(Q10).Prove that the following are irrational.

3 + 2√5

Let us assume 3 + 2√5 is rational.

So we can write this number as.

3 + 2√5 =
a
b
 ........1

Here, a and b are two co-prime numbers and b is not equal to .

Simplify the equation (1) cross multiplying on both sides, we get

2√5 =
a - 3b
b
√5 =
a - 3b
2b
Here, a and b are integers so,
a - 3b
2b
 is a number.

so √5 should be a number

But it contradicts the fact that √5 is a number.

Therefore, 3 + 2√5 is number.