

So, `sqrt(5)` = irrational – which is true

But this means the original sum `sqrt(2) + sqrt(5)` cannot be rational, otherwise it leads to irrational = rational, a contradiction.