Question

$5 - \sqrt{3}$
show that 5-✓3 is a rational number​

Answer 1

$\huge {\tt{ǫᴜᴇsᴛɪᴏɴ:}}$

$\sf{5 - \sqrt{3}}$

$\sf{\huge {\tt{ᴀɴsᴡᴇʀ:}}}$

Let $\sf{5 - \sqrt{3}}$ be a rational number .

$\sf{\implies{\sf{5 - \sqrt{3}}}=\dfrac{p}{q} ~~(p~and~q=~co~primes~and~q≠0)} \\\\ \sf{\implies 5-{\dfrac{p}{q} ={\sqrt{3}}}} \\\\ \sf{\implies{\dfrac{5q-p}{q}={\sqrt{3}}}}$

LHS =$\sf{\dfrac{5q-p}{q}}$ is a rational number as 5q - p is an integer & q is also an integer .

RHS = $\sf{\sqrt{3}}$ is irrational .

Rational ≠ irrational

This contradiction came due to our wrong assumption that $\sf{5 - \sqrt{3}}$ is rational .

Hence , $\sf{5 - \sqrt{3}}$ is irrational .