Consider a right triangle whose two short sides have a length of foot each. By the Pythagorean Theorem the long side has a length of feet. It turns out that is not a rational number .
This remarkable fact can be seen by a classic argument usually attributed to Eudoxus of Cnidus (approx. 406-355BC). We assume that is in fact rational and then derive a contradiction. Since there are no contradictions in mathematics our assumption must be false and so must be irrational.
So let us suppose that can be written as a ratio of two integers and :
We start with the equation . Squaring on both sides gives
Because of the factor on the left side, the right side, i.e., , must be even. For this to be true, itself must be even. If is even then is divisible by . Hence is also divisible by which means that must be even. That implies in turn that must be even. Thus and must be both even, which contradicts our (legitimate) assumption that and have no factor in common. Our (doubtful) assumption that is rational therefore can't be true -- the square root of is not a rational number.
The following facts regarding irrational numbers are beyond the scope of this class, but you can read about them in the book "What is Mathematics" by Courant Robbins, or I would be pleased to tell you more if you are interested: