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Before complex numbers are introduced in senior high school courses, should we emphasise that solutions (e.g. to quadratic equations) are real solutions?

If we do, then when non-real numbers finally arrive, students are (in theory) already somewhat primed for the fact that there are different kinds of numbers. This is already established by distinguishing natural numbers, integers, rational numbers, and irrational numbers. But without the context of complex numbers, saying that the set of rationals and irrationals make the set of real numbers means that no proper meaning is given to the term “real”. And therefore, I speculate, students will either dismiss the word real, or conclude that we’ve finally reached the real set of all numbers (integers and rationals are not really all the numbers).

I wonder whether we shouldn’t instead talk about all possible numbers on the continuous number line. We can still say just “numbers” for short, but this longer qualification has immediately meaningful value (to Year 11/12 students) and doesn’t create any misconceptions about the word “real” that have to be undone when “imaginary” numbers are eventually presented.

To be clear, I’m asking about the language we should be using well before imaginary and complex numbers are introduced, and whether it matters much. Maybe most students who are beginning to learn complex numbers can probably handle a simple explanation about the terminology, and how everyday English meanings and mathematical meanings are not the same.

Edit: And to be clearer, I think what I wanted to ask was about how best to help students avoid developing misconceptions around the term “real numbers”. But as usual, I was consumed in my thinking at the time. To be honest, I have learned (over the last two years) to introduce formal terms early, even if the larger context of those terms remains to be taught much later.

lukejanicke
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    "Real" serves to distinguish these numbers from "rational" rather than from "complex". It is necessary to give some name to whatever field one uses when discussing irrational numbers. Generally it is not viable to speak of specific field extensions, e.g. $\mathbb{Q}[\sqrt{2}]$, and doing still still does not give a place for $\pi$ or $e$ to live. – Dan Fox Nov 29 '19 at 09:23
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    @DanFox: Good remark! Perhaps you could expand it into an answer? – J W Nov 29 '19 at 13:53
  • lukejanicke: Solutions to quadratic equations can be complex, as well. E.g.: $$x^2 + x+1 = 0$$ has solutions $$x = -\frac 12 \pm \frac{\sqrt 3}2 i$$ – amWhy Nov 29 '19 at 16:20
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    Personally, I think the terms natural number, integer, rational number, irrational number, real number and complex or imaginary number ought all to be introduced as soon as possible. There is little to be gained and much to lose by delaying their use. – James S. Cook Nov 29 '19 at 17:21
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    lukejeanicke: I couldn't help but notice your answer, in support of teaching about real numbers, posted Why is it possible to teach real numbers even before rigorously defining them?. So I'm curious why you support the less-rigorous teaching of/mentioning of/exposing students to real numbers, but at the same time seem to argue that such numbers should not be named as such, at least until complex numbers are taught? – amWhy Nov 30 '19 at 01:03
  • And lukejeanicke, you answered the question I link in my preceding comment, in the same hour you asked this question?? – amWhy Nov 30 '19 at 01:43
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    What else are you going to tell them ℝ stands for? – Andrew Morton Nov 30 '19 at 16:32
  • For those of us in different countries, could you expand on what "senior high school" is? – Michael Kay Dec 02 '19 at 08:24
  • @MichaelKay In the United States, students typically start in elementary school (grades K through either 5 or 6; ages 5--11 or so), then move onto either middle school or junior high (grades 5 or 6 through 8 or 9; ages 11--14 or so), and finish their primary education in high school or senior high school (ages 14-18 or so). Students in high school in the US typically take two years of algebra (intro to the use of variables, solving equations, working with polynomials, intro to functions etc), geometry, trigonometry, and (perhaps) some statistics or calculus. – Xander Henderson Dec 02 '19 at 15:03
  • @MichaelKay Year 11 and 12 for the purposes of my question. – lukejanicke Dec 03 '19 at 18:16
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    @DanFox I think I get your intent but “real” doesn’t distinguish irrationals from rationals. But from the students’ perspective, the new thing (irrationals) makes new whole (reals). Do elaborate more in an answer if you have the time. – lukejanicke Dec 03 '19 at 18:29
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    @Namaste Not at all advocating less rigorous teaching. My two posts are related. On that other post, my point is that even without a formal definition of real numbers, students have the basic conceptual foundation from high school as a good enough basis for learning more about them at university level, whether or not they’ve been named “real”. I probably should have noted in the OP that I use a name if there’s a name. So this Q is about the misconceptions that could arises from the use of the term “real” before the larger context is taught, and how to prevent them forming. – lukejanicke Dec 03 '19 at 18:37
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    @AndrewMorton Ah, too right! I’m not going to not teach the notation $\mathbb{R}$, since I’ve already taught $\mathbb{N}$, $\mathbb{Z}$ and $\mathbb{Q}$, so I better call $\mathbb{R}$ “real”. There are a lot of great replies below I have to read through carefully, but your point really makes it a no-brainer! – lukejanicke Dec 03 '19 at 18:50
  • Your posts in the link, suggests that real numbers be presented as such in high school; here are you arguing they should not yet be named. "A real number, by any other name, is still a real number." I think you've just tried to leverage more attention to your question, despite already having clarified that that you think there is no way around reference to real numbers prior to the introduction of imaginary numbers. Contradicting yourself, for the sake of greater attention (greater # of votes) with this question is game-playing, nothing more, nothing less. – amWhy Dec 03 '19 at 19:21
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    @Namaste I’m here for the mathematics pedagogy friend. I’m not a super regular SE user and I’m not very fussed about my SE reputation/points etc. I’ve been reading some recent popular books about teaching in high school, especially mathematics (e.g. Craig Barton, Mark McCourt). A hot topic is misconceptions and how they form. I personally have had to spend time redressing misconceptions around the language of complex numbers. And I’ve taught Year 7/8, where the language in question is first encountered. – lukejanicke Dec 04 '19 at 03:33
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    My comments on the other post do not contradict what I am asking here. Students do arrive at university with a workable conception of real numbers (the nuances of irrationals and maybe even transcendentals). They were asking about the didactics of teaching the formal definition of reals before teaching other things about them at university level. Here, I am asking about the misconceptions that could arise from school Year 7 onwards about the language we use. I appreciate the good replies seen here. Yours, for example, And @Ben Crowell’s, which you also liked. – lukejanicke Dec 04 '19 at 03:43
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    I just saw this from you in your reply on that other post: ‘ Finally, you present the real numbers just as "the union of rational and irrational numbers". This is a very simple and informal, although very precise and sufficient definition.’ Very well said! I would say your whole reply there is better than the current top answer. It seems I didn’t add anything new let alone better when I joined that conversation. I appreciate your contributions and SE community vigilance. – lukejanicke Dec 04 '19 at 03:57

8 Answers8

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Short Answer

You should not avoid use of the term real numbers. This is a term-of-art in mathematics, and it is important for students to learn the correct jargon. However, this technical term should be introduced as such—emphasize that "real" in mathematics does not mean the same thing that "real" means in everyday vernacular English.

Long Answer

The answer provided by Dave L Renfro is quite excellent. I would, however, like to approach things from a slightly different point of view.

First off, you should absolutely refer to the set of real numbers as the set of real numbers. I think that hiding this language, or trying to get around it, has the potential to lead to great confusion down the road. As Dr.(?) Renfro points out, simply using the word "number" is ambiguous, and "an element of the continuous number line" is clunky. Moreover, students will encounter the proper names of various sets of numbers in the future, so it seems wise to introduce the correct terminology early on.

That being said, I entirely understand the concern—I cannot count the number of times that I have had a student in a college algebra or precalculus class tell me that $\sqrt{-1}$ doesn't exist because "it is imaginary." The terms "real" and "imaginary" are, I think, poor terms which reflect the biases of Enlightenment natural philosophers (i.e. proto-mathematicians). However, for better or worse, we are stuck with these terms-of-art for the long run.

So... what can we do about it?

My strategy is to emphasize (early and often) that the words "real" and "imaginary" are used in mathematics to refer to specific sets of numbers, and that the use of these words is technical, and is different from the way that these words are used in vernacular English. Often, I try to make this point via a spiel such as the following (this usually chews up most of a 50 minute lecture, as there are examples at every step of the process, plus time for back-and-forth with the students):

I am a strict 5-ist. I do not believe that there is any actual number larger than $5$—in fact, there are only five numbers which actually exist: $1,2,3,4,5$. Every other number is a fiction created by people. And I think that $4$ and $5$ are a little suspicious—I believe that they exist, but only just barely.

Don't believe me?

Try this: have a friend pick out some number of essentially indistinguishable objects (pennies, marbles, popsicle sticks, etc). Then have this friend arrange these objects in no particular manner—in fact, have them just drop them from a short height so that they fall at random. Close your eyes while your friend does this, then, once they have arranged the objects, open your eyes and determine how many objects there are as fast as you can. If there are $3$ or fewer objects, you will likely be able to determine the number almost instantaneously. If there are $7$ or more objects, you will likely have to count them—at the very least, it will likely take you significantly longer to determine the number. Studies with functional MRI have shown that $5$ is about the upper limit for the number of objects that the vast majority of human beings can instantaneously recognize, regardless of how they are arranged.

Therefore, any "whole number" bigger than $5$ is obviously fiction.

Unfortunately, mathematics would be pretty boring and useless if we couldn't use numbers larger than $5$, so we have to start inventing some new numbers. One of the nice properties of the numbers $1$ through $4$ is that we can always "add one more" and get a bigger number.

So what do we get when we "add one" to $5$?

The utterly fictional number $6$!

And what if we "add one" to that?

We get $7$, which is also a pretend number.

And so on.

By this process, we get a whole infinite set of pretend numbers which, out of some kind of sadism, we call the natural numbers. I don't see anything "natural" about these numbers, but here we are. These are the counting numbers. These numbers can be used to count objects, and can be added (to, for example, count the number of objects in two different groups, then lump them together) or multiplied (to, for example, combine several identical groups of objects). Indeed, for a very long time, these numbers were sufficient to do most of the things that humans needed to do. Of course, people invented these numbers, and they are completely imaginary, but that doesn't prevent them from being useful.

On the other hand, these so-called "natural" numbers don't let us handle debts very well. For example, if you have a sheep that you want to sell me for $10$ chickens, but I only have $7$ chickens, you might give me the sheep anyway, but expect me to give you $3$ more chickens in the future. So, how many chickens do I have after this interaction? Clearly, I have no chickens, but I still owe you $3$ chickens. I have a $3$-chicken debt! The natural numbers are no good for describing this situation, so we have to invent a new set of numbers, called the integers.

The integers consist of all of the natural numbers, plus "negative" natural numbers, as well as a special number called "zero". These "negative numbers" and "zero" are completely fictional, and the product of fevered human imaginations, but they are tremendously useful, so I guess we're stuck with them.

Of course, there are still things that you can't do with integers. For example, if I split a pie evenly among six friends, how many pies does each friend get? They clearly get more than $0$ pies each, but fewer than $1$ pie each. But there are no integers between $0$ and $1$, so the integers don't cut the mustard.

Enter the rational numbers. A rational number is the ratio of two integers. The are called "rational" because they are thought of as ratios—this has nothing to do with their being "based on or in accordance with reason or logic". Indeed, they are completely unreasonable and fictitious, but they serve a useful role, so I suppose we should keep them.

For most practical purposes, the rational numbers are actually good enough. Calculators and other computers really only deal with certain rational numbers, and nearly every computation that you will ever do in your life is going to be done using rational numbers. However, mathematicians are often interested in constructing more powerful tools and techniques for describing the world. This often requires the use of more esoteric kinds of numbers, such as "algebraic numbers" (you need these if you want to give meaning to $\sqrt{2}$), and "computable numbers" (these things are quite esoteric).

In order to do calculus, we require a continuum of numbers. What this actually means is somewhat technical, but it leads to the introduction of the so-called real numbers. The "real" numbers include all of the rationals (which include the integers (which include the naturals))), the algebraic numbers, the computable numbers, and a bunch of "filler" which is really hard to describe. Of course, the "real" numbers are not at all real. Again, they are a human invention, and are just as "real" (or "unreal") as the integers. They are a very useful fiction.

Finally, it should be noted that the "real" numbers are often (and incorrectly, in my opinion) contrasted with the imaginary numbers. "Imaginary" numbers are just as real as the "real" numbers (or, again, just as fictitious), but are called "imaginary" for historical reasons. In some sense, we invented the integers so that we could subtract, and we invented the rationals so that we could divide. Similarly, we invented the imaginary numbers (or, really, the complex numbers) so that we could take roots (specifically, roots of negative numbers).

The moral of the story is that the terms "real" and "imaginary" are technical terms in mathematics. Both the "real numbers" and the "imaginary numbers" are a set of numbers which are defined via technical mathematical construction. These are equally "valid" types of numbers, and each kind of number is just as real or imaginary as the rest.

I'll note that this approach to numbers is far from Platonic (and I know that there are a lot of Platonists out there pulling their hair out in my direction right now). If you are uncomfortable with a non-Platonist point of view, you might want to consider a different spiel. ;)

A more relevant reference for this discussion might be the book Where Mathematics Comes From by Lakoff and Nuñez. There are fair criticisms of this text, but the early chapters and discussions of how the brain understands quantity are interesting.

Finally, I do tend to test these concepts. For example, I often give exam questions of the form "True / False; justify your answer". One such question is

The square root of negative one (that is, $\sqrt{-1}$) does not exist.

My intended answer is something like "False: $-1$ does not have a square root in the real numbers, but it does have a square root in the complex numbers", though I have given credit to students who say that it is true (e.g. "True: the square root of negative one is complex number, and the complex numbers don't really exist.").

Another pair of T/F questions which test a related concept:

The equation $x^2 + 4 = 0$ has no solutions.

vs

The equation $x^2 + 4 = 0$ has no real solutions.

The expected answer to the first question is something like "False: the equation has complex solutions," while the expected answer to the second is something like "True: there is no real number $x$ such that $x^2 = -4$."

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Xander Henderson
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    and that the use of these words is technical --- I used the phrase "complex number" in a paper for a college English class once, and the teacher circled the phrase and wrote in red ink "better phrase?". This was a 4th year graduate student working on her Ph.D. in English, so you would have thought she'd remember something this major from high school algebra 2 and precalculus, even if she'd (understandably) forgotten all the details. (Anyone this advanced academically was almost certainly one of the top 2 or 3 students in her HS, and thus surely took all the college-prep math offered.) – Dave L Renfro Nov 29 '19 at 19:55
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    As I neared the end of your lecture, my mind just kept going. Once we reach quaternions, we have hopefully gotten in the habit of surrendering quickly and just letting the mathematicians have their fun. Fortunately, the properties of the octonions are bizarre enough that most mathematicians will let you leave them alone. – Cort Ammon Nov 29 '19 at 23:29
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    @CortAmmon-ReinstateMonica Alternatively, you introduce the $p$-adic absolute value, and discard with the real numbers entirely. Then you really confuse folk by introducing an "infinite prime", and declaring that $\mathbb{R} = \mathbb{Q}{\infty}$, where $\mathbb{Q}{p}$ represents the $p$-adic numbers. Obviously, $\mathbb{Q}_{\infty}$ makes sense, because there must be a prime number at infinity, right? – Xander Henderson Nov 29 '19 at 23:35
  • The conflation of the word integer with the set of negative whole numbers is inaccurate and causes misconceptions starting from about six-year-olds. Integers are whole numbers, whether negative or positive or neither. – Nij Nov 30 '19 at 07:24
  • @Nij: How is The integers consist of all of the natural numbers, plus "negative" natural numbers, as well as a special number called "zero" such a conflation? (BTW, this appears throughout all the edits made thus far.) Maybe you're referring to The natural numbers are no good for describing this situation, so we have to invent some more numbers, called the integers, which I agree could be worded better, since it appears to be equating the set of additional numbers invented with the set of integers. – Dave L Renfro Nov 30 '19 at 08:14
  • "The natural numbers are no good for describing this situation, so we have to invent some more numbers, called the integers." clearly separates the naturals from the integers. It doesn't matter that detailed clarification is provided later, the introduction is what the student will remember. @DaveLRenfro – Nij Nov 30 '19 at 08:18
  • @DaveLRenfro "Anyone this advanced academically was almost certainly one of the top 2 or 3 students in her HS" I think you'd be surprised. The level a person performs at academically can vary significantly over time - there's a reason why nobody really cares about your undergraduate performance once you've completed some graduate studies. – nick012000 Nov 30 '19 at 08:36
  • @nick012000: True, and more significantly, I was talking about high school (HS), and not even a particularly large one. – Dave L Renfro Nov 30 '19 at 08:59
  • @Nij The goal is to provide a brief (40-50 minute) overview of the different sets of numbers which might show up in a college level precalculus class (though the same discussion ought to be appropriate for a high school class), with the most important take-away being that there are a bunch of technical terms in mathematics which we need to be careful about. I am not trying to give a rigorous construction of this various sets. – Xander Henderson Nov 30 '19 at 13:44
  • I was surprised, but pleasantly, to see you answer this. Not to nitpick, but it seems to me that the reason the ancient mathematicians called "imaginary numbers" imaginary was that they could 'see' real-world quantities as a model for the abstraction of real numbers, but they could not see any meaning for imagining that real numbers could be extended to include square-root of negative one. We of course now know that such fields exist, but they of course didn't, hence "imaginary" actually is accurate for them. – user21820 Nov 30 '19 at 16:39
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    I like the short answer. To me it has two aspects. First even ordinary English can be ambiguous. Consider "Each of us saw her duck." Second every academic discipline or job has its own jargon. You have to learn the jargon. // As another example think of a cook reading a recipe. The recipe states "Beat the eggs." That doesn't mean "Get your baseball bat and whack them." // Part of the problem here is that English doesn't have good way to differentiate between a noun phrase that is "descriptive," and a noun phrase that defines a particular thing. Poor example "white house" vs "White House." – MaxW Nov 30 '19 at 17:20
  • The takeaway that many will have w.r.t. integers is that they are strictly the negative wholes, and all because of sloppy language that isn't even necessary in this explanation. – Nij Nov 30 '19 at 17:28
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    @Nij The point of this lecture is not to clearly and precisely define all of the various sets of numbers that are used in mathematics. The point is to get students to recognize that words like "natural", "real", and "imaginary" have special meaning in mathematics, which is distinct from how they are used in spoken English. Moreover, I would expect that the audience here is made up of qualified educators, and that they will fill in the gaps in their own lectures (for example, none of the relevant board work is in this post). – Xander Henderson Nov 30 '19 at 18:17
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    I also have to admit that I have never had a student with the misconception that the integers are just the negative naturals. I've had students tell me that $3+2i$ is imaginary (rather than complex), and I've had students tell me that $\pi$ is rational (because it is the ratio of the circumference to the diameter). – Xander Henderson Nov 30 '19 at 18:23
  • You're repeating what you just said, and still missing the point: some students will go away thinking that the special maths meaning of integer is "negative whole number", either confusing them because the above explanation is ambiguous or confirming what they already (wrongly) believed. There is no "gap to fill" because you've already stuffed it with a misconception by using sloppy language. Lucky you to not have to deal with this problem before - more likely, you just never knew which students had it because it never came up for them. – Nij Nov 30 '19 at 20:02
  • @Nij I have not recorded my lecture to students word-for-word in the above. Nor have I conveyed any board work, nor any of the additional technical details that I might give. The audience here is other educators, not students. I expect people here to understand that this is an outline, and not a complete transcript. That being said, I don't know what could be more clear than "The integers consist of all of the natural numbers, plus "negative" natural numbers, as well as a special number called "zero". I will also emphasize, again, that this is an informal discussion. – Xander Henderson Nov 30 '19 at 20:18
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    I've had students tell me that $3+2i$ is imaginary (rather than complex) --- Actually, in older literature (mostly more than 20 or 30 years ago, and it's increasingly more frequent as you go back further) authors would refer to $3 + 2i$ as an imaginary number. I'm reminded somewhat of the older use of "single-valued function", which is now redundant, but 50-60+ years ago it was fairly common. Personally, I've always tried to use complex for $a + bi$ numbers and pure imaginary for $bi$ numbers $(b \neq 0$ real, although sometimes it's convenient to include $b=0).$ (continued) – Dave L Renfro Dec 01 '19 at 08:59
  • One of the things I've noticed with mathematical terms and notation since the rise of the internet, and especially since the rise of Wikipedia, is that these types of terms (e.g. complex number) are becoming more standardized, although sometimes with new names (e.g. Thomae's function rather than ruler function, which I believe used to be the most common name you'd see for this function in English-language real analysis texts). – Dave L Renfro Dec 01 '19 at 09:03
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    I like most of that spiel, except for calling zero fictional. I'd think any excessive time required for determining there are zero objects is purely because leading questions like "how many objects are there" prime the subject to think there are some objects to be counted in the first place. "Are there any objects" OTOH might be answered just as quickly. – muru Dec 02 '19 at 04:40
  • @muru In some sense, I agree with you. On the other hand, zero is a much more recent invention than one, two, and three. Cross-culturally, nearly every society on Earth (that we know of-a search of the HRAF for terms like "counting", "numeration", or "zero" is enlightening) can count "three"; many societies / language don't possess words for larger quantities (beyond group quantifiers, such as "many" or "lots"). The notion that "nothing" is a quantity akin to "one" is a relatively recent innovation in human history. – Xander Henderson Dec 02 '19 at 14:51
  • Of course, some authors include zero in the natural numbers, so I would say that this is entirely a matter of taste. I'll also note that I am "playing up" or exaggerating my skepticism of numbers other than one, two, and three for comic effect. Since the particular question of the existence and nature of numbers is a question of mathematical philosophy, rather than mathematics itself, I don't feel that playing up this imaginary aspect of numbers causes harm. – Xander Henderson Dec 02 '19 at 15:06
  • I’m stealing your “5-ist” lecture :) Anyway, marking this as answer and I think para 2 of your Long Answer is the core of it. – lukejanicke Dec 03 '19 at 19:01
  • @user21820 A few other comments/replies have touched on this idea. So I’m going to trawl the internet and my history books for any sources to back up this idea. But it certainly make sense. You don’t have to simply imagine 5 apples, half a loaf of bread, negative temperatures on a thermometer, or the length of a diagonal. You can make (or imagine! lol) real physical representations of real numbers: anything counted or measured. Not so much for complex numbers and beyond though. That is a new/novel take for me. – lukejanicke Dec 03 '19 at 19:08
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For 95% of high school students, this sort of thing is of no interest. But:

  1. The 5% do need to be served well and helped to achieve their potential.

  2. The 95% may find such things confusing if they are never explained, so it makes sense to offer them at least some brief explanation.

  3. Even the 5% are in no position at this point to understand fully what is meant by the real number system. This would require appreciating the completeness property of the reals, which is really not even treated very carefully or completely in most calculus classes.

I think the way most algebra 1 and algebra 2 textbooks treat this is pretty reasonable in view of these facts. They introduce all the elementary axioms of the reals, but not completeness. They say that there is a number system, called the real numbers, which satisfies these axioms, and which also includes our old friends such as $\pi$ and $\sqrt{2}$ (because students have almost certainly encountered these things before high school). At appropriate points in the development of the subject, they mention briefly that these numbers do exist in the real number system, but that we can't prove that based on the axioms we've listed.

They may or may not mention "real" vis a vis "complex," or provide any brief characterization of the complex numbers as including $\sqrt{-1}$. I think it would be fine to do this, but it should be very brief. Students should not be led to believe, incorrectly, that a symbol like $\sqrt{-1}$ has meaning just because we say it does. For the 5%, it might not hurt to point out to them that a system including $\sqrt{-1}$ is not going to be compatible with the real-number axioms about ordering.

If I was writing a textbook at this level, I would probably provide an appendix on the complex numbers, and then in the main text just point to the appendix. Then students who are in the 5% have something they can look at if they're curious, and they'll get a careful, clear presentation rather than something brief and garbled. If you're a teacher who is using a textbook that doesn't do this, you could point them to something like James Nearing's nice treatment in ch. 3 of this free online book: http://www.physics.miami.edu/~nearing/mathmethods/ .

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I recall having been taught different classes of numbers (in maths at school) way before we were introduced to complex numbers. Main reason was to distinguish

  • natural numbers
  • integers
  • rational numbers
  • finally ... real numbers

It's reasonable to teach students things like the coverage of numbers on the number line:

  • Why are real numbers continuous while rationale ones are not?
  • Cardinality and set theory (German: Maechtigkeit, I hope the translation is correct).

And, finally, it is reasonable to teach quite basic things like why pi ($\pi$) is not 3, 3.14, or 22/7, and that those are just approximations.

Conclusion: Yes, tell the students. There will be some who feel they won't need it, but that's a poor reason to not teach the ones who are willing to learn. Additionally, students who will have an issue of thinking that "imaginary numbers do not exist" can also question the existence of real numbers. After all, they are all just constructions based on mathematical axioms. And for those thinking "but imaginary numbers are not real" - well they are correct :)

amWhy
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Regarding all possible numbers on the continuous number line, in my opinion you're overthinking this. The vast majority of students aren't concerned with what the numbers are called, but how to solve the problems they have for homework and on tests. As for the use of real being unnecessary in the absence of knowing about complex numbers, I disagree. For them, the word number by itself could mean a positive integer, an integer, a rational number, or any real number, depending on context, and most good writing will not rely on context, but rather use the appropriate qualifying word. In particular, by saying real number we are telling the reader/listener that "anything goes" so to speak, no restrictions. Regarding why we use the word "real", students naturally might wonder about this, so tell them this word has been in use for a few hundred years to distinguish (if needed) from imaginary/complex numbers, something they'll learn about in a later math class.

The reason I say "overthinking", and here I'm thinking of my U.S. school classmates from the early to mid 1970s, is that surely nearly all have heard about complex numbers being one of the differences in solving quadratic equations in Algebra 1 vs. in Algebra 2. Whether eating with friends in the cafeteria, attending daily sports practices after school, on long bus/van trips for various extra-curricula events after school, sitting and talking with people in homeroom or elsewhere after getting to school but before the first homeroom bell rings, ... (the list is endless) --- students talk. And talk. Often about teachers (their quirks, how hard/easy certain ones are, etc.), and sometimes even about what is covered in their classes (especially while quickly trying to complete unfinished homework). Anyone who is the least bit curious about whether real numbers consist of all possible numbers will surely have heard someone mention complex numbers at some time. And for the very few who have an interest in math, they surely would have seen complex numbers show up when flipping through someone's more advanced mathematics text, flipping through more advanced texts in their school library, looking up "number" in encyclopedias in the library, from their high school math teacher if asked, etc. And all this discussion was restricted to before the internet and our present nearly instant and total access to knowledge about most anything a student is likely to think of.

I think your last sentence is on the mark. Students will have seen many uses of words in math and in other subjects that differ from their everyday use. There are many words used in math classes that they will have seen and used before, but which have much more precise and specific meanings in math, such as "variable", "constant", "average", "positive", "degree", "origin", "exponential growth", "similar", etc. So they will be familiar with this sometimes mismatch between everyday meaning and subject-specific meaning for some words, and it's simply a matter of saying that "real number" is another example, mentioning briefly how "real" came to be in use, although I suspect that most any student today who is remotely interested in the "how" could easily find out with a 5 second google search leading to a Wikipedia article.

Incidentally, to answer your question at the beginning, I'd say "yes", and briefly explain why real is used. In my case, I often said that complex numbers live in a plane, but for us we're only finding solutions that live on the (real) number line.

Dave L Renfro
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    I think that your assumption that most people have heard of "complex numbers" (e.g., that someone with an English PhD would surely have taken precalculus), is seriously flawed. Lots of universities nowadays don't even need basic algebra I skills to receive a diploma and advance. I've had college students even tell me they'd never heard of "ten thousand" before. It would be nice to have survey statistics on that. – Daniel R. Collins Nov 30 '19 at 08:29
  • @Daniel R. Collins: This was in 1977, and the English program was probably among the top 10 to 15 in the U.S. at the time. She didn't count off for this, and I remember that when I mentioned the comment to her, I didn't want to come off as superior or anything. In fact, I assumed this was just some kind of brain-slip (like seeing someone you know well from work at a grocery store, and not recognizing them because of the "wrong" setting -- at least, this is something I've done several times), (continued) – Dave L Renfro Nov 30 '19 at 08:47
  • but actually it was clear when I mentioned it to her that she'd completely forgotten this from her high school math. She went to a high school near mine, so I know very well what her HS Junior-year and Senior-year math classes involved. I even remember her name, and googling just now I see that a few years after 1977 she married someone "local" (lived within 25 miles of where I grew up) and then moved to UC-Santa Barbara for a faculty position (which it appears she no longer has, but maybe she's retired). – Dave L Renfro Nov 30 '19 at 08:50
  • @ DaveLRenfro The cultural context of Alg 1 vs. 2 was very interesting to hear about. It might not be universal, as @DanielR.Collins says, but point well made. I can imagine my embarrassment as a teacher if my student hears from another student in the cafeteria, “Oh, you’re learning about REAL numbers.” – lukejanicke Dec 03 '19 at 18:56
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Here's something that the students might easily grasp and that could also be entertaining:

  • What's "natural" about the natural numbers?
  • What's "rational" about the rational numbers?
  • What's "real" about the real numbers?

Every one of these different sets of numbers is a mathematical idealization of something encountered in the "real" world. None of them is more "natural", "rational", or "real" than the others, and none is less so, they just serve different mathematical purposes for modelling the real world.

These words that we mathematicians use in a technical sense are just being borrowed from ordinary human language, because we mathematicians are humans after all. And each of these borrowed words has its flaws, because we mathematicians are humans after all.

So we should try to keep these things in mind as we study new number systems, such as the complex numbers, which are not particularly more "complex" than the natural, rational, and real numbers, not after you get used to them, anyway.

Lee Mosher
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    A comment on my OP touched on this idea too. I think you’ve made an important point. The regular language meanings of these words can in fact be used to reinforce correct understanding, and protect against future misconceptions. I will use those discussion questions you’ve suggested. – lukejanicke Dec 03 '19 at 19:18
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In my opinion if some numbers are "imaginary" it doesn't mean they don't exist.

It is needed to distinguish somehow between the two kinds of numbers, the "real" that we easily see im our daily life and that make sense to everyone, and the "imaginary" numbers (or complex in general) that are there in life, real of course with no doubt, but they're not easy to "see" in the daily normal life.

So I think it is just the job of the teacher to emphasize the idea of complex numbers and explain their importance in many fields, I don't think there is a problem in the naming.

Nij
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Fareed Abi Farraj
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I'll take the dissenting opinion: don't use the phrase "Real Number" until you're prepared to teach what an imaginary number is.

"Today in science, we're going to be using an optical microscope to look at culture slides."

... which immediately begs the question from any remotely curious student:

"Wait, so there are non-optical microscopes?"

And you now have two choices:

  • Option A - "Yes, but we're not going to learn about them today" / "Yes, it's called an electron microscope. But we don't learn about them until next semester."

Which feels unfulfilling to the student. You quashed their curiosity, and they're probably wondering why you bothered saying "Optical" in the first place (if you weren't going to teach them the alternatives.)

  • Option B - Actually talk for awhile on what an electron microscope is and explaining its basics.

But that's the problem - you weren't planning on talking about Electron Microscopes in your lesson.

Back to the math example - if you're not willing to at least go into a little about imaginary numbers at the time, you probably shouldn't use the phrase 'Real Number'. (And, honestly, imaginary numbers is a subject that can take awhile for students to get their heads around - it's not something you should try to do a quick little 5-minute blurb about.) All in all, by throwing out "Real Numbers" before you're ready to go into Imaginary Numbers, you'd basically be opening a door that you're hoping nobody is curious enough to try to go through.

Kevin
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    The analogy is completely logically flawed. There is nothing false in saying "we are going to be using a microscope" if we are in fact going to use an optical microscope. On the other hand, it is absolutely false to say "there is no $x$ such that $x^2 + 4 = 0$", unless we specify precisely what $x$ is, such as "real $x$". – user21820 Nov 30 '19 at 16:44
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    Do you teach 7th graders that the cube root of 8 isn't just 2, but also -1+Sqrt(-3) and -1-Sqrt(-3) - before they learn about imaginary numbers, let alone complex number algebra? Or do you just teach them that the cube root of 8 is 2? Again, unless you're prepared to go into just what real/imaginary mean, using that terminology is begging a question you don't want to try to answer. – Kevin Nov 30 '19 at 19:07
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    That is even more wrong. Even in modern mathematics, the cube root of 2 is not a bunch of values. Either you take a principal branch-cut, or you say "a cube-root", not "the cube-root". And for high-school and real analysis, we define the cube-root to be the inverse of the cube operation on the reals. And there is no "begging". Who says I don't want to answer? If a bright student asks, further discussion can and should be done separately if it would not be suitable for the rest of the class. – user21820 Nov 30 '19 at 19:25
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    Geez. My answer was "completely flawed", and I was somehow able to be "even more wrong" with a subsequent comment. I can understand disagreeing with the analogy, though I think you're wrong and missing the point. But the way you're expressing your disagreement is pretty darned aggressive (let alone to someone who's posting on this site for the first time.) – Kevin Nov 30 '19 at 20:21
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    Dear @Kevin, welcome to the site. (@user21820, the rules are, we are kind to one another here.)

    Personally, I do like saying something about the lay of the land, ie saying a bit about "other microscopes types" or other number systems bigger than the one we're playing in.

     – Sue VanHattum Dec 01 '19 at 01:14
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    @SueVanHattum: Thanks for your comment, but just because I am critical of something does not imply that I am unkind. I commented only because it is a disservice to mathematical education to draw false analogies and conclude that using the term "real number" is "basically opening a door that you're hoping nobody is curious enough to try to go through.". I want this site to be a good pedagogical resource, not filled with opinions that may engender poor teaching. – user21820 Dec 01 '19 at 07:21
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    @Kevin The problem, I think, with your analogy, is that many students mistakenly believe that the equation $x^2+4=0$ has no solutions. When discussing an equation like this, it is necessary to specify that this equation has no real solutions (though it might have solutions in some larger space). By contrast, "We're going to use microscopes today" says nothing about the existence or non-existence of other kinds of microscopes (though, personally, I wouldn't have any problem mentioning electron microscopes, then letting Google finish the job). – Xander Henderson Dec 01 '19 at 17:11
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    @Kevin You make a convincing argument that I could side with. Respect for choosing the (apparently) dissenting side. As you can see in the comments, it can be risky to do so. But you tackled the heart of my question head on. I do think sometimes it is wise to be judicious and deliberate about when and how you roll out new language. I am particularly sensitive to this as half of my students are EAL. – lukejanicke Dec 03 '19 at 19:28
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I wouldn't. You feel the gap because you know what's coming. But qualifying like that before the students have the context leads to bafflement.

It's sort of a routine issue where people who already know the stuff want to present it perfect (complete). But this is not pedagogically sound. Suited to careful math explanation, but not to learning.

guest
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    You offer an opinion (sentence one.) Then in two sentences, you make two claims, neither of which you support with any sort of sound argument. In your last sentence, you again state another opinion. Please support, with research, the opinions and claims you make in answers. Nor do you provide any reason for anyone to believe you are a math ed "expert", nor even that you have experience in math ed. – amWhy Nov 29 '19 at 17:36
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    @Namaste the trouble is does guest = guest ? Previous "guest" have indicated some experience, but who is to say one guest is the same as the next ? – James S. Cook Nov 29 '19 at 17:47
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    Previous "guest" has shown insight not displayed here. I suspect there are a few "guest"s that recreate their accounts. Whether experienced or not, this answer, @JamesS.Cook, reads as mere opinions and claims. I am not focusing on the answerer, so much as the answer. I find this answer sub-par, and if answers could be closed as "opinion-based" like questions can, this would be a good answer to close for that reason. – amWhy Nov 29 '19 at 19:29
  • If this site is like the other SE sites, usernames aren't unique (there are multiple "Kevin"s that answer on Workplace, for instance. This particular Guest is new. – Kevin Nov 30 '19 at 05:55
  • @Kevin That's not likely true. This site has a number of "guest"s who never register, but rather recreate accounts which they perpetually abandon to create them anew. Just because a user on SE appears with 1 in rep, does not logically imply that said user is "new"! Note that this particular recycler of "guest" has not registered, which is all guests' pattern here. – amWhy Dec 01 '19 at 00:01