Prove that is an irrational number.

MATHEMATICS · Multi-part · CBSE · Mathematics · Real Numbers · Real Numbers

Problem

Prove that

\sqrt{6}

is an irrational number. (i) Suppose that

\sqrt{6}=\dfrac{p}{q}

where

p

and

q

are positive integers with no common factor. Show that

6q^{2}=p^{2}

. (ii) Examine the parity of

p

and

q

obtained in part (i). Explain why this leads to a contradiction and hence establish that

\sqrt{6}

is irrational.