Prove that the following numbers are irrational numbers:

Let us consider that 3/2√5 is a rational number.

Let 3/2√5 = a/b for b ≠ 0 ……………… (i)

Where a and b are co-prime integer numbers.

From (i) we can write as follows:

3b/2a = √5

Now from the LHS we know that a and b are integral numbers and therefore the fraction will also be rational number which implies that √5 has to be a rational number. But we know that √5 is an irrational number.

So our assumption at the beginning of the problem is proved false.

So 3/2√5 is an irrational number.