Q27 of 34 Page 1

Prove that √3 is an irrational number.

Let √3 be a rational number.


Then √3 = p/q HCF (p,q) = 1


Squaring both sides


(√3)2 = (p/q)2


3 = p2/ q2


3q2 = p2


3 divides p2 � 3 divides p


3 is a factor of p


Take p = 3C


3q2 = (3c)2


3q2 = 9C2


3 divides q2 � 3 divides q


3 is a factor of q


Therefore 3 is a common factor of p and q


It is a contradiction to our assumption that p/q is rational.


Hence √3 is an irrational number.


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