This story may not be true, as the Pythagoreans had many enemies, who might have spread this rumor. Here's the proof. So, we have proved that the square root of two cannot be expressed as a fraction, i.e. it is irrational. In the third step, above, we are dealing with integers m and n. The fifth step is the same.
Closure is a fairly important principle in algebra.
The positive integers are closed under addition, for example. That means that a positive integer added to another positive integer gives us a positive integer. Closure allows us to deduce some simple properties of numbers. Integers are not closed under division. This complicates the situation, because it shows us that we have a whole new set of numbers, the irrationals.