Question

Prove that `sqrt(11)` is an irrational number.

Answer 1

Given: The number `sqrt(11)`.

To Prove: `sqrt(11)` is an irrational number.

Proof:

1. Assume the contrary:

Suppose `sqrt(11)` is rational.

Then it can be expressed as a fraction of two integers a and b with no common factors other than 1, i.e., `sqrt(11) = a/b, a, b ∈ ℤ, b ≠ 0, gcd (a, b) = 1`.

2. Square both sides:

`11 = a^2/b^2`

⇒ a² = 11b²

3. Divisibility by 11:

Since a² = 11b², a² is divisible by 11.

By prime factorisation properties, if 11 divides a², it must divide a.

4. Let a = 11k for some integer k:

Substitute back into the equation:

(11k)² = 11b²

⇒ 121k² = 11b²

⇒ 11k² = b²

5. Similarly, b² is divisible by 11:

Hence 11 divides b.

6. Contradiction:

Both a and b are divisible by 11, which contradicts the initial assumption that gcd (a, b) = 1.

Our initial assumption is wrong.

Therefore, `sqrt(11)` is irrational.