Prove that 2 + √3 is an irrational number, given that √3 is an

Q) Prove that 2 + √3 is an irrational number, given that √3 is an irrational number.

Ans: Let us assume that 2 + √3 is a rational number

Let 2 + √3 = $\frac{p}{q}$ ; q ≠ 0 and p, q are integers

$\therefore$ √3 = $\frac{p - 2q}{q}$

Since, p and q are integers; Therefore p – 2q is an integer

Therefore, $\frac{p - 2q}{q}$ is a rational number

$\therefore$ √3 is a rational number

But it contradicts given condition that √3 is an irrational number

Therefore, 2 + √3 is an irrational number

Contact us