# Thread: [SOLVED] real analysis test question

1. ## [SOLVED] real analysis test question

Alright so tomorrow is my first exam, and one of the proofs I need to memorize is for the square root of 2 being real. The proof we were shown in class would probably take me an hour to re-write. Any help on a simpler proof would be awesome.

The Proof that the square root of 2 is real.

2. Originally Posted by trojanlaxx223
Alright so tomorrow is my first exam, and one of the proofs I need to memorize is for the square root of 2 being real. The proof we were shown in class would probably take me an hour to re-write. Any help on a simpler proof would be awesome.

The Proof that the square root of 2 is real.
if you can prove that $\sqrt{2}$ is irrational, then you are done.. and there are lots of proofs.. the proofs depends on what level you are..

you can read several proofs here... Square root of 2 - Wikipedia, the free encyclopedia

3. once i prove that the square root of 2 is irrational what do I say? or can I just leave it at that?

4. well, being a rational or an irrational number is being a real.. so it should be enough.

5. Originally Posted by trojanlaxx223
Alright so tomorrow is my first exam, and one of the proofs I need to memorize is for the square root of 2 being real. The proof we were shown in class would probably take me an hour to re-write. Any help on a simpler proof would be awesome.

The Proof that the square root of 2 is real.
If you want to prove that this number is real, you can either show that it is rational or irrational, or alternatively, show that the number exists.

We all know the Pythagorean Identity for Right Angled Triangles (RATs)...

For any RAT of side length $a, b, c$ (with $c$ as the hypotenuse) $a^2 + b^2 = c^2$.

So think of the Right Angled ISOSCELES Triangle. $a = b = 1$.

All you'd have to do is find the length of the hypotenuse.

$a^2 + b^2 = c^2$

$1^2 + 1^2 = c^2$

$1 + 1 = c^2$

$2 = c^2$

$c = \sqrt{2}$.

If you show a picture of the triangle and use Pythagoras, you've shown without a doubt that $\sqrt{2}$ is a real number.