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This question is related to another question, Do we really need reals?, and could be considered a duplicate, so I would not be surprised if it will be put on hold. But I'm especially interested in the teaching aspects of the problem so I ask it in the following form.

An anecdote.

Years ago, when I was a high school teacher, I used to introduce real numbers showing first that $\sqrt{2}$ is irrational and that there are infinitely many algebraic numbers of the same type. Then I used to add (obviously without any proof) that there are other numbers, such as $\pi, e, 2^{\sqrt{2}}$ (said transcendental) that are not algebraic. All of these new numbers have a non periodic representation and, added to the rationals, form the set of real numbers.

To taste the beauty of mathematics, I was then used to sketch the Cantor's diagonal proof, to show that the real numbers are much more numerous than the rationals and form a set called continuous.

Once a student asked me if were the transcendental numbers (as $\pi, e ...$) that make the set of reals continuous. I was a bit uncomfortable and I thought about it for a while before I gave an answer; finally the answer was: NO, we don't really know the numbers that make the reals continuous because those numbers are not computable. The student was a bit astonished by that answer and he commented that mathematics was not such an exact knowledge as he hoped.

After that day I was convinced that students have to be exposed with caution to the mysteries of real numbers.

Now the question.

What is the minimal extension of the rational field that we need to teach (and learn) the calculus at a beginner level?

My guess is that is enough an exponential extension $\mathbb{E} / \mathbb{A}$ of the algebraic numbers field $\mathbb{A}$ considered as a subfield of the complex numbers $\mathbb{C}$ and constructed as in Exponential extension of $\mathbb{Q}$$.

As shown in that post, such a field is countable and all its elements are obviously computable. As far as I know, we don't know if $e$ is an element of that field, but however if we add it to $\mathbb{A}$ (possibly with some other helpful transcendent numbers) the field closure is anyway countable and its exponential estension is entirely computable.

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Emilio Novati
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    You maybe should have asked the student what she or he meant by the reals being “continuous” (a property usually attributed rather to functions)? Maybe she or he meant “connected”, and in this regard, was sort of right as the reals are the metric/order completion of the rationals yielding them connected (as opposed to totally disconnected – the state in which the rationals are). – k.stm Dec 28 '14 at 20:09
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    On some profound level we know all the reals pretty well. It's just that giving some (even most) reals a name is impossible. It's like fish. We know a lot about fish, but most fish we will never ever see, but we still know fish. – Ittay Weiss Dec 28 '14 at 20:21
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    So rereading the question, I think you need to clarify a few things: (1) What do you mean by the reals being “continuous”? (2) What do you mean by a number being “computable” (having a finite $b$-adic representation for some base $b = 2, 3, …$)? (3) What kind of calculus are you trying to teach? (Is it high school level still?) The intermediate value theorem will be false if you take any proper subfield of the reals, so you should state what results/mathematics you want to impart. – k.stm Dec 28 '14 at 20:22
  • @k.stm well. for "continuous" see http://www.encyclopediaofmath.org/index.php/Continuous_set. Not much used but expressive. – Emilio Novati Dec 28 '14 at 20:27
  • for "computable" see http://en.wikipedia.org/wiki/Computable_number – Emilio Novati Dec 28 '14 at 20:29
  • For the level, yes, I think to a high school. and about the intermediate value theorem you can see the discussion in Do we really need reals? cited. – Emilio Novati Dec 28 '14 at 20:32
  • You could always teach discrete calculus instead of the traditional calculus on $\mathbb{R}$. – ml0105 Dec 28 '14 at 20:58
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    Okay, I now get your question – interesting. It seems to me that you/your student were unpleased with that you cannot list, compute or even define all of the numbers which are responsible for the reals being connected (which are, of course, all real numbers). But I don’t think you can say we don’t know the numbers which make the reals continuous: It’s not like the question of whether a given real number is responsible for the reals being continuous is undecidable – the answer is always “yes”. – k.stm Dec 28 '14 at 21:04
  • @k.stm Yes. I think that the impossibility to compute ( in some manner) almost all the number that is using is e bit astonishing. – Emilio Novati Dec 28 '14 at 21:07
  • @EmilioNovati do you need to be able to catch all fish in order to eat some fish? – Ittay Weiss Dec 28 '14 at 21:54
  • @IttayWeiss: no, but if you tell me the totality of fish is uncountably infinite, and that there's no way you could possibly talk to me about the fish I do catch, other than in terms of their membership of an uncountably infinite set, then I'm at least going to question why you can't make do with a smaller model of "all the fish". That is to say, I will remain a bit astonished until shown the gruesome details ;-) – Steve Jessop Dec 29 '14 at 00:02
  • ... which is to say, the analogy is great once you're satisfied that there's a good reason not to worry about all the fish you can't catch, but I fear the analogy isn't sufficient to persuade someone to ignore this gigantic stack of unobtainable fish you've apparently invented just to vex them! Note that these aren't just fish I haven't caught, they're fish that cannot ever in principle be caught. – Steve Jessop Dec 29 '14 at 00:09
  • What if it's really easy to describe the collection of all fish, but rather difficult to describe the collection of all those fish you can potentially actually meet (this is in fact exactly the case in real life). It's still useful information to say "all fish need oxygen to live" even though you will never encounter most of those fish, rather than say "all fish in this incredibly complicated to describe subset of fish which you can actually meet need oxygen to live". In other words, it is easier to describe the set of all reals than it is to describe only those reals you can describe. – Ittay Weiss Dec 29 '14 at 00:38
  • @IttayWeiss: As against that, what if it's really easy to claim that the number of fish in the Pacific Ocean, right now as we speak, is infinite? Sometimes you avoid the easy route for some reason (in the case of infinitely many fish, that reason would be because it isn't true, but of course that's only an example). Just because something is easy doesn't mean there's no burden on the person who introduces it, to explain why it's necessary. Especially if it appears to be unintuitive and bizarre, as uncountable sets are to many people. – Steve Jessop Dec 30 '14 at 19:18
  • @SteveJessop I don't quite understand. It's very easy to describe the uncountable set of all reals but it's very difficult to describe the countable set of the describable reals. Unless you can come up with a better calculus, one based on the describable reals, the burden of proof is upon you to substantiate that the easier route is somehow wrong. Yes, cardinalities are somewhat counterintuitive, but hey, it's a great way to improve one's intuition and learn something new. The fact the it's counterintuitive does not suggest it is wrong. – Ittay Weiss Dec 30 '14 at 23:40
  • @Ittay Obviously I agree that there is not wrong in real numbers. I'm only searching for a gradual introduction that gives to beginners all the the numbers they needs for elementary calculus, before to tackle the Cauchy sequences. I think that, basically, that's what all mathematicians have made in their training and I'm only searching good and simple definitions that trace a path toward all the complexity of reals. – Emilio Novati Dec 31 '14 at 09:56
  • Then it seems the simplest approach is to define all the reals. There are numerous definitions and they are all relatively simple, some very elementary. Attempting to define the describable reals in any clear and simple way seems impossible. – Ittay Weiss Dec 31 '14 at 10:15
  • @IttayWeiss My guess is to introduce, as an intermediate step between the algebraic numbers A and the reals, an exponential extension of A, in which we can calculate any exponential and logarithmic function ( something as in http://math.stackexchange.com/questions/992463/exponential-extension-of-mathbbq end/or in http://math.stackexchange.com/questions/665406/exponential-function-properties-for-rational-numbers) and then add some transcendental number as $e,\pi$, using some succession. This way naturally open the route towards general successions of rationals and to real numbers. – Emilio Novati Dec 31 '14 at 17:05
  • About fish: as noted by @Steve Jessop, the question is not about my poor knowledge of all the set of fish, but the fact that even the most skilled fisherman will never catch any one of those fish I don't know. – Emilio Novati Dec 31 '14 at 17:06
  • Emilio, are you aware of http://matheducators.stackexchange.com ? – Incnis Mrsi Apr 21 '15 at 08:04
  • @Incnis: Thank you. I did not know it! I explore... – Emilio Novati Apr 21 '15 at 08:19

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For the basic notions of the calculus, like continuity and limits, you don't need the reals if you are happy to substitute them for something abstract. There are two ways this can be done. One is topology, but this is almost certainly not going to appeal to someone who did not already know enough calculus. The axiomatics of topology allows you to speak rigorously of the basic notion of calculus without mentioning the reals. Another possibility is to generalize metric spaces. Classically a metric space takes values in the reals, but you can replace the reals by what is called a value quantale. This axiomatization is much more easy to digest, so it can be used to introduce metric spaces without the reals, and again introduce the common notion of the calculus.

You are what is the minimal extension of the rationals needed to speak of calculus. Well, it would seem that a crucial property to have is that whatever the extension is it must be a complete lattice. Any complete lattice extension of the rationals must contain the reals, so the minimal such would be the reals.

Ittay Weiss
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I would suggest the following set:

All the numbers that can be calculated using a formula which contains a finite amount of:

  • Natural numbers
  • The basic arithmetic operations ($+,-,\times,\div$)
  • The infinite-repetition operator (e.g., $\sum\limits_{n=1}^{\infty}$ or $\prod\limits_{n=1}^{\infty}$)

In fact, you only need $\left[1,+,-,\sum\limits_{n=1}^{\infty}\right]$ but I wanted to keep the definition above simple.

In any case, this set contains all the algebraic numbers, as well an infinite amount of transcendental numbers (including $\pi$, $e$, etc).

I'm pretty sure that this amount is countable, since we are using a finite amount of symbols in order to represent every element in the set, but I'm not sure how to prove it.

UPDATE

After positing a related question, I have realized that such set has already been defined (the credit goes to a comment made by @PeterFranek).

It is the set of computable numbers, which contains many of the specific real numbers that appear in practice, including all real algebraic numbers, as well as $e$, $\pi$, and many other transcendental numbers.

You may want to focus on the section which refers the rather philosophical question of whether or not the computable numbers can be used instead of the real numbers.

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barak manos
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  • Such a field/number system contains $ℚ$ and if it’s closed under $\sum_{n=1}^∞$ (which is the same as taking limits as far as I can tell), doesn’t this imply it contains all the reals? – k.stm Dec 28 '14 at 20:26
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    You can write every real as an infinite sum of rationals. –  Dec 28 '14 at 20:28
  • @k.stm: My point is, that there are real numbers which we cannot represent in any known method (the uncountable "majority" of $\mathbb{R}$. I might be wrong in explaining that, and will be happy to know in what way (if you can explain). – barak manos Dec 28 '14 at 20:28
  • @MikeMiller: Using a finite number of symbols from the list above? – barak manos Dec 28 '14 at 20:28
  • Explain what you mean by a finite number of symbols. How do you exhibit $e$ using only finitely many such symbols? –  Dec 28 '14 at 20:30
  • @MikeMiller: $e=\sum\limits_{n=0}^{\infty}\frac{1}{n!}$ (and of course, we can represent the "$n!$" with $\prod$). Also, can you please read my comment to k.stm above? – barak manos Dec 28 '14 at 20:33
  • I don't agree that that contains finitely many symbols, since the $\prod$ you're using depends on $n$. If you allow me to use finitely many symbols for each $n$ then I can define each $a_n$ to be an appropriate rational - all of which are individually definable with finitely many of your symbols. –  Dec 28 '14 at 20:37
  • I guess we probably want to demand that each $a_n$ can be defined using some bounded number of symbols (ie, the maximum number of symbols is independent of $n$). –  Dec 28 '14 at 20:38
  • @barakmanos Okay, I get what you mean. This set (say $\mathbf L$) is certainly countable because $D → \mathbf L$ must be a surjection with $D ⊂ Σ^*$ being the set of strings actually defining a real number and $Σ$ being the finite/countable set of symbols you allow. This seems like a valid suggestion, but you should clarify this in your answer. – k.stm Dec 28 '14 at 20:39
  • @MikeMiller: Yes, but the $\prod$ itself is only counted as one symbol. And even if it counted as $n$ symbols, the total amount would still be a finite amount. – barak manos Dec 28 '14 at 20:40
  • @k.stm: This issue has been built in my head only in the past few weeks, and this specific set (coincidentally) has only occurred to me yesterday. I've been actually thinking with myself how to form it as a question here, since I'm not really sure about the whole thing. It seems like the "majority" of real numbers are in fact values that we can never "lay our hands on" - some sort of "unreachable numbers in another dimension" (like particles that go undetected in every sensor that we could possibly build, if I may use metaphors here). But like I said, I'm still very unsure of it. – barak manos Dec 28 '14 at 20:45
  • I fear that counting the symbols is not a good strategy. See http://math.stackexchange.com/questions/1083804/well-defined-uncomputable-numbers – Emilio Novati Dec 28 '14 at 20:47
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    @EmilioNovati: First of all, you provided a post that has received no answers or voting, so I think that it is hardly something to rely upon at this point. Second, your $P_F$ there is by itself a set of real numbers if I understand correctly. I did not allow the use of those symbols in my definition. You can use only what you're able to compute (even $\pi$ and $e$), but you cannot use, for example, "for every real number $r\in[0,1]$". – barak manos Dec 28 '14 at 20:51
  • I believe what you’re getting at here is what is known as “constructible reals.” They are, of course, countable. For a proper definition, though, you need to specify a language (in the CS sense) for describing the numbers, and in my experience, they are not really easy to work with. – Christopher Creutzig Dec 28 '14 at 22:08
  • @ChristopherCreutzig: Thank you. I've got a similar comment (I believe) for a related question that I have subsequently posted. I believe that the set that I was trying to describe is called the set of computable numbers. – barak manos Dec 29 '14 at 12:28
  • @EmilioNovati: Please see updated answer (the new section might be exactly what you're looking for). – barak manos Dec 29 '14 at 12:28
  • @barak Thank you! for the useful information. Now i'm searching some other source about computable analysis. (Wiky is poor). – Emilio Novati Dec 29 '14 at 15:16
  • @EmilioNovati: You're welcome. BTW, I believe that it might also partially answer your original question of whether or not we really need real numbers. – barak manos Dec 29 '14 at 15:20
  • I believe that the original question is more difficult. I'm convincing myself that we really need all the real numbers to do functional analysis. – Emilio Novati Dec 29 '14 at 15:30
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    @barak It's an open question to see if the construction that you proposed build up all the field of computable numbers. The usual algorithmic definition of such numbers is not so easy for beginner students. But I accept your answer. – Emilio Novati Dec 29 '14 at 15:36
  • @EmilioNovati: Thank you. I've got a few more thoughts about the other group of numbers (the non-computational ones). I might post them later on... – barak manos Dec 29 '14 at 17:27
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It is possible to teach a form of differential calculus entirely algebraically. For functions $f, g$ of one variable define as follows:

Let $x'=1$ (for $f(x)=x$)

and impose linearity so that $(af(x)+bg(x))'=af'(x)+bg'(x)$

And the product rule $(f(x)g(x))'=f'(x)g(x)+f(x)g'(x)$

The difficulty here is in making it useful, because there is no natural interpretation (gradiant of graph) to hand. [It is also important to ensure that the definition is consistent - for example if $h=fg=de$ then the two products give the same result.]

It is easy enough to show that this will detect double roots of a polynomial. It is also possible to show that $f(x)$ is monotonic near $x$ when $f'(x) \neq 0$, and that you can recover a polynomial from its derivatives (Taylor Series).

But this algebraic definition comes unmotivated, and generally appears rather later in mathematical development when the motivation is clearer.

Mark Bennet
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  • While I’m a fan of differential algebra (which is the term I learned for what you described), it’s probably not a substitute for calculus in a school setting. You really want things like the mean value theorem, and without reals (or, if you take a rather complicated route, constructible reals), it’s just plain wrong. – Christopher Creutzig Dec 28 '14 at 22:11