An ancient school of mathematicians held one idea above all others: that every length in the world could be written as one whole number divided by another. A fraction. Two lengths, they believed, always shared some tiny common measure that fitted a whole number of times into each.
Then someone looked at the simplest figure available: a square whose sides are one, and its diagonal.
By the rule about right-angled triangles, that diagonal multiplied by itself must come to exactly two. So the question is whether any fraction, multiplied by itself, gives two. It is a short argument to show that none does: suppose such a fraction exists in its simplest form, and you can prove that both its numbers must be even — which means it was not in its simplest form after all. The assumption destroys itself.
The diagonal has a perfectly definite length. You can draw it with a ruler. It simply cannot be written as one whole number over another, and no amount of searching will ever produce one.
The number has a name now and a decimal expansion that never repeats and never ends. Writing more digits of it is not progress towards a fraction; there is no fraction to reach.
The legend says the man who found this was punished for it, and the legend is probably not true. What is true is that the school's central belief was false, and that the counterexample was sitting in the corner of every square anyone had ever drawn.