Question

show that 5-√3 is irrational
​

Answer 1

$\large{\underline{\underline{\red{\sf{\hookrightarrow Given:- }}}}}$

• A Irrational Number is given to us.
• The Number is 5 - √3 .

$\large{\underline{\underline{\red{\sf{\hookrightarrow To\:Prove:-}}}}}$

• It is a Irrational Number.

$\large{\underline{\underline{\red{\sf{ \hookrightarrow Proof:-}}}}}$

On the contrary let us assume that is Rational number , so it can be expressed in the form of p/q where p and q are integers and q is not equal to 0. Also HCF of p and q is 1.

So , as per our assumption ,

$\tt:\implies 5 - \sqrt{3}=\dfrac{p}{q}$

$\tt:\implies 5 - \dfrac{p}{q}=\sqrt3$

$\underline{\boxed{\red{\tt \longmapsto \:\:\sqrt3\:\:=\:\:\dfrac{5q - p}{q}}}}$

Now we arrived at a contradiction , √5 is a Irrational but 5q - p / q is Rational [ As per assumption ] .

But Rational ≠ Irrational .

Hence our assumption was wrong , 5 - √3 is a Irrational Number.

$\blue{\boxed{\red{\bullet \bf Hence\:\:\:\:Proved}}}$