## Why do we teach complex numbers?

32

9

In algebra II, USA, we teach our students complex numbers. However, after algebra II, they never use complex numbers until pretty much complex analysis. The whole point of teaching them complex numbers is to find the roots of polynomials... but, that's all we ever do with them. Sure, we do some algebraic manipulation of them, just to get a feel, but that's it. Nothing ever too deep, and often times there isn't even a geometric understanding of them.

And by the time students ever reach a point where they need to use complex numbers... it'll be very far in the future.

As far as real world applications, it'll probably be finding the real roots of something for anyone who doesn't take complex analysis or similar courses that deal with complex numbers. And at that point, you probably don't sit down and factor your polynomial. I would think most would either see it as a quadratic or go straight to a calculator (one that can solve such things with complex numbers or numerical methods)

So, my question is if its really worth it teaching students about complex numbers. Most students will forget about complex numbers even, as complex analysis is usually a far way off, if they ever get there.

Maybe to be more specific,

Why do we teach students about complex numbers if most will never reach a course that uses them? When do laymen use complex numbers in real world applications?

19Disagree with a couple of assumptions here: That "the whole point of teaching them complex numbers is to find the roots of polynomials", and that mathematics is inherently about applications. Or, if it needs be said, passing a particular high school test.Daniel R. Collins 2016-09-24T14:23:40.620

8How could you not talk about complex numbers, at least a little? In the quadratic formula, sometimes the discriminant is negative!Adam 2016-09-24T16:14:09.780

28I would ask a different question: "why don't we teach more complex numbers?" They have vast application in trigonometry, two-dimensional vector analysis etc... much of this ought to come before calculus.The Chef 2016-09-24T21:29:21.687

@JamesS.Cook Good point. Maybe later as a separate question.Simply Beautiful Art 2016-09-24T21:38:21.437

3@JamesS.Cook: and, the formulations of physics of the last 100 years all require them in an essential way, be it electromagnetism or quantum mechanics.Martin Argerami 2016-09-25T01:44:17.517

1http://insti.physics.sunysb.edu/~siegel/history.html "Complex numbers don't appear till the end of quantum mechanics (itself toward the end of the book). Sin's, cos's, and complicated trigonometric identities abound, especially for electromagnetism. "Count Iblis 2016-09-25T06:53:32.887

2Because you can't draw Mandlebrot Sets without them. – None – 2016-09-25T13:20:57.347

That. Is. A very good point. Definitely need to draw my Mandlebrot Sets :DSimply Beautiful Art 2016-09-25T14:09:25.090

2They taught me complex numbers at high school, in the second year, and I've never stopped using them since then.Massimo Ortolano 2016-09-25T17:07:13.423

1@MartinArgerami: the formulations of physics of the last 100 years all require them in an essential way, be it electromagnetism or quantum mechanics Huh? I've seen more than one formulation of E&M, and none of them required complex numbers. Maxwell's equations don't, and the covariant formulation doesn't. I would actually be interested if you knew of a formulation that did require complex numbers.Ben Crowell 2016-09-25T18:36:26.750

1@CountIblis: [quoting Warren Siegel:] "Complex numbers don't appear till the end of quantum mechanics (itself toward the end of the book). Sin's, cos's, and complicated trigonometric identities abound, especially for electromagnetism." Complex numbers really are central to quantum mechanics. If freshman physics textbooks postpone them, it's to try to make the subject less scary for students, but it doesn't reflect the basic logic of the subject. Siegel's web page, which you linked to, is bitterly criticizing this fact about most freshman texts.Ben Crowell 2016-09-25T18:38:12.547

2Never judge a fish by its ability to climb trees. – None – 2016-09-25T18:55:03.723

3>

• Complex numbers make elementary differential equations much less painful. In particular, in physics you quickly get used to thinking about complex valued "solutions" to the equations which characterize circuits built out of inductors, capacitors, resistors, and power supplies. But this is still relevant in the math course. 2. I'm not sure I agree with all these points about the fundamental theorem of algebra, because if your algebra II student asks you "why is the FTA true?", you probably can't tell them the answer.
• < – Ian 2016-09-25T23:46:10.563

2@Adam: in the Netherlands they don't mention complex numbers in high school (or they didn't use to 25 years ago, anyway), and the sign of the discriminant was used to figure out if the formula has roots. Negative discriminant -> no roots.RemcoGerlich 2016-09-26T08:12:07.793

1If the only things we taught were the things we could be sure pretty much everyone would need to know, school would consist of nothing but driver eduction and basic computer skills.barbecue 2016-09-26T16:40:00.460

Also, from my experience (admittedly many years ago) complex numbers featured prominently in my calculus, analytical geometry, and diff. eqs. courses.barbecue 2016-09-26T16:41:54.840

3Just a general comment that many forms of math (euclidean geometry, trig, calculus, and yes complex numbers) didn't just teach me math - they taught me how to solve problems in a different way. Geometry taught me to justify my reasons for everything; trig taught me to convert the problem from one form to another; calculus taught me to take things to the extreme and see what shakes out; and complex numbers taught me there's always another dimension to things to take into consideration. I never use complex numbers - ever - but my brain circuits are better at solving problems because I did them.corsiKa 2016-09-26T16:56:45.293

We used complex numbers in trig/pre-calc (Euler's formula and all that), which was the coarse right after Algebra II in my HS math curriculum.pwcnorthrop 2016-09-26T18:03:24.190

@corsiKa IMO, I think you summed up the reason why we teach Algebra.Simply Beautiful Art 2016-09-26T19:42:56.140

@SimpleArt While incredibly important in learning to think, I didn't include Algebra in the list because quite honestly, I use Algebra in every day life (my home monthly budget, for example) and at work (allocating disk space for servers based on usage trends), while I haven't exactly used "opposite angles are congruent" very much...corsiKa 2016-09-26T20:20:59.090

1As an engineer by degree but not by profession, I was hoping for a good old-fashioned $i$ vs. $j$ argument somewhere in the comments on this page. Then I remembered I'm on matheducators SE.shoover 2016-09-26T21:10:46.947

This is country-dependent, but for me the schedule of high school math topics was driven by the requirements of the other subjects; and when you teach electromagnetism and alternating current in physics, you'd want students to have covered complex numbers at that point, otherwise you'd have to teach them in physics class instead.Peteris 2016-09-26T21:10:58.300

Because it is a concept you come across early in algebra and might as will define it and give it a name. Sure only a fraction will use it for higher level math.Paparazzi 2016-09-27T02:38:21.937

51

Some of your students will become engineers, and engineers use complex numbers all the time, e.g., to represent impedance. This kind of thing is by far the most common application. Complex numbers are also used in quantum mechanics.

after algebra II, they never use complex numbers until pretty much complex analysis.

I assume you mean "they never use complex numbers in a math course until..." Many of them will use complex numbers in an engineering or physics course, and they will simply never take complex analysis.

I'm also not convinced that STEM students will never see a complex number in a math course until complex analysis. For example, in freshman calculus they may learn to do an integral like $\int e^x \sin x dx$ using complex numbers. Complex numbers come up in linear algebra as eigenvalues.

When do laymen use complex numbers in real world applications?

I'm confused by your usage of "laymen" here. When I first read the question, I took this to mean non-mathematicians (i.e., 99.999% of students in Algebra II). But in your comment on Daniel R. Collins's answer, you seem to be talking about "laymen" as including bartenders and truck drivers...? Does "laymen" include engineers? The majority of the population never understands or uses any math beyond arithmetic (even if they are forced to go through the motions of taking a course such as algebra). The reason we force college-bound students to take a course like Algebra II is that it indicates that they have the ability to do abstract thought.

1I used them in Calculus quite a bit before I ran into them again in Physics, myself, but I fully agree that they are used in engineering.The Nate 2016-09-24T21:00:33.953

2If I had not understood complex numbers I would have had trouble reading engineering and scientific programs that used Fortran's complex and double complex data types.Patricia Shanahan 2016-09-25T21:48:53.733

3Lots of people don't understand or use even relatively simple arithmetic. I don't know how many times I have explained to people how to calculate the value of a percentage of a value, and I'm not a math teacher by a long shot. When calculating something like 18% of 200 is beyond them, I expect that complex numbers definitely are beyond them...Michael Kjörling 2016-09-26T12:47:21.793

1Wait a minute, where do they teach how to solve $\int e^x\sin x~{\rm d}x$ using complex numbers in a freshman calculus course? o.OSimply Beautiful Art 2017-11-29T23:44:18.613

24

We owe students a presentation of the Fundamental Theorem of Algebra -- that every nonconstant polynomial has a root; or, equivalently, the marvelous fact that every polynomial of nth degree has precisely n roots (including multiplicities). When made, it serves as a capstone and culmination of all the work that the student has done in elementary algebra. Of course, this statement can only be made in the language of complex numbers.

Complex numbers are the answer to what's the algebraic completion of polynomial roots?

Complex numbers serve as an introduction to higher-degree metric spaces and matrices (i.e., point the way towards linear algebra).

Complex numbers are used in many real-world applications (esp. any case where a 2-dimensional measurement is neatly packaged thus): Control theory, fluid dynamics (flow in two dimensions), electrical engineering (impedence), signal analysis, fractals, etc. (https://en.wikipedia.org/wiki/Complex_number#Applications).

In the Preface to his Visual Complex Analysis, Tristan Needham writes:

If one believes in the ultimate unity of mathematics and physics, as I do, then a very strong case for the necessity of complex numbers can be built on their apparently fundamental role in the quantum mechanical laws governing matter. Also, the work of Sir Roger Penrose has shown (with increasing force) that complex numbers play an equally central role in the relativistic laws governing the structure of space-time. Indeed, if the laws of matter and space-time are ever to be reconciled, then it seems very likely that it will be through the auspices of the complex numbers.

Yes, I know about the applications of complex numbers, but why do laymen need to know them when they take Algebra? I can take your entire wikipedia link to most people (in my area), and I would think that a) hardly any of them would recall what a complex number is, b) wouldn't understand any of those example applications (besides what the job is and knowing a bit about them) or c) use them because of their job, upon which I would not classify them as laymen.Simply Beautiful Art 2016-09-24T14:29:14.887

5@SimpleArt: Note that the main thrust of my answer is not about applications. That is a secondary issue. Mathematical truth is greater than workaday applications, and people have a mangled understanding of the discipline if that is not shared with them.Daniel R. Collins 2016-09-24T16:28:39.133

22

I am not a mathematician, nor a teacher, but I can give an example of how learning complex numbers in school made me fall in love with mathematics.

It was the first time I realized it was possible to be creative with numbers. "What if we just define sqrt(-1) to be something". I remember realising that I could have thought of that. It wasn't hard, it wasn't some professor somewhere thinking up rules, I could have tried that. Up until this point, math had always been a set of rules that were supplied that could never be broken. And this one made me realize that these rules can sometimes be questioned and broken. I could try something and see what happens. It could be a creative, experimental, process.

Later on, I also realised that certain "what ifs" can make calculations easier. Sometimes the r-theta space makes the numbers easier than the x,y space. Sometimes the Laplace transform is where its at. Sometimes using complex numbers is easier than not using them. Someone, somewhere, said "what if I try this..." to create the mathematical tools I use today as an Engineer.

Complex numbers is an example of how a very simple "what if" can create a new way of answering previously unanswered questions. It is an example of how math can be a creative experimental thing.

1A side effect of this creativity I soon appreciated was that mathematicians value consistency. I could say sqrt(-1) is any number I bloody well please, and be valid (as you say: creative with numbers). However, the "cannon" complex numbers have a huge body of very consistent behaviors that make it valuable. This sort of "you can say anything you like, as long as it is consistent" attitude proved very helpful in later life (as an Engineer) understanding many really "special" formulations which exist only because they are useful, and consistent.Cort Ammon 2016-09-26T01:25:07.277

13

I am surprised that nobody has mentioned differential equations. If you know about complex numbers and Euler's formula then there is a beautiful unified theory of linear differential equations with constant coefficients in terms of the characteristic equation. Without complex numbers the theory becomes somewhat ad-hoc, with different solutions depending on whether or not the polynomial has irreducible quadratic factors. This is often taught to sophomore math/engineering students who haven't had any previous college-level exposure to complex numbers.

Furthermore, complex numbers are just beautiful. If a student were to ask me why they should study them I would be tempted to quote to them the following passage from Peter Høeg's novel "Smilla's Sense of Snow"

Do you know what the foundation of mathematics is? The foundation of mathematics is numbers. If anyone asks me what makes me truly happy, I would say: numbers. Snow and ice and numbers. And do you know why? Because the number system is like human life. First you have the natural numbers. The ones that are whole and positive. The numbers of a small child. But human consciousness expands. The child discovers a sense of longing, and do you know what the mathematical expression is for longing ... The negative numbers. The formalization of the feeling that you are missing something. And human consciousness expands and grows even more, and the child discovers the in between spaces. Between stones, between pieces of moss on the stones, between people. And between numbers. And do you know what that leads to? It leads to fractions. Whole numbers plus fractions produce rational numbers. And human consciousness doesn't stop there. It wants to go beyond reason. It adds an operation as absurd as the extraction of roots. And produces irrational numbers ... It's a form of madness. Because the irrational numbers are infinite. They can't be written down. They force human consciousness out beyond the limits. And by adding irrational numbers to rational numbers, you get real numbers ... It doesn't stop. It never stops. Because now, on the spot, we expand the real numbers with the imaginary square roots of negative numbers. These are numbers we can't picture, numbers that normal human consciousness cannot comprehend. And when we add the imaginary numbers to the real numbers, we have the complex number system. The first number system in which it's possible to explain satisfactorily the crystal formation of ice. It's like a vast, open landscape. The horizons. You head toward them and they keep receding.

2great quote. is it explained why the complex numbers are needed to explain the crystal formation of ice?Jonah 2016-09-26T13:25:30.890

1No, though the book is an interesting read. The protagonist is a somewhat autistic female math graduate student living in Denmark (but originally from Greenland, which is a Danish colony) who gets involved with a murder mystery. Math isn't the focus of the story, but there are occasional references to it, with the quoted passage being the longest.John Coleman 2016-09-26T14:01:50.873

12

I think the fundamental tenant of this question is simply false. Here are some of the many encounters that undergraduate college students have in my classes:

• In Calculus II, as an application of power series, we discuss Euler's identity: $$e^{i\theta} = \cos(\theta)+i\sin(\theta).$$
• Still in Calculus II as an application of the preceding example, we derive formulae for the trig expansions of $\sin(\alpha+\beta)$ and $\cos(\alpha+\beta)$. I further mention this application as the "easy way" to derive these identities any time I need to use one in a class.
• In Calculus III after polar coordinates, we discuss the $n^{\text{th}}$ roots of a real number, i.e. the solutions of $x^n-a=0$.
• In linear algebra and differential equations, we discuss the interpretation of complex eigenvalues.

6

Most people are not going to use most of what they use in high school. High school (in the US, at least) is about building a broad foundation for students to be able to jump into any specialization when they go to college.

Some examples of things I haven't had to do since high school:

• Name the 5 (I think 5, maybe it changed?) biological kingdoms
• Diagram a sentence
• Explain the meaning of the scaffold in "The Scarlet Letter"
• Discuss the differences between the Indus Valley Civilization and Mesopotamian Civilization
• Etc.

(Aside: I don't consider having had taken those courses a negative by any means, just that they aren't "useful" in my line of work, and, I assume, in most people's daily lives)

As far as complex numbers explicitly, they show up again in Trigonometry, which immediately followed Algebra II in my HS curriculum, and appeared from time to time in succeeding courses in math and physics. However, I really had to start using them in an application setting was in Control Systems, which was junior year of college. By that point you don't want to want to waste time explaining what complex numbers are, you want to be able jump in and use them.

Even if you're not proficient with complex numbers at that point, you are comfortable with their existence, which is better than getting to that point thinking sqrt(-1) is meaningless and learning that your whole mathematical life is a lie.

1Well, the Indus River Valley was very much watery, providing a good basis to their civilization. They even had a sewage system. Mesopotamia on the other hand...Simply Beautiful Art 2016-09-26T20:02:38.913

4

There are many subjects in High School that are taught which some will never use. Accounting, Drama, Band, Woodshop etc. These are entire subjects that students learn and most will never use. Does that mean we shouldn't teach them? Of course not! These are valuable subjects and just having exposure to the subjects will cause some to pursue them and others to appreciate them. It makes them more creative and able to do different things or at least not afraid to try.

I don't believe High School is only about giving students things that all of them will use. Students need to be exposed to subjects to figure out what they do want to do in life, whether it's college or trade school or some other path.

Complex numbers is just one part of mathematics, and some will be intrigued enough to pursue such topics later. Maybe THAT's the subject that really drew them in. No, you won't use complex numbers in shop or accounting (although some firms do use imaginary numbers -- nudge nudge wink wink), and at that age their brains are still developing and exposure to areas they find more difficult is actually beneficial to them. How many people do you know that hated to practice piano and didn't and now regret it? How many DID practice and are grateful for it now?

Doing hard things, making them think is good for them both for character and brain development. These are established results from both psychology and neuroscience. While I wouldn't stuff High Schoolers full of these types of things, complex numbers are important enough that exposure is necessary for those who will need it later, and common enough in many various fields that merit at least some attention.

If instead of complex numbers you were teaching algebraic topology or string theory, I would be in complete agreement.

1True, but there is an opportunity cost. For every topic in math that you do teach, there's also a topic that gets crowded out of the syllabus.200_success 2016-09-26T19:19:06.950

2This reminds me of when I first join SE. I was intrigued to understand $2^i$, and so I learned Calculus on my own using math.SE.com as my crutch. :)Simply Beautiful Art 2016-09-26T20:00:49.167

1"subjects they'll never use" "accounting" You just made the manager of the local Money mart smile.corsiKa 2016-09-26T20:24:04.183

4

Just another question from a math guy showing his ignorance of math as a service course.

1. 2nd order diffyQ with constant coeffiecients (most important diffyQ for applications) has complex roots in the characteristic equation. [IOW very standard part of standard ODE course; OFTEN part of even a second semester calculus course during the diffyQ survey...was when I took AP, part of Granville, etc.]

2. Complex numbers are normal part of AC circuits. REQUIRED part of survey course for ALL engineers (not just EEs). Sometimes part of calculus based intro physics as well. Also for physicists needed for E&M junior year course. (Note: all these uses occur even if complex analysis is NEVER studied).

3. For college chemistry solution of the quadratic equation is normal part of many equilibrium, rate, and stoichiometry problems. While real roots are only ones usually needed, having familiarity with complex versus real makes it easier to grasp for students when we exclude some solutions of the quadratic.

4. Solutions of harmonic oscillator (i.e. the second order constant coefficients diffyQ above) is normal part of required courses for junior year quantum (all physics majors), P-Chem (all chem majors), weapons systems controls (all military academy students), and probably a bunch of required courses for engineers (fluids, etc.) which I don't know in detail.

May I ask what you mean by "service course"? I am not familiar with the term.Simply Beautiful Art 2017-11-29T23:36:55.993

A class that a department teaches that is needed by non-majors. For math departments this would be calc 1, 2, 3, ODE and typical probability and statistics. The vast majority of students will be non math majors. Freshman chem is a service course too (vast majority will be non chemists: medical types or any science/engineering). The point is that having a lens of "needs of the math major" is obtuse since they are a tiny minority of the audience...and it ignores needs of the non majors who are vast majority.guest 2017-11-30T00:54:29.120

Good luck with college. 'zoics (physics), EE, CS all good choices. Lots of other cool things too. If I had it to do over again would do geology or geophysics. Sciencey but messy enough so that nobody really understands it. And a connection to industry. And to outdoors. But whatever you like.guest 2017-11-30T00:58:05.327

I think under math, some sort of applied math either the weird diffyQ stuff or even just operations research or statistics would be cool. But if you love pure math, that is fine. (Someone has to...LOL.)guest 2017-11-30T00:59:30.687

3

As Ben Cromwell mentioned engineers and physicists use complex numbers. His example of impedance is a common one but here are some another examples along with easily recognized applications.

Signal and Image Processing: Analzying images and audio signals is commonly done using a technique called a Fourier transform. https://en.wikipedia.org/wiki/Fourier_transform Complex numbers are crucial to this and it has a variety of applications. For example, a lot of operations in Photoshop(Gaussian blurs, interpolation) typically use Fourier analysis in some way. When your cell phone transmits a signal, it encodes the outgoing signal on a "carrier" wave in a process called modulation. This also requires complex numbers. DJs help improve the quality of their sound by compensating for something called the "frequency response" of the room, the frequency response is a Fourier transform of the room's acoustic properties.

Controls Theory: Controls theory is closely related to signal processing(using a more generalized form of the Fourier transform called the Laplace transform) and it used in a lot of robotics and automation. Cruise control and aircraft autopilots all use controls theory.

I should also point out that I studied these topics and college and ran into complex numbers as soon as second semester, freshman year.

These are just the areas I'm very familiar with but there are certainly many more. Any of the engineers working on these projects need complex numbers. As for the rest of the population; there are numerous other responses suggesting that not everybody will use everything we learn in high school, so it doesn't seem fair to select complex numbers as opposed to anything else.

2

I think your premise is flawed "Why do we teach students about complex numbers if most will never reach a course that uses them?"

As soon they get a calculator they will test $sqrt(-1)$. They are going to see and answer and ask you what it means.

Also, " When do laymen use complex numbers in real world applications?" Check out this answer.

2Welcome to the site. Could you summarize the answer from Dr. Math?Dag Oskar Madsen 2016-09-24T20:57:20.233

5The majority of calculators, even scientific/engineering ones will give you an error, not $i$.Ruslan 2016-09-25T06:58:42.167

@Ruslan yes...what does that error mean.Jp McCarthy 2016-09-27T11:47:24.437

@JpMcCarthy is it a question? The error means that the result can't be represented/approximated as an N-digit real number. You get the same error on overflow or division by zero.Ruslan 2016-09-27T12:35:45.650

@Ruslan --- yes I know this: I am saying that the students will ask what the "error" meansJp McCarthy 2016-09-27T14:33:13.353

@JpMcCarthy I suppose students do know that no real number has negative square. This has been known to people far earlier than complex numbers were invented.Ruslan 2016-09-27T15:48:22.897

1

Speaking of complex numbers, why do we not teach them Geometric Algebra, or even appreciate it ourselves? In there, we can give a geometric meaning to imaginary numbers, as oriented segments of a plane (bivectors). We also can use them to e.g. rotate vectors conveniently! They are very easy to handle! GA gives a lot of new connections btw physics and math too, like new interpretations of Pauli and Dirac matrices. I was blown away when I first stumbled on it! I'm studying physics at master level, and nobody told me about this...

0

I highly suggest you obtain a (used) copy of the CME Project Algebra 2 (orange) textbook and look at its section on complex numbers. They should be thought of as very geometric objects - they apply to computer graphics, optimization, and all sorts of other areas of mathematics.

Applied Mathematics (which is what my graduate degree is in) is not just about "the real world." It is about using the power of mathematics to model whatever scenarios come to mind. You don't have to model "real world" things -- you can do a linear programming situation based on goblins and unicorns, for example. You can model social interactions at a party. You can use complex numbers to model/design PacMan. Whatever you want. One person's application is another person's dull bore. That's why it's not about the -specific- applications you are teaching. You should be teaching a mindset, an approach, a worldview. I find all of the examples listed using physics as reasons for complex numbers completely exhausting and couldn't care less; however, I don't need to care about those -- I need to understand why they exist and why physicists must employ complex numbers to model the situations they encounter. THAT is what we're teaching, not just specifically complex numbers to find zeroes.

I'm still with most people that I don't agree with your assumptions nor what it means to be a "laymen."

-2

A complex number is not a number at all, it is a vector representing a magnitude and a rotation about the origin in Cartesian coordinates. It is the root of the equation x^2 + 1 = 0. Nothig more nothing less

3This does not answer the question nor do I feel it mathematically accurate. To say complex numbers are nothing more and nothing less is inaccurate.Simply Beautiful Art 2016-09-27T19:52:25.487

8

There was a time when some scholars said that irrational numbers were not numbers at all. There was a time when they said that negative numbers were not numbers at all. There was a time when they said that zero was not a number. Without abstraction and generalization, mathematics never would have advanced beyond the tally stick

james large 2016-09-28T02:30:55.357