Real Numbers

The construction of the real numbers, R, is much more complicated than the construction of the rational numbers, and can be done in at least two different, though equivalent, ways. Instead of using ordered pairs to construct our new numbers, we actually need an infinite sequence of rational numbers. These infinite sequences, called Cauchy sequences, have to have the special property that the further along the sequence you go the closer the numbers get to each other. Infinite sequences are tricky things, so expressing that condition in a mathematically precise way is a little awkward: in words the formulation might go like this: for any positive rational number, no matter how small, there is a point in the sequence beyond which the difference between any two terms is always smaller than that number. As with the previous extensions of the number system, it is possible to define the basic operations of addition and multiplication on such sequences, and to prove that the resulting sequence is another sequence of the same type. We can also formulate ways of comparing such sequences, and saying which corresponds to a larger or smaller value. Finally we can find sequences that behave like rational numbers (for example the sequence where every term is p/q is the obvious choice for the rational number p/q), and so once again embed our previous number system, Q, in our new system R.

Gains and Losses

By the time we get to the real number system, we have what is known as a complete ordered field. This means that any set that is bounded above or below has a least upper bound or a greatest lower bound. So in some ways we are better off than with the rationals - we can in a way be more precise when answering questions about the bounds of a set. On the other hand, the real numbers have some paradoxical properties. If we go back to the natural numbers for a minute, it is easy to see that the list 0,1,2,3,... exhausts the set - in other words that any natural number, will, sooner or later, appear in the list. For the integers, we can also come up with such a list: 0,1,-1,2,-2,... Rather surprisingly we can also come up with a way of listing, all the rational numbers, and in fact we can come up with a way of listing all the algebraic numbers, or in fact any set of numbers which we can specify in a finite way. Yet this does not even scratch the surface of how many real numbers there are. It is easy to see that there is no list that includes all real numbers. This is the famous Cantor Diagonal Proof. The set of real numbers is said to be uncountable or non-denumberable.

For example mathematicans can prove, quite easily, that almost all "real" numbers are what is called "normal". Yet few, if any, numbers are actually known to be "normal", and a great many, including almost all the numbers we use in everyday life, are definitely not "normal". Furthermore, it is almost certainly impossible to prove that any particular number that is thought to be normal actually is so!

What is perhaps even worse, is that we can find sets of real numbers, in some sense as small as we please, that still have uncountably many members. For example, there is the set of numbers between 0 and 1, such that the first 999,999 out of very 1 million decimal places is 0, but the one millionth is either 1 or 2. And as soon as a set is uncountable, we can find a way to put each member of the set into a one to one correspondence with the points in a square or even a cube, of whatever size we choose.