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Prove that (√5) is an irrational …

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Prove that (√5) is an irrational number

Posted by Sonam Sharma 6 years, 10 months ago

CBSE > Class 10 > Mathematics

2 answers

Khushboo Giri 6 years, 10 months ago

Let √5be a rational number say p/q where q is not equal to zero and p and q are coprime. √5=p/q Squaring both side 5=p²/q² 5q²=p²----------(1) 5q²is divisible by 5 p² is divisible by 5 p is divisible by 5 let p=5a putting the value in equation 1 5q²=p² 5q²=(5a)² 5q²=25a² q²=5a² 5a² is divisible by 5 q² is divisible by 5 q is divisible by 5 Therefore both p&q have common factor 5 We supposed p&q are coprime. Our supposition is wrong √5 is irrational.

Yogita Ingle 6 years, 10 months ago

let root 5 be rational
then it must in the form of p/q [q is not equal to 0][p and q are co-prime]
root 5=p/q
=> root 5 ×q = p
squaring on both sides
=> 5×q×q = p×p ------> 1
p×p is divisible by 5
p is divisible by 5
p = 5c [c is a positive integer] [squaring on both sides ]
p×p = 25c×c --------- > 2
sub p×p in 1
5×q×q = 25×c×c
q×q = 5×c×c
=> q is divisble by 5
thus q and p have a common factor 5
there is a contradiction
as our assumsion p &q are co prime but it has a common factor
so √5 is an irrational

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