Prove that the following numbers are irrational numbers:

us consider that 2 + √2 is a rational number.

Let 2 + √2 = a/b for b ≠ 0 ……………… (i)

Where a and b are co-prime integer numbers.

From (i) we can write as follows:

a/b – 2 = √2

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 √2 has to be a rational number. But we know that √2 is an irrational number.

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

So 2 + √2 is an irrational number.