Prove that 1/√11 is an irrational number.

Prove that 1/√11 is an irrational number.

Solution:

Let us consider 1/√11 be a rational number, then
1/√11 = p/q, where ‘p’ and ‘q’ are integers, q ≠ 0 and p, q have no common factors (except 1).
So,
1/11 = p² / q²
q² = 11p² …. (1)
As we know, ‘11’ divides 11p², so ‘11’ divides q² as well. Hence, ‘11’ is prime.
So 11 divides q
Now, let q = 11k, where ‘k’ is an integer
Square on both sides, we get
q² = 121k²
11p² = 121k² [Since, q² = 11p², from equation (1)]
p² = 11k²
As we know, ‘11’ divides 11k², so ‘11’ divides p² as well. But ‘11’ is prime.
So 11 divides p
Thus, p and q have a common factor 11. This statement contradicts that ‘p’ and ‘q’ has no common factors (except 1).
We can say that, 1/√11 is not a rational number.
1/√11 is an irrational number.
Hence proved.

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