I am currently teaching a course in geometry for teachers (Euclidean, nonEuclidean, projective, the whole ball of wax), and we were recently discussing the need for the Dedekind axiom for plane geometry, which guarantees in effect that the points on a geometric line behave the same , way as real numbers on a real number line. What was interesting to me was that, even after all we’d said about all the ways that geometry might behave in unexpected ways if we don’t make certain assumptions, somehow the idea that geometric lines were real number lines was more deeply ingrained. The idea that there might be a world were there were no line segments with length was harder to imagine than the idea of a world where there are multiple lines through a point parallel to another point.
It got me thinking, why is that?
What it comes down to, I think, is that many people think of “number” as “real number”. Oh sure, sometimes we’re only talking about integers, sometimes only fractions, sometimes we are even including complex numbers, but when the chips are down, when push comes to shove, in the bottom of the ninth, when I run out of cliches, numbers are real numbers, at least in the typical psychology.
This may seem a little bit stranger when you ask what real numbers even are, and realize what a mysterious question that is to answer. The rational number system is much easier to define. Even, say, the constructible number system, consisting of all the numbers you can write with integers, addition, subtraction, multiplication, and division, is simpler to visualize concretely than the real number system. Or maybe it’s because real numbers seem like “all” the numbers, because there’s no natural larger number system . . . except that that’s not true, since we know about the complex numbers. (And don’t go thinking I’m backing the complex numbers as the natural largest number system, because they embed in the quaternions . . . and the quaternions embed in the even larger system of octonions . . . )
If I asked you to actually define what a real number is, you’d probably say something about decimal expansions. A real number is a finite string of digits, then a decimal point, then an infinite string of digits (so-called “terminating” decimals end in all zeroes, if you like). Of course, for this to make sense, you have to understand how to add up the infinite series represented by infinite decimal expansions. And it’s pretty clear that most people don’t understand this process. If people did, we wouldn’t . Even if we did understand it, the fact that some (but not all) numbers have two decimal expansions suggests that decimal expansions are probably not the ideal definition of a real number. Another issue is that the concept seems to depend on the fact that we used based 10. If another civilization used base 7 and defined real numbers using base-7 expansions, would that be equivalent to our concept of real numbers? The answer is yes, but it’s very far away from obvious.
So why do the real numbers feel so right as the “official” notion of what a number is? Well, I can’t answer that, not fully. But I can give you a clearer perspective on the construction of the real number system, in the style of Dedekind.
I will assume that you understand the rational number system (fractions, positive, negative, and zero). There is a natural order on these numbers (an idea of when one number is greater than/less than/equal to another), which I will also assume you understand.
Then a Dedekind cut is any way of dividing the rational numbers into two nonempty groups so that every element in one group (“the small group”) is less than every element of the other (“the large group”).
Then in some sense we can define a real number to be a Dedekind cut. To phrase it more naturally, for each Dedekind cut, create a number to be at least as large as the numbers in the small group and no larger than the numbers in the large group.
If the groups are, say, the nonnegative rational numbers and the positive rational numbers, then the number 0 is at the “cut point”, and that was rational already. On the other hand, if one group is all the numbers that are negative or whose square is less than 2, and if one group is all the numbers that are positive and whose square is greater than 2, then the cut point is , which did not previously exist in the rational number line.
In summary, a real number is completely described by which fractions are less than and which are greater.
You might wonder what would happen if I tried the repeat the process. When I began with rational numbers and took Dedekind cuts, the result is the real numbers. If, however, I start with real numbers and take Dedekind cuts, then the new and improved number line is . . . *spoiler alert* . . . the real number system. There’s nothing new. All the ways of dividing the real numbers up into “small” and “large” numbers come from dividing them at a real number.
This property of the real number line (as opposed to the rational number line, etc.) is called completeness, and it’s fundamental to our understanding of the number line. It’s what guarantees that, if you have a sequence of numbers which are getting closer and closer together (if you wish I were being precise, research the phrase “Cauchy sequence”), there number be an actual number that they’re all getting closer to. In other words, if you have a list of numbers, each one larger than the previous, then the only possibilities are that the numbers converge on a genuine real number, or else they go off to infinity.
If the reason you believe that exists sounds anything like “there are numbers which square to something less than 2, and there are numbers which square to something greater than 2, so right in between, between all the numbers that are too small and all the numbers that are too big, is a number whose square is 2″, then your inner mathematician is relaying on completeness and Dedekind’s principle. The real number line is the one you have to use if you want that kind of argument to make sense.