Prove that root 2 or √2 …

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Prove that root 2 or √2 is irrational no.

Posted by Uttpan Patnaik 5 years, 4 months ago

CBSE > Class 09 > Mathematics

1 answers

Sia ? 5 years, 4 months ago

Suppose {tex}\sqrt2{/tex} is a rational number. That is , {tex}\sqrt2{/tex} = {tex}\frac{p}{q}{/tex} for some p{tex}\in{/tex}Z and q {tex}\in{/tex}Z. We can assume the fraction is in lowest fraction, That is p and q shares no common factors.

Then {tex}\sqrt2q=p{/tex}

Squaring both side we get,

{tex}2q^2=p^2{/tex}

So {tex}p^2{/tex} is a multiple of 2,

let's assume {tex}p=2m{/tex}

Then, {tex}2q^2=\left(2m\right)^2{/tex}

{tex}2q^2=4m^2{/tex}
Or {tex}q^2=2m^2{/tex}
So {tex}q^2{/tex} is a multiple of 2,
{tex}\therefore{/tex} q is multiple of 2
Thus p and q shares a common factor.This is contradiction.
{tex}\Rightarrow {/tex}{tex}\sqrt { 2 }{/tex} is an irrational number.

1Thank You

ANSWER

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CBSE > Class 09 > Mathematics