# (How) Do American undergraduate math programs teach complex numbers?

What kind of exposure to complex numbers can you expect in mathematics majors at American colleges?

I teach at a very large public university. It occurred to me that it is possible to graduate in their math majors with the exposure to complex numbers being exactly two lessons in calculus.

This seems a bit odd to me as far as math majors are concerned. For example, complex eigenvalues are never mentioned at my university.

Is this normal by US standards?

• Probably, and it's a shame. Gauss would be sad. Apr 30 '20 at 3:41
• I attended a public university in america and for the math major it was required to take a full semester-long course on complex analysis; covering usual things like cauchy-riemann equations, cauchy integral formula, analytic continuation and some things briefly like open mapping theorem, connection to PDEs Apr 30 '20 at 5:09
• Not an answer, but pretty relevant nonetheless: Common Core State Standards for complex numbers. Jun 18 '20 at 18:36
• Ergo, most US college students today in math major courses will have seen complex numbers in that sense. Whether they see them in a future course depends a lot on the emphasis of the major - a lot of math careers today have nothing to do with electrical engineering or the various sources of complex eigenvalues, so it's possible to justify them not showing up in the curriculum. I doubt that any faculty member plans a major without complex numbers, though, and personally I lament this. (We definitely talk about complex numbers in my math history course!) Jun 18 '20 at 18:37
• No clue about the US. In Spain we had a good amount of classes in high-school dedicated to complex numbers (basic arythmetic, polar form and a few more things). Then at college we went a bit deeper into them on the first year in an analysis class. In the 4th year, there was a mandatory subject dedicated exclusively to complex analysis Jul 13 '20 at 12:11

The following is anecdotal, but based on experience as a student and instructor in American high schools in three states, as well as my undergraduate and graduate experiences.

• High School: In the high school curriculum, complex numbers first appear in algebra courses. The usual curriculum first introduces complex numbers as a way of interpreting the quadratic formula when the discriminant is negative, i.e. as a way of giving meaning to the result $$ax^2 + bx + c = 0 \iff x = \frac{-b\pm \sqrt{b^2 - 4ac}}{2a}$$ in the case that $$b^2 - 4ac < 0$$. Good instructors will interpret this graphically, and note how the graph sits above or below the $$x$$-axis if the discriminant is negative.

Perhaps motivated by this, there is typically a short chapter or section (1–2 weeks of lecture, maybe) on the complex numbers. This might consists of a discussion of how complex numbers may be added, subtracted, multiplied, and divided, but likely not much more than this.

It is also worth noting that the complex numbers are often treated as some kind of mysterious, vague, not really useful or "real" system. They are a "trick" used to solve certain problems, but are only really useful as a trick. This is, of course, incorrect, but is reflective of the fact that most high school teachers in the US have not taken much mathematics beyond calculus (e.g. it is possible to teach high school mathematics with two semesters of calculus, a handful of watered-down upper division courses in mathematics, and not much else).

The high school curriculum usually tops out in a course titled "Precalculus" (or something similar). This course is typically a hodge-podge of mathematical concepts: real functions and the associated notation, some discussion of limits and continuity, basic trigonometry, exponential and logarithmic functions, simple linear algebra (e.g. vector and matrix operations), and so on.

The complex number system sometimes comes up again in this class, often in the context of vectors. After a discussion of trigonometric functions, vectors in $$\mathbb{R}^2$$ are introduced, and students are shown how to give vectors in their polar form. At this point, you might see the complex plane introduced, and the students might be given a few exercises converting back and forth between the Cartesian and polar forms of a complex number, i.e. $$z = a + ib = r \big(\cos(\theta) + i\sin(\theta)\big).$$ Some instructors might go a little deeper than this, but I think that such discussion is uncommon.

• Lower Division College Classes: The complex numbers typically don't come up very much in lower division college classes, which include precalculus (which is generally much the same as the high school course, but with an accelerated calendar), calculus (of one and several variables), linear algebra, and differential equations. Some instructors might discuss the complex numbers in a differential equations class, in the context of complex roots of the characteristic polynomial. However, the discussion is usually elided, and students are instructed to write their solutions in terms of trigonometric functions. For example, the differential equation $$u'' + u = 0$$ has characteristic polynomial $$r^2 + 1 = 0$$, which has roots $$\pm i$$. Hence a general solution to this equation is of the form $$u(t) = C_1 \mathrm{e}^{it} + C_2 \mathrm{e}^{-it},$$ where $$C_1$$ and $$C_2$$ are arbitrary constants (which will ultimately depend on some initial condition). However, via the principle of superposition (linear combinations of solutions are also solutions), the existence of a minimal spanning set of solutions, and the facts that $$\cos(t) = \frac{\mathrm{e}^{it} + \mathrm{e}^{-it}}{2} \qquad\text{and}\qquad \sin(t) = \frac{\mathrm{e}^{it} - \mathrm{e}^{-it}}{2i},$$ it is possible to rewrite the general solution as $$u(t) = k_1 \cos(t) + k_2 \sin(t),$$ where $$k_1$$ and $$k_2$$ are constants. Again, it is normal to spend very little time on the complex analysis, and instructors often assume that students already know the materials.

The complex numbers might also come up in linear algebra in the context of complex eigenvalues. However, again, the discussion is often very brief (if it exists), and most of the examples students are given involve only real eigenvalues. As the main goal of lower division linear algebra is often to get to the diagonalization of a square matrix (rather than the Jordan canonical form, or, better yet, singular value decomposition), complex eigenvalues are often ignored as something too difficult to deal with.

• Upper Division College Courses: There are upper division classes where complex numbers become a major focus of the course or are important examples: complex analysis (duh?), Fourier and/or harmonic analysis, proofs-driven courses in differential equations and linear algebra, abstract algebra, etc. However, these courses are typically elective courses for mathematics majors. The general assumption in these courses is that the students are (1) bound for graduate school, probably in mathematics or some closely related field, (2) already familiar with the complex number system from previous courses (which courses? ¯\_(ツ)_/¯ ), and (3) relatively mathematically mature and therefore capable of filling in gaps in their knowledge on their own.

Moreover, upper division courses in which a more comprehensive treatment of the complex number system might appear (e.g. complex analysis or differential equations) are typically electives, and not required even for mathematics majors. It is therefore very common for students to complete an undergraduate degree in mathematics knowing next to nothing about the complex numbers.

In short, the answer to the original question is "No, based on my (anecdotal) experience, it is not at all uncommon for US institutions to graduate mathematics majors who have had very little experience with the complex number system."

• It has been long enough since I got my degree that I don't remember how much we focused on complex numbers in my math courses, but in my engineering courses we did A LOT of complex analysis. It seems weird to me that math majors wouldn't have as much exposure. I know that by differential equations when we worked on Fourier transforms, there probably wasn't anyone left that wasn't already confident with complex numbers. May 1 '20 at 20:13
• I'm no math major (I understood through calculus, I could follow the steps for a couple of classes beyond that but never really got it) but your first two points certainly match my experience. I was familiar with them before college but we never dealt with them in college. May 2 '20 at 2:39
• I think the "elided" is important. We don't have time to "teach complex numbers" so we sort of assume people know enough to get by. But that isn't necessarily the same as actual "exposure" to complex numbers. Jun 18 '20 at 18:40

At my institution we do not discuss complex numbers at all in calculus, but we assume that students are somehow familiar with them in differential equations and linear algebra (to analyze real-linear mappings with complex eigenvalues). A study of single variable holomorphic functions is an upper level course, and most math majors will graduate without ever meeting a holomorphic function.

• Or at least they won't know they've met one. Apr 30 '20 at 17:04

I teach physics and occasionally a little math at a community college in California. My students come from all over the place, so I think their exposure to math is somewhat of a representative sample of what kids in the US learn in high school and the first couple of years of college. In our three-semester physics survey course for STEM majors, I review complex numbers with my students in both the second semester electricity and magnetism class (so they can use them for complex impedances and such) and in the third semester (for use in the baby quantum mechanics that we do).

I say "review" because I have never seen a student who had had no previous exposure to the topic. I think they must see it in 9th grade algebra when they learn the quadratic formula. Also, the catalog description for my school's trigonometry class says that it includes de Moivre's theorem, clearly implying that they are already expected to have seen complex numbers before that point.

However, their exposure seems always to have been very shallow and poorly motivated. They do not recognize notations for magnitude and argument, and they don't understand the interpretation of multiplication and division in terms of the polar representation. A calculation like $$\operatorname{arg}(1/(1+i))$$ takes them a long time to figure out. They tend to cling to the cartesian representation even when it's the wrong tool for the job.

I teach at a very large public university. It occurred to me that it is possible to graduate in their math majors with the exposure to complex numbers being exactly two lessons in calculus.

This seems a bit odd to me as far as math majors are concerned. For example, complex eigenvalues are never mentioned at my university.

Is this normal by US standards?

So no, in my experience this does not seem normal or even possible here in California, because the state tries to make every kid take algebra in 9th grade, and they get some basic exposure to complex numbers there.

It also seems weird to me that students could graduate as math majors without ever seeing complex eigenvalues. This could not have happened at my undergrad school (UC Berkeley). I still have my textbook from my lower-division linear algebra course there, and it barely mentions complex eigenvalues. However, a math major there is required to take a core of four upper-division required courses, which are analysis, linear algebra, abstract algebra, and complex analysis. Three out of four of these deal with complex numbers, and the upper-division lin alg course would certainly cover complex eigenvalues. Maybe there are vast differences between more and less selective schools in the US in terms of their requirements for an undergraduate math degree.

• Different schools have different focuses for their math majors. The one I went to had a strong mechanical-engineering emphasis, so more differential equations and numerical methods, and fewer complex numbers and symbolic techniques.
– Mark
Apr 30 '20 at 22:37

The basics of complex numbers are really covered in high school algebra. (Basics means easy/simple/prioritized, though.) Complex numbers come up occasionally in other topics (as needed, often real numbers is all that is needed). But obviously roots of the second order linear homo ODE is one area in math. As is reactance versus resistance in physics and EE (ELI the ICE man).

I guess in a way it is sad that math majors don't have a required unit in complex analysis, but you have to realize it is a BS degree, not a grad degree. And there needs to be some space for kids that are NOT going to go to math grad school. A LOT OF SPACE for kids like that. They are more the norm than the exception.

And we can clearly think of other topics that in some way feel like they should be required (e.g. a PDE class), but that are not a strict requirement, but an elective instead. But as always with these sorts of questions on MESE, what I find lacking is people saying what they will get rid of, when they consider to add a requirement.

If I look at the math courses at USNA, which of the required math classes should I get rid of if I make complex analysis required? The one baby real analysis class? Probability and statistics? I don't think so. I think a math major lacking a basic prob/stat course is a bigger gap than missing complex analysis.

https://www.usna.edu/MathDept/_files/documents/majorMatrices/SMA.pdf

But realistically a lot of them will probably select a complex analysis as one of their electives anyhow. Not all, but many. And let some freedom exist also. Not everything needs to be required.

• What course to sacrifice for complex analysis ? Easy, any non math course. If it must be techincal, get rid of a chemistry or physics lab. Those are a joke. Some "universities" force really elementary computer courses to be taken by everyone. Kill that course. Honestly, a good look at nearly any degree completion plan will reveal that MOST of the required nonmajor courses could be skipped with little lost. In Europe, with a 4 year degree you get way way more technical credits. Undergrad in the US is an epicallly bad deal for people who actually want to learn. Apr 30 '20 at 3:48
• It would be stronger if you could propose changing out a math course. As it is, your "easy" response has a dependency on changing the entire structure of US colleges to be majors only institutions. Note also, that this ignores the differences in US high schools versus, for instance, German gymnasiums in terms of tracking and prep and the like. So...add changing high school to the "easy" bucket. ;-) Apr 30 '20 at 5:09
• Re, "...basics of complex numbers..." My own personal anecdote: My U.S.A. high school taught me that complex numbers could be solutions to real polynomial equations, and it taught me how to add, subtract, multiply, and divide them. They taught me that complex numbers could be represented in polar coordinate form, but left me with no clue as to why anybody would want to represent them in that way. When I asked whether there were any practical applications, one math teacher suggested that maybe they had some use in aeronautical engineering, but he couldn't say any more than that. Apr 30 '20 at 11:52
• But your teach should have mentioned EE, not AE as the classic application area. Apr 30 '20 at 12:06
• As someone with an electrical engineering degree, I strongly appreciate this answer—we used “j” all the time. Weird to think it might have been more important for us than to an actual math major... May 1 '20 at 2:38

My background includes attending one liberal arts college for undergrad; graduate school at a relatively small university; and two years as a mathematics professor at another liberal arts college.

While I wouldn't describe only two lessons as "normal," neither would I describe it as surprising. What I would expect is a few weeks of complex numbers between a couple of early or intermediate courses, typically ones labelled "Discrete Math", "Linear Algebra", or "Differential Equations" or the like. Then there will be an advanced course in Complex Analysis, but it will probably be optional and not necessary for math majors.

I'm actually a bit surprised that complex numbers were mentioned in a calculus class; I'm used to the only mention of them at that level being something of the "and square roots of negative numbers is something we will save for a more advanced course" variety.

• In second semester calculus if you do a lot of Taylor series, it's pretty natural to at least mention how the series for the exponential and trig functions are connected this way, so it's not totally surprising. But probably most people are not doing much serious with it in first-year calc, as you say. Jun 18 '20 at 18:39