Proof that√5, is irrational number

1.	Proof that√5, is irrational number
Answer» 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/qOn squaring both the sides we get,⇒5=p²/q²⇒5q²=p² —————–(i)p²/5= q²So 5 divides pp 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 5Hence, 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

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