#### 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 = p^{2} / q^{2}

q^{2} = 11p^{2} …. (1)

As we know, ‘11’ divides 11p^{2}, so ‘11’ divides q^{2} 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

q^{2} = 121k^{2}

11p^{2} = 121k^{2} [Since, q^{2} = 11p^{2}, from equation (1)]

p^{2} = 11k^{2}

As we know, ‘11’ divides 11k^{2}, so ‘11’ divides p^{2} 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.

**More Solutions:**

- Insert a rational number between and 2/9 and 3/8.
- Insert two rational numbers between 1/3 and 1/4.
- Insert two rational numbers between – 1/3 and – 1/2.
- Insert three rational numbers between 1/3 and 4/5.
- Insert three rational numbers between 4 and 4.5.
- Find six rational numbers between 3 and 4.
- Find five rational numbers between 3/5 and 4/5.
- Find six rational numbers between 1/2 and 2/3.
- Prove that, √5 is an irrational number.