Prove that √5 is an irrational number

Q) Prove that √5 is an irrational number.

Q 29 – Question Paper – Set 2 – Maths Standard – CBSE 2026

Ans:

Step 1: Let us assume that √5 is a rational number

Let √5 = $\frac{p}{q}$ ; where q ≠ 0 and let p, q are co-primes.

By squaring on both sides, we get:

(√5)² = $(\frac{p}{q})^2$

∴ 5 = $\frac{p^2}{q^2}$

∴ 5 q² = p²………………. (i)

Step 2: From equation (i), it is clear that p² is divisible by 5

∴ p is divisible by 5

∴ we can write that p = 5 a, where a is an integer……. (ii)

Step 3: Substituting this value of p in equation (i), we get:

5 q² = (5 a)²

5 q² = 25 a²

q² = 5 a²

It means that q² is divisible by 5

∴ q is divisible by 5

∴ we can write that q = 5 b, where b is an integer…… (iii)

Step 4:

Now, from equations (ii) and (iii), we conclude that p and q both share a common factor of 5,

∴ p and q are not co-primes.

This contradicts our assumption.

Therefore, √5 is an irrational number………… Hence Proved !

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