Question

Prove that ( √2 +1)² is an irrational number given that √2 is an irrational number.​

Answer 1

Answer:

First of all, it should be noted that $(\sqrt{2} + 1)^{2}$ is a rational number.

As $(\sqrt{2} + 1)^{2}$  = 3.

So, we prove that $\sqrt{2} + 1$ is an irrational number.

Let us assume that $\sqrt{2} + 1$ is a rational number.

So, we find integers a and b such that $\frac{a}{b} =$ $\sqrt{2} + 1$

∴ $\sqrt{2} = \frac{a}{b} - 1$

Since, a, b, and 1 are integers, $\frac{a}{b} - 1$ is rational and thus, $\sqrt{2}$ is also rational.

But, this contradicts the fact that $\sqrt{2}$ is irrational.

This contradiction is due to the wrong assumption that $\sqrt{2} + 1$ is rational.

So, $\sqrt{2} + 1$ is irrational.