Question

prove that ✓3 is irrational​

Answer 1

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The ans...is in the attachment...

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Answer 2

let us assume that root 3 is a rational number so we can write root 3 in the form of p by q where A and B are coprime and no other factor other than one .

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

by squaring both the side ,

$( \sqrt{3} )^{2} = ( \frac{a}{b} ) ^{2}$

$3 = \frac{ {a}^{2} }{ {b}^{2} }$

$3 {b}^{2} = {a}^{2}$

$\frac{ {a}^{2} }{3} = {b}^{2}$

Hence is divided by a square .

so3 divided with a also ---eq (1)

so,

$\frac{a}{3} = c$

where c is a integer .

a=3c

3b^2 =(3c)^2

3b^2 =9c^2

b^2 = 3c^2

b^2/3 = c^2

Hence 3 is also divided with be so 3 with the also .

3 divided both A and B so a and b is a factor of 3 therefore A and B are not coprime our assuption is wrong by contradiction root 3 is irrational number .