# Thread: Proving the irrationality of the sum of non perfect square roots?

1. ## Proving the irrationality of the sum of non perfect square roots?

I know how to prove that the $\displaystyle \sqrt{2}$ is irrational, and that $\displaystyle \sqrt{3}$ is irrational. But how does one go about using the tricks involved in the $\displaystyle \sqrt{2}$ proof on something like proving that $\displaystyle P$ is irrational if:

$\displaystyle P = \sqrt{2} + \sqrt{3}$

?

2. You can prove this by contradiction. Assume P is rational.

then try squaring the equation

$\displaystyle P = \sqrt2 + \sqrt 3$

Use the facts that a square of a rational is a rational, and rational number addition and scalar multiplication is closed under the rational field.

Fiddle around with the squared equation and you'll just have to prove $\displaystyle \sqrt 6$ is irrational using those tricks you mentioned, which leads to a contradiction (a rational number = an irrational number)

3. If I'm not missing something you have posted this before, and it has received a considerable attention in here.

4. My bad, yes I must've posted it before. I had thought it was different problem. I wasn't sure. Wont happen again

5. Oh, and thanks for the help Gusbob, I figured out the polynomial that represented the squared version and had no trouble proving the irrationality of 6^(1/2). That was a clever idea to use the square of the sum of the square roots. I hadn't thought of that before, I know I used it once along time ago, but I forgot about it. Thanks for the help.