Prove root 5 is an irrational number


We need to prove that√5 is irrational


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

On squaring both the sides we get,
⇒5q²=p² —————–(i)


So 5 divides p
p is a multiple of 5
⇒p²=25m² ————-(ii)
From equations (i) and (ii), we get,
⇒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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