Question

Answer correctly, Class 10!

Answer 1

Step-by-step explanation:

Given: √5

We need to prove that √5 is irrational

Proof:

Let us assume that √5 is a rational number.

So it can be expressed in the form p/q where p,q are co-prime integers and q≠0

⟹ √5 = p/q

On squaring both the sides we get,

⟹ 5 = p²/q²

⟹ 5q² = p² —————–(i)

p²/5 = q²

So 5 divides p

p is a multiple of 5

⟹ p = 5m

⟹ p² = 25m² ————-(ii)

From equations (i) and (ii), we get,

5q² = 25m²

⟹ q² = 5m²

⟹ q² is a multiple of 5

⟹ q is a multiple of 5

Hence, p,q have a common factor 5. This contradicts our assumption that they are co-primes. Therefore, p/q is not a rational number

√5 is an irrational number.

Hence proved

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