Question

prove that 2+√
3 is an irrational number


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Answer 1

Step-by-step explanation:

assume 2+√3 as irrastional

•2+√3=a/b

2-a/b=√3

√3=a/b-2

√3=a-2b/b

a and b are positive integer

a-2b/b is rational

√3 is rational

but we know that √3 is irrational

hence, 2+√3 is irrational

Hence proved,

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Answer 2

$\huge\mathfrak{\underline{\underline{\red {Answer}}}}$

$\sf \: let \: us \: assume \: 2 + \sqrt{3} \: as \: rational$

$\sf2 + \sqrt{3} = \frac{a}{b}$

$\sf2 - \frac{a}{b} = - \sqrt{3} \: or \: \sqrt{3} = \frac{a}{b} - 2$

$\sf \sqrt{3} = \frac{a}{b} - 2$

$\sf \sqrt{3} = a - \frac{2b}{b}$

$\sf \: a \: and \: b \: are \: positive \: integers$

$\sf \: a - \frac{2b}{b} \: is \: rational$

$\sf \: \sqrt{3} \: is \: rational$

$\sf \: but \: we \: know \: that \: \sqrt{3} \: is \: irrational$

$\sf2 + \sqrt{3} \: is \: irrational$