State whether the following statement is true or not:7

Foundation

Mathematics Foundation

Properties of Real Number

Irrational Number

Rational Numbers

Question

State whether the following statement is true or not:

7 - \sqrt{2}

is irrational.

True

False

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Let us suppose

7 - \sqrt{2}

is rational.

Therefore,

=&gt;7-\sqrt 2

is in the form of

\frac{p}{q}

where

p

and

q

are integers and

q \neq 0

=&gt;\sqrt2=-\dfrac pq+7

=&gt;\sqrt2=\dfrac{-p+7q}{q}

p, q

and

7

are integers therefore,

\frac{- p + 7 q}{q}

is a rational number.

Hence,

\sqrt{2}

is a rational number.

But we know that

\sqrt{2}

is an irrational number.

So this is a contradiction.

This contradiction has arisen because of our wrong assumption that

7 - \sqrt{2}

is a rational number.

Hence,

7 - \sqrt{2}

is an irrational number.