 Hipposas, born around 500 BCE, was a Greek philosopher of the Pythagorean School of Thought. He is widely regarded as the first person to recognize that a square's diagonal cannot be expressed as the ratio of two integers. At this time in Greek society, numbers were intimately connected to their religion, so Hipposas' finding was considered heresy. 200 years later, Euclid published his proof. Here's how it goes. First, assume that there is such a rational number, and show a resulting contradiction that negates the assumption. Suppose p over q is a rational number expressed in its lowest terms, meaning they have no common factors except the number 1, such that p over q is equal to the square root of 2. We can square both sides of the equation and multiply both sides by q squared. This shows that p squared is an even number, and therefore p must be an even number, because an odd number times itself would be an odd number. Since p is even, there exists a number t, such that p is equal to 2 times t. If we substitute this in for p and divide both sides by 2, we see that q is also an even number. In other words, both p and q have 2 as a factor. But our stipulation was that they had no factors in common except for the number 1. We have a contradiction, and it shows that the statement, the square root of 2 can be expressed as a rational number, is false. Therefore, it cannot be expressed as a rational number. In fact, the nth root of any number that isn't a perfect n square is irrational. Add to that the fact that any irrational times irrational will be irrational, and you can see that the set of irrational numbers is infinite. The rational number line is dense, but it is not continuous. The union of the set of rational numbers and irrational numbers creates the set of real numbers. But to prove the basic number properties for a number line that includes irrational numbers turned out to be quite the problem. In 1872, Richard Dedekind defined cuts in the rational number line that expose the holes created by irrational numbers. He then proved that the set of these cuts is equivalent to the set of real numbers. This extended the rational number line into the real number line in a manner that preserved all the properties of the rational number line. In addition, it is not only dense, it is continuous, it has no holes. The real number line is the foundation from which all the rest of our math will flow.