Question

how to prove theoretically that square root of 2 is irrational

Answer 1

google it and find one of the many many proofs that you
a) like
b) understand

Answer 2

they all start by supposing that
\[\sqrt{2}\] is rational, then get a contradiction

Answer 3

usually first step is " if it is rational then
\[\sqrt{2}=\frac{a}{b}\]"
second step is
\[2=\frac{a^2}{b^2}\] and third step is
\[2b^2=a^2\] and that last statement will lead to a contradiction.

Answer 4

Here, I helped you find an easy one online. Satellite is completely right over googling the one you'll either like or understand. They all are the same flavor though. http://www.homeschoolmath.net/teaching/proof_square_root_2_irrational.php

Answer 5

make sure you understand these steps first, because they pretty much all start out that way.

Answer 6

oreostar sent you the 'even/odd' version. easy enough to understand if you can keep track of the statements that say 'this is even so that is even" etc. you can also use the fundamental theorem of arithmetic to arrive at a contradiction.

not that you asked but it seems that all the methods use something called "infinite descent" the idea being that if you can find the answer then there is a smaller one, and this is a contradiction because you cannot keep pulling smaller and smaller POSITIVE INTEGERS out of a hat forever

Answer 7

There's one here olus an interesting bit of history if you like that sort of thing.

http://en.wikipedia.org/wiki/Hippasus

Answer 8

here is a nice proof that used infinite descent directly, without reference to "even/odd"
http://www.math.hmc.edu/funfacts/ffiles/30005.5.shtml