Answer to Question #292875 in Calculus for Micky

Question

Prove that √(1+√3) is irrational, assuming that √3 is irrational

Answer 1

Solution:

Given, $\sqrt3$ is irrational.

To prove that $\sqrt{1+\sqrt3}$ is irrational.

Proof:

Assume $\sqrt{1+\sqrt3}$ is rational, so that $\sqrt{1+\sqrt3}=\dfrac ab$, where a,b are co-primes and $b\ne0$ .

$\Rightarrow 1+\sqrt3=\dfrac {a^2}{b^2} \\ \Rightarrow \sqrt3=\dfrac {a^2}{b^2}+1$

On left side, $\sqrt3$ is irrational as it is given.

On right side, $\dfrac {a^2}{b^2}+1$ must be rational because $\dfrac ab$ is rational by assumption.

Now, an irrational cannot be equal to rational.

We get a contradiction.

So, our assumption is wrong.

Thus,$\sqrt{1+\sqrt3}$ is irrational.