Prove that root 5 is irrational number

Given: √5

We need to prove that √5 is irrational

Proof:

Let us assume that √5 is a rational number.

Sp it t 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

5 Comments

  1. Thank you so much for your help

  2. Mohd shakib khan

    Thanks

  3. Thanks for helping

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