Question

prove that 2 +√3 is an irrational number with indirect method
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Answer 1

Answer:

${\tt{\red{\underline {\underline {\huge{Question}}}}}}$:-

To prove that 2+√3 is irrational

Step-by-step explanation:

$\huge\mathfrak{\color{orange}{\underline{\underline{soloution♡}}}}$

let us assume that 2+√3 is a rational

number and it's simplest form is

$\frac{p}{q}$

${\small{\mathcal{\pink{Then,}}}}$ p and q are integers having no common factor other than 1 and q≠0.

$\huge{\underline {\sf{\blue{Therefore,}}}}$

$\frac{p}{q} = 2 + \sqrt{3}$

$\frac{p}{q} - 2 = \sqrt{3}$

$\frac{p - 2q}{q} = \sqrt{3}$

Since, p and q are integers and q ≠0.

$so \: \frac{p - 2q}{q} is \: a \: rational \: number$

Therefore, √3 is also a rational number.

But, this contradicts the fact that √3 is a irrational number.

The contradiction arises by assuming that 2+√3 is rational.

Thus, our assumption is incorrect.

Hence, 2+√3 is irrational.

${\tt{\black {\underline {\underline {\huge{proved♥}}}}}}$

Answer 2

Answer:

2+root3 is a rational number

Step-by-step explanation:

p/q = 2+root 3

p/q -2 = root3

p-2q/q =root3

since p and q are integers and q equal not zero

so p-2q/q is a rational number