Should high school teachers say “real numbers” before teaching complex numbers?

Question

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.

Answer 1

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 $x^2 = -4$."

Answer 2

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/ .

Answer 3

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 :)

Answer 4

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.

Answer 5

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.

Answer 6

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.

Answer 7

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.

Answer 8

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.