Prove that (2-√3)/5 is an

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

Ans:

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.

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