Question

prove that 5 minus root 3 is an irrational number​

Answer 1

Given:-

$5 - \sqrt{3}$

To prove:-

It is a irrational number.

Concept Used:-

[a,b are integers]

$\frac{5b - a}{b} is \: a \: rationl \: number$

Solution:-

Let us assume on the contrary that 5 - √ 3 is rational. Then, there exist coprime positive integers a and b such that

$5 - \sqrt{3 } = \frac{a}{b} \\ ⇒5 - \frac{a}{b} = \sqrt{3} \\ ⇒ \frac{5b - a}{b} = \sqrt{3} \\ ⇒ \sqrt{3} is \: rationl$

Answer:-

This contradicts the fact that √ 3 is irrational. So, our assumption is incorrect . Hence, 5 - √ 3 is an irrational number.

Answer 2

Answer:

let 5-

$let \: 5 - \sqrt{3 \: } \: is \: a \: rational \: no \\ so \\ \frac{a}{b \: } \: = 5 - \sqrt{3} \: (where \: a \: \: and \: b \: are \: co - prime \: no \: and \\ b \: is \: not \: equls \: to \: 0) \\ \frac{a}{b } - 5 = - \sqrt{3 } \\ \frac{a}{b } - 5 \: is \: a \: rational \: no \: but \sqrt{3} \: is \: an \: irrational \: no \\ \: our \: supposition \: is \: wrong \: so \: 5 - \sqrt{3} is \: an \: irrational \: no \\ i \: hope \: it \: will \: help \: you$