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Prove the root 5 is irrational

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Prove the root 5 is irrational

Posted by Sanjeev Singh 6 years ago

CBSE > Class 10 > Mathematics

2 answers

Sneha Tyagi 6 years ago

Let root 5 is a rational no. Root 5=p/q (where p and q are co prime) P=root5 q Squaring both side P^2=5q^2 (let eqn. I) 5 is a factor of q^2 5 is a factor of q also , P= 5c for some integer c Put the value of eqn. ( 1 ) 5p^2=(5c)^2 5p^2=25c^2 P^2=5c^2 5 is a factor of p^2 5 is a factor of p also, (Where p and q are integers ) Therefore , our assumption is wrong Root 5 is irrational no.

Yogita Ingle 6 years 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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