Question

Prove that √3 + 2 is an irrational number.​

Answer 1

Answer:

we have to prove that √3+2 is an irrational number .

let's √3+2 is rational no.

we know that rational no. can be written in the form of p/q where q is not equal to 0 .

√3+2 =p/q

√3 = p/q -2

√3 = p-2q/2q

p-2q/2q - √3

hence we can say that √3+2 is an irrational no.

hence proved

Answer 2

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$\large\bf{To~prove:}$

$\bf{{\sqrt{3}}+2}$ is an irrational number

First let us assume that $\bf{{\sqrt{3}}+2}$ is a rational number.

Then we get :

$\bf{{\sqrt{3}}+2={\frac{p}{q}}}$, where p and q are integers and q ≠ 0

$\therefore\bf{{\sqrt{3}}={\frac{p}{q}}-2={\frac{p-2q}{q}}}$. p and q being integers, $\bf{{\frac{p-2q}{q}}}$ represents a rational number.

But in LHS, it's $\bf{{\sqrt{3}}}$ which is irrational.

Thus LHS ≠ RHS

$\implies$ our assumption is wrong. $\bf{{\sqrt{3}}+2}$ is not rational, it's an irrational number..

　　Hence proved..!!

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$\huge\bf{{\color{navy}{Hope}}~{\color{blue}{it}}~{\blue{helps..!!}}}$