It is true that the real numbers are 'points on a line,' but that's not the whole truth. This web page explains that the real number system is a Dedekind-complete ordered field. The various concepts are illustrated with several other fields as well. Version of 11 Nov 2009 by Eric Schechter. If you find any errors, or see anything that isn't explained clearly enough, or have any other comments about this page, please write to me.

What are the "real numbers," really?

The short, simple answer used in calculus courses is that a real number is a point on the number line. That's not the whole truth, but it is adequate for the needs of freshman calculus. The freshman calculus course (at most universities nowadays) follows the 17th century style of Newton and Leibniz, emphasizing computations and omitting many proofs. The omitted proofs depend on a careful explanation of what the "real numbers" really are. That explanation and those proofs were not discovered until the 19th century, after Newton and Leibniz were long dead.

A proper explanation of the real numbers nowadays is covered, if at all, in a course in "real analysis" in the junior or senior year of students who are majoring in mathematics. Surprisingly few students take such a course; perhaps that's because it is too algebraic for the analysts' taste and too analytic to please the algebraists.

In this web page, I'll discuss the mathematical meaning of "real number." Before that, I want to discuss this more elementary question: where did the name "real" come from? (It turns out to have little to do with the deeper properties of real numbers.) To answer that question, I first need to talk about complex numbers.

Treating points in the plane as numbers

There is a natural way to "add" or "multiply" two points in the Euclidean plane. By "natural" I mean that the definitions have turned out to be useful for many applications, and that the definitions are fairly simple. Unfortunately, the definitions take their simplest forms if we use different coordinate systems for the addition and multiplication operations. When addition and multiplication are defined as above, then the points in the plane are called complex numbers, for reasons that will be discussed a few paragraphs from now.

Since (a,0)+(c,0)=(a+c,0) and (a,0)×(c,0)=(ac,0), the points along the horizontal axis have an arithmetic just like "ordinary" numbers; we will write (a,0) more briefly as a. For instance, (5,0) will be written as 5. The points along the vertical axis also have a shorter notation: the point (0,b) will be written more briefly as bi; for instance, (0,5) will be written as 5i. The i stands for "imaginary", for reasons explained below.

Important exercises. Using either the formula (a,b) × (c,d) = (ac−bd, ad+bc) or the definition in terms of polar coordinates, the beginner should now verify that i2 = −1. That will be important in the discussion below.

Here are the answers to those two exercises: Using the Cartesian coordinate system, we compute i2 = (0,1) × (0,1) = (0•0−1•1, 0•1+1•0) = (−1,0) = −1. Or, using polar coordinates: The number i has radius 1 and angle π/2. Hence the number i2 has radius 1•1=1 and angle (π/2) + (π/2) = π; the complex number with those polar coordinates is −1.

What's "real" about the real numbers?

Probably the simplest way to understand "complex numbers" is to start with points in the plane, as I have done in the preceding paragraphs. However, by a historical accident, the simplest explanation was not the first explanation discovered. Indeed, the geometric, points-in-the-plane viewpoint wasn't discovered until the 19th century, long after the algebraic computations had been investigated. As early as the 16th century, mathematicians were devising new "numbers" as a way of solving polynomial equations; they were thinking in terms of algebraic formulas rather than pictures. They were particularly interested in the third and fourth degree equations at that time, but they even had new insights into the quadratic equation. The attitude that they took was something like this:

We all know that there isn't really any "number" p that can satisfy the equation p2 = −1. Such a "number" can only exist in our imagination. But if it somehow did exist, what kind of arithmetic rules would it have to follow?

You have to admire the genius of the 16th century mathematicians: They correctly worked out the arithmetic rules of the complex numbers despite their lack of the simple geometric model; they calculated with "numbers" whose existence they didn't even believe in!

Their terminology was unfortunate, however. There is nothing fictitious or dreamlike about rotations of engines, but the name stuck. The points on the vertical axis are now called imaginary numbers, despite the fact that they have very tangible applications. The points on the horizontal axis are (by contrast) called real numbers. All the points in the plane are called complex numbers, because they are more complicated -- they have both a real part and an imaginary part.

Thus ends our tale about where the name "real number" comes from. But we have barely begun investigating the mathematical properties associated with that name.

Getting rid of the pictures

The "point on a line" answer is not a fully satisfactory answer, because it is not axiomatic or algebraic. It relies on pictures that we don't really understand. For instance, the set of real numbers and the set of rational numbers have essentially the same picture, but their algebraic properties differ in ways that are very important for analysts.

Imagine studying that picture of a line under a super microscope. If you could magnify the line at a very high power -- say at a magnification of a googolplex, or better yet a magnification of infinity -- would it still look the same? Or would you see a row of dots separated by spaces, like the dots in a picture in a newspaper? (It turns out that, in some sense, the real numbers would still look like a line under infinite magnification, but the rational numbers would be dots separated by spaces. But that is only a vague and intuitive statement, not anything precise that we can use in proofs.)

The only way to get rigorous answers to these questions is to set up a very careful system of axioms about geometry ... but that amounts to the same thing as setting up a careful set of axioms about the algebraic properties of the real numbers. It turns out that the latter is a little easier, so we may as well concentrate on the algebraic aspects of the situation. To answer questions like this, ultimately we have to get away from the pictures; we have to understand the real numbers entirely in terms of formulas.

As a preview, here is the definition that we're going to end up with: the real line is a Dedekind-complete ordered field. That's complicated, so we'll work our way up to it in stages. We'll discuss:

Groups and fields

First of all, a group is a mathematical object; it is a triple (X,e,*) with these properties: Exercises:

The group is said to be abelian (or commutative) if it also satisfies this property:



Now, a field is a quintuple (Y,0,+,1,×) with these properties:

(Exercise: A few mathematicians do not include the requirement that 0≠1. Prove that there is only one "field" in which 0=1. For that field, the set Y has only one member.)

Here are a few examples:

Following is one more example. We will present a finite field -- that is, a field with only finitely many members. For the set Y, we'll use Y={0,1,2,3,4}. For its addition and multiplication operations, we'll use ordinary addition and multiplication, modified by this rule: If the result of addition or multiplication results in a number greater than 4, subtract 5 or 10 or 15, to get a number in the set Y again. In other words, we'll use these tables for addition and multiplication:

+ . 012 34
× . 012 34

This field is sometimes called arithmetic modulo 5. (Exercises: Show that a similar field can be given with 5 replaced by any prime number. Show that there is also a field with 4 elements, and a field with 9 elements, but there is no field with exactly 6 elements. Much much harder: It can be shown that there is a field with exactly n elements, for some integer n, if and only if n is of the form pr for some prime number p.)

Ordered fields

Next, we need to define an ordered field. This is a sextuple (Y,0,+,1,×,<) where The reals and the rationals, with their usual orderings are two familiar examples of ordered fields. A slightly less familiar example is given by the set of all numbers of the form p+q√2, where p and q are rational numbers. (Exercise: Show that that set is an ordered field.)

It can be shown that every ordered field contains, as a subset, an isomorphic copy of the rational numbers -- i.e., a set that is identical to the rational numbers in all its arithmetic operations; it may differ only in the names of some things, via a change in labeling. If you relabel things a bit, you can say that the rational numbers are a subset of every ordered field.

In particular, every ordered field contains infinitely many members. Therefore, the field of the arithmetic modulo 5 cannot be made into an ordered field by defining < in some clever way.

It can also be proved that, in any ordered field

Since i2 = −1, it follows that there is no way that we can make the complex numbers into an ordered field, no matter how we define <.


This next part is optional -- i.e., you can get through the definition of the real numbers without ever thinking about infinitesimals. But I think this next part is interesting, and also makes the definition of the real numbers easier to understand.

About 300 years ago, Newton and Leibniz invented calculus. Well, that's an oversimplification. Some of the ideas of calculus were already around, but they cleaned it up and knitted it together with what we now call the Fundamental Theorem of Calculus. Newton also showed some of the ways calculus can be used -- he worked out many of the basic laws of physics, and showed how to compute the orbits of the planets much more simply and accurately than anyone had ever done before. In doing so, he contributed greatly to the beginning of the Age of Enlightenment -- an age in which people realized that they can accomplish quite a lot through reasoning, and that they don't have to just live in fear, superstition, and confusion. This may have indirectly contributed to things like the industrial revolution and the birth of democracy.

Anyway, Newton and Leibniz knew how to do many of the computations that we now teach in calculus, but they didn't know how to do satisfactory proofs of the theory behind calculus. They tried to do proofs, but their explanations were a bit lacking. Many of their explanations were based on infinitesimals -- i.e., numbers that are infinitely small but not zero. For instance, in their explanations, dy/dx did not represent a limit of changing numbers. It represented a quotient of unchanging numbers, but those numbers were infinitesimals.

The computations of Newton and Leibniz were accepted by other mathematicians, but the proofs were not. The explanation of infinitesimals didn't entirely make sense, and mathematicians were uncomfortable with it. In the following centuries, Cauchy and Weierstrass produced the epsilon-delta proofs that we now find in calculus textbooks. Those proofs involve numbers that are of "ordinary" size (not infinitesimal), but the numbers would vary through many different ordinary sizes; thus we take the limit as epsilon changes toward zero. In our textbooks, dy/dx represents the limit of a changing quotient of two ordinary numbers. In the late 19th century, Dedekind finally gave a clear explanation of the real numbers (which we'll sketch at the end of this web page), and we can prove that in Dedekind's number system there are no infinitesimals. Arguments with infinitesimals were no longer needed and fell out of favor. Ultimately, infinitesimals were discredited and discarded by mathematicians (though they continued to be mentioned in some physics books many decades later).

In the 1960's, mathematician Abraham Robinson finally figured out how to make sense out of infinitesimals. Thus nonstandard analysis was born. It involved some nonstandard real numbers, among which we can find some infinitesimals. In the paragraphs below, I will give an example of an ordered field that has some infinitesimals. The discussion below is based on 20th-century ideas, not just on those of Newton and Leibniz. I should mention, however, that the example that I will present is not the approach preferred by the nonstandard analysts. They prefer an approach that is more complicated but also more powerful. (It involves making careful logical analysis of a formal first-order language, but we don't need to discuss that here.)

Some of the nonstandard analysts now actually feel that infinitesimals yield a better understanding of calculus. After all, it gave Newton and Leibniz the intuition that they needed. We can actually make rigorous mathematics, with only slight adjustments in the ideas of Newton and Leibniz. (For instance, the derivative should be the standard part of that quotient of infinitesimals; this term is explained in a later paragraph below.) But most mathematicians still prefer the epsilon-delta approach, which they feel is simpler. (Both methods are correct, and both yield the same results.) At any rate, some discussion of infinitesimals may be helpful in our explanation of ordered fields.

Definitions. Suppose that Y is an ordered field. An infinitesimal member of Y is a member r, other than 0, that satisfies all of these infinitely many conditions:

−1 < r < 1 ,      −1/2 < r < 1/2 ,      −1/3 < r < 1/3 ,     ...

Two members of Y are said to be infinitely close if their difference is an infinitesimal.

Some ordered fields have infinitesimals, and some don't. The ordered fields that have no infinitesimals are called Archimedean fields; we'll see later that the real number system (i.e., Dedekind's number system, also known as the standard real numbers) is Archimedean. The ordered fields that do have infinitesimals are called non-Archimedean fields; we'll give an example of such a field in the next few paragraphs.

The example will be based partly on rational functions. By a rational function in the variable t, we will mean a function of the form p(t)/q(t), where p(t) and q(t) are polynomials with standard real coefficients, and q is not the constant polynomial 0. Note that each real number can be viewed as a rational function -- for instance, the number 7 can be viewed as 7/1, where 7 and 1 are both polynomials of degree 0. Thus the set of real numbers is a subset of the set of rational functions. (Of course, to make sense of this, we have to assume that we already have some understanding of the real numbers. But we won't need a very deep understanding; the "points on a line" conception will suffice for now.)

We define addition and multiplication of rational functions in the usual fashion, as in high school algebra. However, we make this one alteration in the usual treatment of rational functions: We will consider two rational functions to be "the same" if they agree except at finitely many values of t. For instance, these two functions



are not really the same, because the first one is defined at t = −2 and the second one is not. But the two functions are identical for all other values of t, so we will view them as "the same" for purposes of the present discussion. With that convention, it can be shown that the set of all rational functions is a field.

Also, the real numbers are a subset of the rational functions. For instance, the constant 1 and the constant 7 are polynomials of degree 0, so the constant 7/1 is a rational function. In this fashion we can view every real number as a rational function.

We can make the rational functions into an ordered field, if we just define the right ordering. To do so, we will make use of the following theorem. (We will omit the proof of the theorem, which is a bit harder, but it just involves some advanced calculus and some college algebra.)

Theorem. Suppose that q(t) and r(t) are given rational functions in the variable t. Then there exists some real number t0 (which may depend on the choice of q and r) such that exactly one of these three cases holds:
  1. For every real number t > t0, the real number q(t) is less than the real number r(t).
  2. For every real number t > t0, the real number q(t) is equal to the real number r(t).
  3. For every real number t > t0, the real number q(t) is greater than the real number r(t).
Furthermore, if case 2 holds, then q(t) = r(t) for all but finitely many values of t.

We now define an ordering on the rational functions, by saying that

q < r     or     q = r     or     q > r

if cases 1, 2, or 3 hold, respectively. In other words, one rational function is less than another if it is eventually less -- i.e., if it is less when we go far enough to the right on the graphs of the two functions. How far to the right we have to go may depend on which two functions we're looking at; but the theorem says that for each choice of two rational functions, there is some point after which one function stays below the other (unless they're the "same").

With this definition of ordering, it turns out that the set of rational functions is an ordered field. But it also turns out that the functions

1/t,     2/t,     1/t2,     etc.,

are infinitesimals. Thus, the field of rational functions is non-Archimedean, when ordered as we have described.

How does this relate to Newton's view of numbers? I'm sure that Newton wasn't thinking of his infinitesimals as rational functions. But we can get some idea of his viewpoint, as follows:

There are no infinitesimals among the standard real numbers. But we could imagine that, with a sufficiently powerful microscope, we might discover some additional "nonstandard" numbers that we had not noticed before. Nestled around each standard real number r, infinitely close to it, are infinitely many new nonstandard numbers. (Then r is the standard part of any of those new numbers.) In particular, nestled around 0 are the infinitesimals. We can also get some other nonstandard numbers by taking the reciprocals of the infinitesimals; those numbers are infinitely large. The collection of all the numbers -- both "standard" and "new", together -- is an ordered field. Its ordering is the same as the ordering of the set of rational functions.

Least upper bounds

Suppose that Y is an ordered field, and S is a nonempty subset of Y, and b is a member of Y. We say that b is an upper bound for the set S if we have s < b satisfied for every s in S.

If the set S has an upper bound, then in general it has many upper bounds. Say B is the set of upper bounds of S, and B is nonempty. Does B have a lowest member? If it does, that member is called the least upper bound of the set S.

The word "complete" has different meanings in different branches of mathematics. Generally, an object is called "complete" if there are no "holes" in it -- i.e., if nothing that seemingly "ought to" be there is missing. This vague description has different meanings for different kinds of mathematical objects -- a complete ordered field, a complete measure space, a complete logic, etc. Here, we will only consider the meaning of completeness for ordered fields.

An ordered field Y is said to be complete, or Dedekind complete, if it has this property, also known as the least upper bound property:

Whenever S is a nonempty subset of Y, and S has at least one upper bound, then S has a least upper bound.

Dedekind completeness turns out to be crucial in analysis, because it enables us to take limits.

Some ordered fields are Dedekind complete, and some aren't. Here are two quick examples of ordered fields that aren't complete:

(Note that, conversely, any complete ordered field must be Archimedean.)

Complete fields

We have used the real numbers in some of our preceding discussions. For instance, the complex numbers are ordered pairs of real numbers, and our example of infinitesimals involved rational functions with real coefficients. In effect, we "borrowed" the real numbers -- we used the reals in examples, even though we hadn't formally defined them yet; we just relied on the informal and intuitive understanding that students already have, based on the geometric line. Trust me, there is no circular reasoning here -- I won't use the "borrowed" concepts when I finally get around to defining the real numbers. You'll see that if you actually work through all the details. (I'm not claiming that this web page is more than an outline.)

The definition of the reals depends on two more theorems, both of which are difficult to prove.

Theorem 1. There exists a Dedekind-complete ordered field.

The literature contains many different proofs of this theorem. I think three are simple enough to deserve mention here:

The other theorem is harder to prove, and I won't even sketch a proof here. In fact, this theorem is even difficult to state:

Theorem 2. Any two Dedekind-complete ordered fields are isomorphic i.e., there exists a one-to-one correspondence between them that preserves, in both directions, the orderings and the arithmetical operations. Thus, any two Dedekind-complete ordered fields are essentially "the same"; one is simply a relabeled copy of the other.

In particular, the decimal expansions, the Dedekind cuts, and the equivalence classes of Cauchy sequences, though they appear to be entirely different, all turn out to have the same arithmetic and algebraic structure -- they are really the "same" object. It is that object which we call the real number system.

Finally, the real definition of the reals

(No pun intended.)

Definition. The real number system is that unique algebraic structure represented by all Dedekind-complete ordered fields.

You might wonder why mathematicians want to use such a complicated definition. Wouldn't it be easier to simply define the real numbers to be the Dedekind cuts, or define the real numbers to be the decimal expansions, or something like that? That is the approach taken in some elementary textbooks, but ultimately it is less productive. When we actually use the real number system in proofs, the properties that we need are not specifically the properties of (for instance) Dedekind cuts or of decimal expansions. Rather, the properties we need are the axioms of a Dedekind complete ordered field. It is much simpler to think in terms of those axioms. To think of "numbers" as being cuts or expansions would just encumber us with extra baggage. The cuts or expansions are models -- they are useful for the job proving Theorem 1, but they are useful for little else. Once they've done that job, we can discard and forget them.

If you wish, you can now think of the points on a line as representing the members of a Dedekind-complete ordered field. It is then correct to say that the real numbers are the points on a line.