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

• September 30th 2010, 03:38 PM
mfetch22
Proving the irrationality of the sum of non perfect square roots?
I know how to prove that the $\sqrt{2}$ is irrational, and that $\sqrt{3}$ is irrational. But how does one go about using the tricks involved in the $\sqrt{2}$ proof on something like proving that $P$ is irrational if:

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

?

• September 30th 2010, 04:26 PM
Gusbob
You can prove this by contradiction. Assume P is rational.

then try squaring the equation

$P = \sqrt2 + \sqrt 3$

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 $\sqrt 6$ is irrational using those tricks you mentioned, which leads to a contradiction (a rational number = an irrational number)
• September 30th 2010, 09:45 PM
TheCoffeeMachine
If I'm not missing something you have posted this before, and it has received a considerable attention in here.
• October 1st 2010, 01:09 PM
mfetch22
My bad, yes I must've posted it before. I had thought it was different problem. I wasn't sure. Wont happen again
• October 5th 2010, 11:39 AM
mfetch22
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.