Question

Prove that
$2 + \sqrt{3}$
is irrational.​

Answer 1

GIVEN:

• 2 + √3

TO FIND:

• Prove that 2 + √3 is Irrational.

SOLUTION:

Let us assume, to the contrary, that 2 + √3 is a rational number, it can be written in the form of p/q where, q ≠ 0

p and q are coprimes

$\rm{\dashrightarrow 2 + \sqrt{3} = \dfrac{p}{q} }$

$\rm{\dashrightarrow \sqrt{3} = \dfrac{p}{q} -2 }$

$\rm{\dashrightarrow \sqrt{3} = \dfrac{ p - 2q}{q}}$

Since, p and q are integers

$\bold{ \dfrac{p - 2q}{q} \: is \: a \: rational \: number. }$

$\bf{\therefore \sqrt{3} \: is \: a \: rational \: number }$

But √3 is an irrational number.

This shows our assumption is incorrect.

So, we conclude that 2 + √3 is not a rational number.

Hence, 2 + √3 is an irrational number.

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