Question

Prove that `sqrt(5)` is an irrational number. 

Answer 1

Let us assume, to the contrary, that `sqrt5` is irrational.

That is, we can find integers a and b (≠0) such that `sqrt5 = a/b`.

Suppose a and b have a common factor other than 1, then we can divide by the common factor and assume that a and b are coprime.

Therefore, b`sqrt5` = a

Squaring on both sides, 5b² = a²  ...(1)

The above implies that a² is divisible by 5 and also a is divisible by 5.

Therefore, we can write that a = 5k for some integer k.

Substituting in (1)

5b² = (5k)²

5b² = 25k²

b² = 5k²

⇒ b² is divisible by 5, which means b is also divisible by 5.

Therefore, a and b have 5 as a common factor.

This contradicts the fact that a and b are coprime.

We arrived at the contradictory statement as above since our assumption `sqrt5` is not correct.

Hence, we can conclude that `sqrt5` is an irrational number.

Answer 2

If possible, let `sqrt5` be rational and let its simplest form be `p/q`.

Then p and q are integers having no common factor other than 1, and q ≠ 0.

Now `sqrt5 = p/q`

⇒ 5 = `p^2/q^2`    ...[On squaring both sides]

⇒ 5q² = p²  ...(i)

⇒ 5 divides p² ⇒ 5 divides p  ...[∵ 5 is prime and 5 divides p² ⇒ 5 divides p]

Let p = 5R for some integer R.

Putting p = 5R in (i), we get

5q² = 25R²

⇒ q² = 5R²

⇒ 5 divides q²  ...[∵ 5 divides 5R²]

⇒ 5 divides q    ...[∵ 5 divides q]

Thus, p and q share a common factor of 5. However, this contradicts the fact that p and q share no common component other than 1.

Hence, `sqrt5` is irrational.

Answer 3

Let `sqrt(5)` be a rational number.

∴ `sqrt(5) = p/q`,

where p and q are co-prime integers and q ≠ 0

On squaring both the sides, we get

`5 = p^2/q^2`

⇒ p² = 5q²

∵ p² is divisible by 5

∴ p is divisible by 5

Let p = 5r for some positive integer r,

p² = 25r²

∴ 5q² = 25r²

⇒ q² = 5r²

∵ q² is divisible by 5

∴ q is divisible by 5.

Here, p and q are divisible by 5, which contradicts the fact that p and q are coprimes.

Hence, our assumption is false.

∴ `sqrt(5)` is an irrational number.