Question

Given that √3 is irrational number. Prove that
5-√3 is irrational number.
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Answer 1

ANSWER -:

5 - √3 is irrational.

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EXPLANATION-:

Let us assume the given number be

rational and we will write the given

number in p/q form

⇒ 5 - √3 = p/ q

⇒ √3 = 5q - p / q

We observe that LHS is irrational and

RHS is rational,

which is not possible.

This is contradiction.

Hence the given number is irrational-:

= 5 - √3 is irrational.

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Answer 2

Given :

• √3 is an irrational number.

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To prove :

• 5 - √3 is irrational number.

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Proof :

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⠀⠀⠀⠀☯ Let's assume that 5 - √3 is a rational number. So,It can be written in form of p/q where p and q are two co - prime numbers.

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$:\implies\sf 5 - \sqrt{3} = \dfrac{p}{q}$

$:\implies\sf \sqrt{3} = 5 - \dfrac{p}{q}$

$:\implies\sf \sqrt{3} = \dfrac{5q - p}{q}$

Here,

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• The LHS i.e. √3 is irrational. (Given)

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We observe that LHS is irrational and RHS is rational, which is not possible.

This is contradiction.

Hence, our assumption that given number is rational is false.

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$\therefore\;{\underline{\sf{ \bf{5 - \sqrt{3}}\;is\; irrational.}}}$

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$\qquad\quad\boxed{\underline{\underline{\purple{\bigstar \: \bf\:More\:to\:know\:\bigstar}}}}$

• Rational numbers are numbers that can be expressed as a fraction or part of a whole number.

• Irrational numbers are numbers that cannot be expressed as a fraction or ratio of two integers.