Good definition for introducing real numbers?

In the first lectures about calculus/analysis, you should introduce real numbers. Let's assume students know that rational numbers are.

What are the advantages or disadvantages in the different "definitions" of the real numbers?

I'm particulary interested in pedagodic arguments: At that point of the lecture, you want to have a good balance in

• not killing the motivation of the students/overcharging the students at that level,
• having a good definition to work with,
• a good agreement with the concept of real numbers the students have from school.

I've seen many books where the introduction of the reals are by far the most difficult chapter at that point (Compare handling Dedekind cuts vs. obtaining convergence of a series) or books where they not really introduce the real numbers, but also assume that students know it already more or less. There are also books where the authors have done a abstract contruction, but state that one should skip that part on first reading. This is okay for a book, but somehow this does not align with the deductive structure of mathematics.

edit: As it was asked in the comments. A typical course would be the first course where you want to define real numbers (i.e., a course where typically math majors are not sharing with engineers, etc.) like an "Analysis I" course in central Europe.

• I see from your profile that you're in Germany. For both your information and that of other readers, it should be pointed out that in the US, "calculus" and "analysis" are generally separate, very different courses. An Analysis course in Germany, even when taken by first-year university students, is comparable to a US analysis course typically taken by third-year university students. American "calculus" classes, in which math majors are frequently mixed together with science and engineering students, rarely offer any definition of the real numbers. Mar 15 '14 at 8:27
• I don't mean to be too US-centric, by the way. It's just that the US and German curricula happen to be the two I know most about (and I know almost nothing about any others). In any case, it would be helpful to specify more about the course you have in mind: what background will students have coming in, and what else will they do in the course? Mar 15 '14 at 8:32

As far as I know, there are four basic possibilities for defining the real numbers:

1. The real numbers can be defined using axioms.

2. The real numbers can be defined using decimal expansions.

3. The real numbers can be defined using Dedekind cuts

4. The real numbers can be defined as equivalence classes of Cauchy sequences of rational numbers.

Approach #1 is probably the easiest, but it is unsatisfying, because you don't actually prove that the real numbers exist. That is, you don't demonstrate the existence of any number system that satisfies the given axioms. I think #1 is probably the best if a course is short on time, or if the students aren't mathematically sophisticated enough to benefit from seeing one of the constructive definitions.

Approach #2 is what the students are expecting, but unfortunately the details are quite cumbersome, and the construction is not very natural (since it depends on a choice of base). It also doesn't give the students much new insight into the nature of the real numbers---it just fills in the details of insight they already have. In my opinion, the only reason to use #2 is if you are independently interested in talking about decimal expansions, e.g. if most of the students in the class are future high-school teachers.

Approach #3 is quite simple and elegant, and is certainly the easiest of the constructive approaches. It's wonderfully intuitive, and really cements in the students minds the idea that the real numbers are obtained by "filling in" holes in the rational number line. The disadvantage of this approach is that it's primarily order-theoretic: students are unlikely to see a construction like this again unless they study order theory, so it might be better to use an approach that's connected to other things the students will be learning.

Approach #4 is complicated, but has three main advantages (1) Students in a real analysis course should probably learn about Cauchy sequences anyway, so the extra time isn't exactly wasted (2) It starts the course off by identifying sequences as the important thing, a theme that is likely to continue for the rest of the course (3) It is a construction that can be generalized to completions of arbitrary metric spaces, which is something that students will need to understand when they learn point-set topology or advanced analysis. The disadvantage, of course, is that this is certainly the most complicated of the four options, and some students will be completely mystified.

My opinion is that the choice of approach should be based on the audience and the nature of the course. No matter what, I think it's worth pointing out that the other approaches exist, and perhaps briefly discussing them in class.

So here's what I think:

1. For a one-semester course in real analysis, it's probably best to stick with an axiomatic approach. There are more important topics to cover than constructions of the real numbers.

2. For a two-semester course in real analysis for typical math majors, it probably makes sense to use either Dedekind cuts or Cauchy sequences. For average students, I think Cauchy sequences are a bit too difficult, so I would start with Dedekind cuts.

3. For a two-semester course in real analysis for future mathematicians (i.e. student planning to go to Ph.D. programs), I would recommend covering both Cauchy sequences and Dedekind cuts, as well as mentioning the other two approaches. This audience will benefit a lot from the Cauchy sequences approach, but they should also know about other possible approaches, since they may be teaching real analysis themselves one day!

4. For an analysis course aimed at future high-school teachers, I would think it would be a good idea to discuss the decimal number system a bit at the beginning. Depending on the length of the course, it might also be helpful to discuss Dedekind cuts and/or the axiomatic approach.

One final note: if you do use the axiomatic approach, there is still some choice in which version of completeness you include as an axiom. The tradition seems to be to use the least upper bound property, but I think there is good reason to use either the convergence of Cauchy sequences or the nested closed intervals property instead. The advantages of using a Cauchy-sequence-based axiom should be obvious, while the advantages of the nested closed intervals property are that it doesn't require any preliminary definitions, it connects well to notions of compactness, and it is perhaps the easiest of the three to understand.

• The option of presenting the construction without the axioms doesn't make sense to me. You'd be presenting a model without saying what theory it was intended to be a model of.
– user507
Aug 2 '14 at 16:51
• Not that I recommend this as a definition of the reals for teaching purposes, but I think it is possible to define them using geometry. If you take distances as existing pre-numbers, then I'm pretty sure you can define the real numbers as the distances from a fixed point on a straight line. Addition and multiplication can be defined via geometric constructions. Aug 4 '14 at 23:06
• Nice answer, but there is an additional problem with the Cauchy sequence approach: since you do not yet have real numbers at hand, to avoid any logical loop you have to introduce first the notions of metric spaces and Cauchy sequence for distance functions taking values in rationals. But then you cannot give a non-trivial example before finishing constructing the real numbers. Certainly, there are ways around this issue, but none that I know of would really feel right and clear. Aug 8 '14 at 20:43
• I upvotes this answer because of your mention of the second possibility. I believe some jobs could benefit from having people who have an insane math ability including the ability to figure out how to write a complete formal proof in ZF that there exists a complete ordered field which is unique up to isomorphism. I never got taught in school what a real number really was but I came up with my own construction of them as described at matheducators.stackexchange.com/questions/7718/…. That type of person with an insane math ability is Jan 15 '19 at 15:56
• probably the type of person who could figure out on their own a proof that a complete ordered field exists rather than just accepting the real number axioms and using them. The decimal expansion is one very simple way to define them. You may think it's bad to define them that way because you haven't proven that it's isomorphic to the one constructed from the rational numbers. That's no problem because it's still true and the student could learn to write proofs only that the decimal construction satisfies certain properties and not proofs that the rational construction satisfies them. Jan 15 '19 at 16:02

I've approached this a couple different ways in the past, and I haven't decided what I think is best. But I'd like to point out some arguments in favor of not including any construction of the real numbers (like Dedekind cuts or limits of equivalence classes of Cauchy sequences), and simply stating the axioms which are needed for the rest of the course. (This may or may not include stating a theorem that there is, up to an appropriate notion of isomorphism, a unique mathematical structure satisfying those axioms.)

You note that the introduction of the real numbers is the most difficult part of many analysis textbooks. Beyond that, I would say that it is in many ways orthogonal to the rest of the course. The mathematical issues and techniques involved in defining the real numbers are mostly rather different from the issues and techniques involved in working with the real numbers. Therefore time and effort spent on defining the real numbers doesn't actually contribute much to students' understanding of real analysis.

You also raise the point that omitting a definition of the real numbers "does not align with the deductive structure of mathematics." But you have to start with something. Most constructions of the real numbers start with the rational numbers. But what are those? Rational numbers can be defined starting from the integers; this construction (and its generalization to integral domains) is frequently taught in algebra courses. But what are the integers? They can be constructed from the natural numbers, which can be constructed in pure set theory… Most people wouldn't think of starting an analysis class with the Peano axioms, let alone axiomatic set theory. (And how many people teaching analysis classes, or writing analysis texts, are even qualified to teach axiomatic set theory?) So an analysis class starts somewhere higher up the ladder, with a lot of structure assumed to be familiar to the students. This admittedly "does not align with the deductive structure of mathematics," but that's a necessary fact of life.

That said, the last time I taught analysis I covered Dedekind cuts, because it seemed to let me get to the real content of the course more quickly than Cauchy sequences.

• I agree completely. The axioms of the real numbers as a complete ordered field characterize them uniquely. It is good to get students to realize that you can wok with an object just by giving a characterization. Mar 15 '14 at 13:19
• I would vote up this answer twice if I could. Note that this principle can be applied to other courses, like integration; it is an important habit to be able to reason from some first principles. In my opinion, constructing the reals belong more to a course about quotients and other similar constructions while constructing a theory of integration is better leaved off to a measure theory course. Aug 8 '14 at 20:48
• Also historically, a construction of the reals was what came probably last in the development of calculus. So one might argue that it is not essential for doing or understanding analysis. Dec 16 '20 at 6:07

Note: The question is broad, I agree with a lot of what Mark Meckes said. Below my take on the Cauchy sequences approach (mainly in favor to complement the other answer).

The approach of constructing the real numbers as equivalence classes of Cauchy sequences of rational numbers modulo sequences with limit zero seems certainly not like the most intuitive one or the one most in line with earlier concepts, but it does not that bad in this regard either, and has some other advantages.

What are the advantages (in my mind):

1. One uses objects and notions that one needs later on, such as: Cauchy sequences, sequences with limit zero, the notion of an equivalence relation in a somewhat non-trivial way, definining operations (sum and product) on the spaces of sequences, checking some operation stays well defined under that equivalence relation. By contrast, a Dedekind cut seems completely confined to this context; granted some of the things I mention also come up in that context but still I feel it is more useful to have it for sequences.

2. The approach generalizes. The same approach of constructing a completion works for any metric spaces. More generally, if one wants, one can use such a course to build up some basic notions of General Topology.

Why it is not that bad regarding intuition:

I think the most intuitive sem-rigorous way to think of a real (non-rational) number is as a sequence of (rational) numbers that approach it in some sense.

Perhaps one has an algorithm to calculate ever better approxiamtions, this would fit this intuition. Perhaps one has "the decimal expansion", this would fit this intuition. Perhaps one has a sequence of ever smaller intervalls containing the number, this would essentially also fit this intuition (considering end-points).

And, then, from a certain point on one should never be too far away from what one wants to approach, whence the Cauchy criterion. And, if a sequecne goes to zero, then it "is" zero. And, two things are "equal" if the difference is zero.

On the problem of what to assume. Two possible arguments.

1. I think there is a point to be made that to assume the rationals but to construct the reals is apt for an Analysis course since the real numbers are in some sense an object of the analytic world. By contrast, in the evoked context often, another, possibly the other, large course followed by students is some (Linear) Algebra course. There they will also use real numbers more often then not working in $\mathbb{R}^n$ and so on, but it is hardly ever really relevant, so I would say to make a 'big deal' in that context about what the real numbers are would feel besides the point.

2. One can start earlier. Starting from (naive) set theory one can build up 'everything', introducing students to proofs by induction and some related things on the way. (For example Schwartz's 'Analyse I' is subtitled set theory and topologie.) If this is feasible depends on the context. In some classical curricula there would be essentially Anaysis 1,2,3,4 spanning the first two years, often thaught by one and the same instructor. In such a context one has quite a bit of flexibility at the start and to start with set thoery, Peano axioms and so on is viable. If the students should know some actual analysis after the first term, then of course less so. I would likely not do it myself even in the former conext, but I was taught it like this: I enjoyed it, many others did not. And, this level of (naive) set theory feels then really like something of a natural baseline as it is this level on which many mathematicians actually build/operate. (Perhaps this is an over generalisation, and things are also changing in this regard but I think that this is/was a justification for such choices.)

1. It takes a lot of time while not being that central/useful.

2. It is not that easy to follow and some students might find it boring or pointless.

3. Turning an argument in favor around. Things like this will anyway be seen later on (say, in a Toplogy course). [But this is not always the case.]

End note: I never had the possibility so far to teach Analysis "like I want/decide" but in one form or another participated in instructions using each of Dedekind cuts, Cauchy sequences, and an axiomatic approach. Depending on time and students I would go either for Cauchy sequences in detail (lot of time, solid students) or the axiomatic approach (possibly offering some reading material for those interested in 'filling the gap').

• In my experience, teaching something now "because they'll need it a few semesters later" does no good. Mystifies now, will be long forgotten when the use comes around. Apr 10 '14 at 8:35
• @vonbrand I am not exactly sure what you want to express. Yet it seems to me you want to express some disagreement with my answer. But please note that I listed under disadvanatages the fact that things like this might be seen anyway later on.
– quid
Apr 10 '14 at 9:10
• In my experience (and I've had plenty, on my own and having to assess courses elsewhere) teaching something without reasonably immediate application, hopefully a lot of it, is just a waste of time. Apr 10 '14 at 9:13

Whether you construct the reals or just axiomatize them, at some point you need to talk about the completeness property or completeness axiom. The students should ultimately learn several versions of this (Cauchy completeness, nested intervals, least upper bounds). I've found the following version, not listed in previous answers (unless I overlooked something) to be relatively easy to formulate, without a lot of prior definitions, and thus useful as a first version to show the students.

Suppose $X$ and $Y$ are two sets of real numbers, and suppose $x\leq y$ for all $x\in X$ and all $y\in Y$. Then there is a real number $z$ such that $x\leq z$ for all $x\in X$ and $z\leq y$ for all $y\in Y$.

There are different ways to introduce the real numbers. Even if you choose an axiomatic introduction, you have to choose an appropriate formulation of the completeness theorem.

Since you are german, I suggest you to have a look the following article: Zur Behandlung der reellen Zahlen im Oberstufenunterricht. In: H. Schröder (Hrsg.): Der Mathematikunterricht im Gymnasium. Schroedel, Hannover 1966, 215-227.

Some 50 years ago, the same question was discussed concerning school. :-) Kirsch preferes the axiomatic introduction; however, one should connect the new knowledge to the decimals which the students already know.

Since often, they think they already know the reals and thus do not pay attention, I would suggest to irritate them in the lecture by asking questions like:

• Is $0.999... =1$?
• What is $0.999...+0.111...$?
• Are there infinitely small numbers?
• How is $\sqrt2^{\sqrt2}$ defined? How can it be calculated?

For an interesting introduction of the reals based on the integers (not the rationals!) you might have a look at "eudoxos reals": http://arxiv.org/abs/math/0405454v1 However, I would not recommend this for your teaching.

I very much liked how the real numbers were introduced to me at university.

The professor introduced the usual axioms of arithmetic. Then defined continuous functions, Cauchy sequences, limits, and so on with all kinds of proofs.

Then started proving the Intermediate Value Theorem. (If f (x) is continuous on [a, b], and f (a) < 0, f (b) > 0, then there is an x in [a, b] such that f(x) = 0). So he takes the set S of reals in [a, b] where f (x) <= 0, S is not empty and has an upper bound, so we take x as the least upper bound of S and f (x) must be 0. Pause. The audience does not like the proof. The audience slowly realises that there is something wrong with that proof. How do we know that S has a least upper bound?

So he sneaked in the Axiom of Completeness - actually used it literally in the proof, word by word as you would find it defined in a text book. And that's the missing axiom that turns the axioms of arithmetic into the axioms of real numbers. Introduced in a way where it totally made sense. If you put that axiom at the start of real numbers, there is no understanding. In the middle of the proof for the Intermediate Value Theorem, it is absolutely obvious.

BTW. I like proving the existence of real numbers by equating them with equivalence classes of Cauchy sequences.

I will add to this answer. I think only the second way or something like it is something everyone can learn in elementary school with no struggle as long as they still teach fractions their current ways. The third one was Dedekind cuts but they probably meant of rational numbers which is very confusing to introduce from scratch using fractions. According to this answer, people 14-18 struggle with fractions.

I read the body of the question How to teach sum of fractions to students?. It was very confusing what they were saying. I thinking giving students a long list of direct instruction on how to add fractions is why they can't learn it. I think the task teachers gave them was to learn how to show their work and learn what the direct instruction on how to add them is. I think not all students can learn how to do that when they're in elementary school. I think the whole system on everything should be changed to not need them to learn it then anymore. Instead, teachers should do something like demonstrate the following example. They could ask "What is $$\frac{3}{4} + \frac{5}{6}$$? Then they will let the students think a little bit. Then they will demonstrate that it can be rewritten as $$\frac{9}{12} + \frac{10}{12}$$. Then they will give homework problems like that but only ask them to give the answer and not ask them to explain it. Naturally, they will ask themselves "I remember there was a way to do it. What was it?" without realizing it. They will find their own method in their head to do it. Maybe they will get the students to challenge themselves to get the answer with them. Then afterwards, they could tell them that what they were doing was taking the lowest common denominator. I'm not sure they will even figure out that that means lowest common multiple of the denominators or what lowest common multiple means. So they will tell them that the new denominator was the lowest common multiple of those numbers and be more direct and not even say "lowest common denominator" yet.

Later on, I'm not sure the best way to introduce irrational numbers if they teach them fractions first. I suppose they could then introduce a discrete graph for each rational number that has the slope of that rational number, that is the function that assigns to each integer the floor function of that integer times that rational number For example, this would be the graph for the rational number $$\frac{4}{7}$$ Then they could demonstrate the following example. Take the following partial graphs whos entirety are the images.

Now take another 2 images. The first takes 1 copy of the first followed by 2 copies of the second and the second takes 1 copy of the first followed by 3 copies of the second to get

Now we can create another 2 images. The first takes 1 copy of the first followed by 2 copies of the second and the second takes 1 copy of the first followed by 3 copies of the second to get

This demonstrates how irrational numbers are possible. Then afterwards, maybe the real numbers could be constructed from the Dedekind cuts of the rational numbers.

However, I think it's easier to deal just with halves quarters, eights and so on at first. Then it will be very easy to define $$\frac{1}{3}$$ as a Dedekind cut of those numbers and then introduce all Dedekind cuts of those numbers. Then they will explain that not all of them can be gotten by dividing an integer by a nonzero integer if negative numbers were introduce before non whole numbers. Then it will make sense. They will be like "I don't see why they have to all be able to be gotten that way." Then if they already know the associativity and distributivity of real number multiplication, maybe they will simplify things and won't introduce fraction notation at all and will instead say "Take the numbers that can be expressed as $$x \div y$$ where $$x$$ and $$y$$ are integers and $$y$$ is not zero." Given 2 numbers expressed that way like $$(2 \div 3) \div (5 \div 7)$$, they will then ask "How do you write it as an integer in its normal form divided by an integer in its normal form?" The answer will be $$14 \div 15$$. They might also later give another question of "How do you write $$(3 \div 4) + (5 \div 6)$$ in that way?" The answer will be $$19 \div 12$$. That's right. They will not use improper fractions and get them to write $$1 + (7 \div 12)$$.