= { p / q : p, q , and q 0 }

The rational numbers, believe it or not, are countable. The rational numbers include, under algebraic addition and multiplication, closure under addition and multiplication, additive and multiplicative identities, and additive and multiplicative inverses. Unfortunately, the rational numbers does not have the Axiom of Completeness.

For example suppose we had a set A, as follows:

A = { x : x^2 < 2, x }

A is the set of all rational numbers less than the square root of 2. An upper bound to this set would be 2. Another upper bound would be 1.5. Another upper bound would be 1.4143. Where does it end? What is the least upper bound of the set?

Why, obviously, the square root of 2 is the least upper bound. But is the square root of 2 a rational number? It's not.

Proof by contradiction. Suppose by way of contradiction, that the square root of 2 was indeed a rational number.

, where p and q .

We notice that there are many natural numbers 1 / 1, 2 / 2, 3 / 3, that are exactly the same number. We will assume that our rational number p / q is already simplified as much as possible.

Since , we can say that .

By multiplying q^2 to both sides, we have that .

Notice that since 2q^2 is even, p^2 must be even. And, since p is an integer, p's prime factorization must contain at least one 2. Therefore, we can write p to be a product of 2 and some integer, such as  , j . Our equation now looks like .

This can be simplified to look like . Since q^2 is even, q is even, and q can be written as the product of 2 and some integer, such as .

Well, wait a second. Our initial claim was that , where p / q is already in its most simplified form, yet when we rewrite p/q with j's and k's, we have , which is very clearly not in a simplified form. Therefore, the square root of 2 is not a rational number.

That means that there is no least upper bound (in the rational numbers) to the set above! We can express the square root of two with an arbitrarily close rational number, but we can never write the square root of 2 as exactly what it is. This is where the rational numbers fall short, there is a gap at the square root of 2.

Likewise, there is a gap at e, pi, at the square root of every prime, at the products and summands of every rational with every irrational number, and the products and summands of most irrationals with other irrationals, etc. In fact, the set of all gaps of , ie, the complement of , , is actually larger than the set itself.

The complement of is not only infinitely large, but it is also uncountable. So, in an informal way, we are introduced to the Real numbers, , where

= .