Why do we teach complex numbers?

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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.

Answer 5

I think the fundamental tenet 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.

Answer 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.

Answer 7

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.

Answer 8

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.

Answer 9

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.

Answer 10

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."

Answer 11

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...

Intro here: https://www.youtube.com/watch?v=srwoPQfWWS8&list=PLLvlxwbzkr7igd6bL7959WWE7XInCCevt

Answer 12

When I read the original question, I focused mostly on why in algebra 2 do we teach complex numbers. And, as I am preparing to teach algebra 2 after several years of having taught other courses, I also wonder if it is the right course to introduce that topic. Of course, I do not disparage the importance and the beauty of the complex set, and when I teach it at the pre-calculus level, I do spend some time (not enough!) to talk about the geometry of the complex plane where I find the beauty of the complex numbers particularly clear. (I also present the complex plane simply as the real plane with a special algebra structure that allows you to add, subtract, multiply, and divide POINTS!) I also love to introduce complex numbers with a simple question that only involves natural numbers but whose solution is so quick if you play with sqrt(-1). I ask my students to express the product of the sum of two square numbers as the sum of two square numbers (e.g. find a and b such that: (12^2+7^2)*(9^2+17^2) = a^2+b^2.) I give them time to think about it while on a corner of the board I quickly solve it. I then scratch my work and simply give them my numbers. I find to be a fun historical analogy with the way 15th century Italien mathematicians were "secretly" using sqrt(-1) to solve cubic equations. But why in algebra 2? Last time I taught that course, I did cover it, but now, as I try to have a coherent and logical understanding of the algebra 2 topics, and especially how the curriculum should flow, I perceive complex numbers as a marginal topic that may hinder this logical flow I'm seeking. In my view, better left for the next course, pre-calculus. (With a greater emphasis than is usually given to that fundamental topic.)

Answer 13

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.

Answer 14

A few reasons for the importance of using complex numbers. I will split them following the age of the students:

Highschool level (in Romania we had a whole year studying complex numbers, 10th grade)

• just like negative, rational, irrational numbers are introduced so that we can have the right context to solve various equations, complex numbers help solve all polynomial equations. In particular all quadratics with real coefficients, all cubics, all quartics.

• many aspects in planar geometry can be turned into algebraic manipulations using complex numbers. Some geometry problems become exceptionnally simple when turned into complex context: (for example proving that the midpoints of a quadrilateral form a parallelogram). Basic geometric transformations (translations, rotations, homotheties) are easily described using complex numbers.

University level (in addition to topics mentioned in the other answers)

• Fourier analysis, signal processing, solving ODEs.

• Whenever a polynomial equation needs to be solved, we want to be able to talk about its roots, without worrying about their existence or not. Therefore, students should be aware that the right context, where every polynomial of degree $n$ has $n$ roots, is the one of complex numbers.

Answer 15

Why do we teach complex numbers?

because $\mathbb{C}$ is in some sense inevitable, naturally inevitable even, and it's also where the magic happens...

the usual way to see the 'progression' of the number systems is to start with $\mathbb{N}$, then move to $\mathbb{Z}$ and $\mathbb{Q}$ to 'do more arithmetic/algebra', then to $\mathbb{R}$ to 'do calculus' and lastly to $\mathbb{C}$ to 'complete the algebra' and be marvelled calculus still quite works; but, instead one could well pass from $\mathbb{Q}$ to $\bar{\mathbb{Q}}$ in their atempt to 'complete the algebra' $-$ and note $\mathbb{Q}[i] \subsetneq \bar{\mathbb{Q}}$ $-$, only to realise they still can't do calculus, then taking the metric completion to arrive at $\mathbb{C}$, again: there is not much (nothing?) more one could ask of a field than being algebraically closed and metrically complete, and $\mathbb{C}$ 'naturally' appears as a (even the, pace p-adics) solution to both problems

so much for inevitability [there's more here!], where's the magic? well, admittedly this falls under "very far in the future" and specially "not for the layperson", but i think it's worth noting: besides one time differentiable functions already being analytic + the nullstellensätze (much better than the real case), there are some very powerful results enabling one to bring analytic tools to algebraic problems and vice versa, which codify/reify the previous intuition that in $\mathbb{C}$ the algebra is very much intertwined to the analysis