Question

2+√3 /5 is an irrational number prove it

Answer 1

Answer:

$\sf{\frac{2 + \sqrt{3}}{5}}$ is Irrational.

Step-by-step explanation:

Let us assume that  $\sf{\frac{2 + \sqrt{3}}{5}}$  is a rational number.

⇒ $\sf{\frac{2 + \sqrt{3}}{5}}$ = $\sf{\frac{p}{q}}$

⇒ $\sf{2 +\sqrt{3}}$ =  $\sf{\frac{5p}{q}}$

⇒ $\sf{\sqrt{3}}$ =  $\sf{\frac{5p}{q}}$  - 2

⇒ $\sf{\sqrt{3}}$ = $\sf{\frac{5p - 2q}{q}}$

In the RHS, '5p', '2q', 'q' are all rational numbers. This makes the whole of the RHS as a Rational Number.

But we know that $\sf{\sqrt{3}}$ is a Irrational number.

But, Irrational ≠ Rational

This result is due to our wrong assumption that $\sf{\frac{2 + \sqrt{3}}{5}}$ is a rational number.

∴  $\sf{\frac{2 + \sqrt{3}}{5}}$  is a rational number.

NOTE: This is a 2 mark solution, for the 3-4 mark solution, you have to prove that $\sf{\sqrt{3}}$ is a irrational number.