Prove that (2-√3)/5 is an irrational number. It is given that √3 is an irrational number.

Question

Q) Prove that $\frac{(2 - \sqrt 3)}{5}$ is an irrational number. It is given that √3 is an irrational number.

Sapling Academy · Accepted Answer

STEP BY STEP SOLUTION
Let's start by considering $\frac{(2 - \sqrt 3)}{5}$ is a rational number.

∴ $\frac{(2 - \sqrt 3)}{5}$  = $\frac{p}{q}$  (here p and q are integers and q ≠  0)

∴ (2 - √3) = 5 $\frac{p}{q}$

∴ √3 = 2 - 5 $\frac{p}{q}$ ..... (i)

Since p and q are integers, so, $\frac{p}{q}$ is a rational number.

Since, in equation (i), LHS = RHS. Therefore, if RHS is a rational number, then LHS is also rational.

Therefore, √3 is a rational number.

But it contradicts the given condition (given that √3 is an irrational number).

Therefore, our assumption that "$\frac{(2 - \sqrt 3)}{5}$ is a rational number" is wrong.

Therefore, it is confirmed that $\frac{(2 - \sqrt 3)}{5}$ is an irrational number.

Please press the “Heart” button if you like the solution.

Contact us