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

• The "addition" of points is described most simply as vector addition. A vector can be represented by a directed line-segment; two vectors are considered equal if they point in the same direction and have the same length. (See diagram.) We can change the representation of a vector by moving it (i.e., "translating" it) to a new position parallel to the original position.
  

  To add two vectors V₁ and V₂, represent them with directed line-segments so that the initial end of V₂ is located at the terminal end of V₁. Thus the arrows in the diagram form a path: start at the initial end of V₁, proceed to its terminal end, then turn a corner and follow V₂ from its initial end to its terminal end. The sum, or resultant, V₁+V₂, is the journey going from the initial end of V₁ to the terminal end of V₂. That sum is represented by a single directed line-segment, the dashed third side of the triangle.
  

  To represent vectors with the Cartesian coordinate system, draw a vector V so that its initial end is at the origin (0,0). Then the coordinates of the location of its terminal end are used as the coordinates of the vector. (See diagram.)
  

  If we use that coordinate system, then the formula for vector addition is very simple: The first coordinate of V₁+V₂ is the sum of the first coordinates of V₁ and V₂, and the second coordinate of V₁+V₂ is the sum of the second coordinates of V₁ and V₂. That is,

  (a,b) + (c,d) = (a+c, b+d)

  

• The "multiplication" that we want to use can also be described in Cartesian coordinates: (a,b) ⋅ (c,d) = (ac−bd, ad+bc). But that's a bit complicated and nonintuitive; it looks somewhat arbitrary and contrived. We get a much simpler, more geometrically appealing definition if we switch to polar coordinates. Let a point be represented by <r,θ> if it has radius r and angle θ -- i.e., if it is located r units away from the origin, and on a ray that is θ radians counterclockwise from the ray that points toward the right. That point has Cartesian coordinates (r cosθ, r sinθ). If you substitute those values into our Cartesian formula for multiplication, and then simplify using some trigonometric identities, you'll end up with this much simpler definition of multiplication:

  If P1 has polar coordinates <r1,θ1> and P2 has polar coordinates <r2,θ2>, then the product P1P2 is defined to be the point with polar coordinates <r1r2, θ1+θ2>.

  In other words, multiply the radii and add the angles. The effect of multiplying points in the plane by P₂ is to rotate the plane through an angle of θ₂ and stretch (or shrink) the plane by a magnification factor of r₂. This concept is very simple, and it's quite useful in engineering, which is often concerned with describing rotations (e.g., of engines).

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?

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

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:

• What is a field?
• What is an ordered field?
• What is a Dedekind-complete ordered field?
• Why do I say that the real line is a Dedekind-complete ordered field? How can that be a definition?

Groups and fields

• X is a nonempty set.
• e is a specially chosen member of the set X. It is called the identity of the group.
• * is a binary operation on X, which we may call the group operation. This means that whenever p and q are members of X, then p*q is also a member of X.
• (p*q)*r = p*(q*r) for all p,q,r in X.
• p*e = e*p = p for every p in X.
• For each p in X, there exists at least one corresponding q in X that satisfies p*q = q*p = e. (It can be shown that there is at most one such q, and thus q is uniquely determined by p; we call q the inverse of p.)

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

• p*q = q*p for all p,q in X.

Examples:

• (Z,0,+) is an abelian group, where Z is the set of all integers
• ({even integers}, 0,+) is an abelian group
• ({-1,1}, 1, x) is an abelian group
• (R+,1,x) is an abelian group, where R+ is the set of all positive real numbers
• (R\{0},1,x) is an abelian group, where R\{0} is the set of all nonzero real numbers. (Here "\" means the difference of two sets.)
• (T,1,x) is an abelian group, where T is the set of all complex numbers that lie along the unit circle centered at 0

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

• Y is a set, 0 and 1 are two specially chosen members of Y, and + and × are two binary operations on Y.
• 0≠1.
• The triple (Y,0,+) is an abelian group.
• The triple (Y\{0},1,×) is an abelian group. (Note that this group has for its set of members, all the members of Y except 0.)
• p×(q+r) = (p×q) + (p×r) for all p,q,r in Y.

Here are a few examples:

• The rational numbers (i.e., numbers like 3/4 and -171/25) are a field.
• The real numbers (i.e., numbers like 87.324116279...) are a field.
• The complex numbers are a field. (Exercise: Verify all the axioms. Also, what is the multiplicative inverse of 3+2i ?)
• The set of all numbers of the form p+q√2, where p and q are rational numbers, is a field; it is a subset of the reals and a superset of the rationals. (Exercise: Verify all the axioms. Also, what is the multiplicative inverse of 3+2√2 ?)

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:

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 p^r for some prime number p.)

Ordered fields

• (Y,0,+,1,×) is a field.
• < is a binary relation on the set Y. This means that for each p and q in the set Y, either p < q is a true statement or p < q is a false statement. That can also be described this way: We are given some subset S of the set of all ordered pairs of elements of Y, and we abbreviate the sentence "(p,q) is a member of S" with the notation p < q.
• For each p and q in Y, one and only one of these three statements is true:
  p < q,     p = q,     q < p.

  (That's called the Trichotomy Law, because we are cutting the possibilities into three cases.)

• For all p,q,r in Y, if p < q, then p+r < q+r.
• For all p,q in Y, if 0 < p and 0 < q, then 0 < p×q.

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

• −1 < 0, and
• if p ≠ 0, then p² > 0.

Infinitesimals

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.

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.)

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

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

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

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:

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:

• The set of rational numbers is incomplete (i.e., not complete). To see this, let S be the set of all rational numbers r that satisfy r² < 5. Then S has many upper bounds -- for instance, 3 is an upper bound, and 2.24 is another upper bound, and 2.23607 is another upper bound. We can keep finding more of these numbers -- whatever rational number we propose for an upper bound for S, it is possible to find another rational number that is still a little lower and that is also an upper bound for S. You can probably see why already: These numbers are converging to
  √5 = 2.23606797749978969640917366873128...

  But that number is not rational. Any rational upper bound for S would have to be slightly higher than √5, and between that rational number and √5 we can always find still another rational number. In the field of rational numbers, the set S does not have a least upper bound.

• If Y is a non-Archimedean field -- i.e., an ordered field that has infinitesimals -- then Y is incomplete. One way to see this is to let S be the set of all infinitesimals. Since some of the infinitesimals are positive, any upper bound for S must be greater than 0. Note that 1 is an upper bound for S, and 1/2 is another upper bound for S, and 1/3 is another upper bound for S, and so on. Suppose (for contradiction) that b were the least upper bound for S. Then b must be positive, and must be less than or equal to all of the numbers 1, 1/2, 1/3, etc. -- thus b must be a positive infinitesimal. Then 2b is also an infinitesimal, so 2b is a member of S. Since b is an upper bound for S, that tells us 2b < b. But b < 2b since b is positive. This is a contradiction.

Complete fields

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

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

• Proof using decimal expansions. Let Y be the set of all infinite decimal expansions -- i.e., expressions such as 3.682951... and −17.311897... . Adopt the convention that 2.719999... is the "same" as 2.7200000..., etc. Use the usual operations of addition and multiplication. Then Y is a complete ordered field, but verifying that fact is extremely tedious. It generally isn't worked out in full detail. One place that you can find it in fairly complete detail is in J. F. Ritt, Theory of Functions, 1946. It is also sketched in M. Rosenlicht, Introduction to Analysis, reprinted by Dover.

• Proof using Dedekind cuts. Let Q be the set of rational numbers; we assume that we already have a good understanding of those. By a Dedekind cut we mean a pair (A,B) with these properties:
  • A and B are nonempty subsets of Q whose union is Q
  • a < b, for every a ∈ A and every b ∈ B
  • A has no highest element.
  The set B might or might not have a lowest element. Here are some examples of cuts in which B has a lowest element: A_-2 = {x ∈ Q:   x < -2},    B_-2 = {x ∈ Q:   x > -2}               A_3.7 = {x ∈ Q:   x < 3.7},    B_3.7 = {x ∈ Q:   x > 3.7} and here is an example of a cut in which B has no lowest element: A = {r ∈ Q:   r < 0    or    r² < 5},      B = {r ∈ Q:   r >0   and   r² > 5}. (That cut would be called (A_√5, B_√5) if we had a √5.) The set of all cuts can be made into a complete ordered field, if we define addition and multiplication the right way. Again, it's tedious; you can find some of the details worked out in W. Parzynski and P. Zipse, Introduction to Mathematical Analysis.

• Proof using Cauchy sequences. Again start from the rational numbers. Say that a sequence r₁, r₂, r₃, ... of rational numbers is a Cauchy sequence if it has the property that

  for each positive integer p there exists a positive integer m (which may depend on p and on the particular sequence being studied) such that, whenever i and j are greater than m, then |ri − rj| < 1/p.

  Now, say that two Cauchy sequences r₁, r₂, r₃, ... and s₁, s₂, s₃, ... of rationals are equivalent if they have the property that

  for each positive integer p there exists a positive integer m (which may depend on p and on the particular sequences being studied) such that, whenever i is greater than m, then |ri − si| < 1/p.

  By an equivalence class we mean the set of all the sequences that are equivalent to some particular sequence. Now, it can be shown that the set of all equivalence classes is a complete ordered field, if we define addition and multiplication on it in the right fashion. This proof, due to Cantor, is a slight modification of a proof that can be found in many analysis or topology books, showing that every metric space has a metric completion.

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:

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

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.