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# Prove that  is an irrational number.

Asked by Topperlearning User 25th July 2017, 4:54 PM

Suppose  is not an irrational number, then it must be a rational number.

So let =q where q is a rational number.

)=

=

=

So here we have an equation with its   left side is a rational number(as the set of rational numbers is closed under subtraction and division i.e the subtraction and division of two rational numbers is always a rational number)and  so its right side must also be a rational number.

is a rational number

Which is a   contradiction. So our basic assumption   must be wrong.

=q where q is a rational number,is wrong

is an irrational number.

Hence proved.

Answered by Expert 25th July 2017, 6:54 PM
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