when should we teach basic complex algebra?

I am teaching Differential Equations this semester. The pre-req is Calculus II, not even Multivariable Calculus. I made my peace with that, can't fight with the big-shots of the department. So, I have been teaching stuff from Multivariable calculus on a need basis (partial derivatives, chain rule, etc.) I also spent one lecture to teach basic matrix algebra, determinants. Then, evidently, I need a little bit of complex algebra as well, so I did that too. Now, I am teaching higher order differential equations, so kids need a bit further explanations for finding complex roots. Well, these kids will eventually take linear algebra, and multivariable. And it is the department's fault not to require these as pre-req. But what about complex algebra/variables? There is no calculus course, single or multivariable, where we introduce complex numbers and complex algebra. Why? What is the fix for this? I mean I am checking calculus books, syllabus from different universities and their calculus courses, no complex algebra whatsoever. How can this be?

• A lot of the stuff that is often taught in US intro Diff EQ classes can be done without complex variables or multivariable calculus. For example, a basic introduction to the terminology (i.e. classification of DEs), techniques for dealing with separable equations, the method of integrating factors, solutions to exaction equations, the Laplace transform, and so on. It might be better to consult with the people in your department that established the curriculum and have taught the class before. It is possible that you are not teaching the class that the department is expecting you to teach. – Xander Henderson Apr 4 '18 at 1:52
• It is my research area, and my department members have no clue about how to teach it. Just like in many universities in USA, they think DE is nothing but a bag of tricks. Well, it is not. – NezPerce Apr 4 '18 at 2:16
• But most intro level classes are usually taught (in the US) as bag-o-trick classes. I totally get where you are coming from---the basic pre-calculus and calculus curricula taught at the several universities that I have had contact with make me cry. But you are not being asked to teach a research level class or even an advanced undergrad class. Again, you probably should discuss this with your department, rather than denigrating them to a bunch of strangers on the internet. – Xander Henderson Apr 4 '18 at 2:38
• I can see why you are frustrated. You clearly believe that there is only one way for the material to be taught, you are openly disrespectful of your department (your colleagues "have no clue"), and you are complaining about fixing "the stupid." With that kind of attitude, you are likely doomed to be frustrated and unsatisfied for quite a while to come. One last time, I would strongly advise that you communicate with your colleagues in an open-minded and honest manner about what the goals of the class are and why the material that you are teaching is not prerequisite. – Xander Henderson Apr 4 '18 at 2:59
• Nez: User @XanderHenderson is trying to help you by taking on their time to explain what they think your situation is. No need to turn your anger about the academic system you are in, towards them. – Did Apr 4 '18 at 21:13

Let me give you a slightly different perspective. I also come from a European background and have had the chance to TA and teach undergrad courses in the US. As a caveat I have not taught DE, but have taught many calc classes including multivariate and have graded for some DE.

First I understand your utter frustration if you come from Europe and are used to the European approach to teaching and the European undergrad students the US is a very very rude awakening. I remember being at one of my first calc I lectures and seeing the professor add infinite sums by terms without as much as a nod to convergence and present it as an ok thing to do and it was actually physically painful as I was clenching my fists in frustration that this is taught to someone. So believe me I understand that you want to teach the right stuff in the right way.

Second Xander is right. Noone is asking you to teach what you intend/try/are teaching. The expectation of the department and of the students and maybe most importantly of the follow up courses is that you will teach the students bags of tricks. The courses in the US do not generally correspond to similarly named courses in Europe.

Everything in the calculus sequence including DE is mostly about tricks and learning to handle the very basic formalism. Calculus students may be taught epsilon-delta proofs but 90% of them do not understand them and they are not really tested on whether they do. In the same way they are not taught real derivatives, no one cares about boundary conditions and pathologies and they are essentially never asked to think for themselves in the problems. Not really think anyhow I'm convinced that all the finals I've seen could be solved by a an easy program with minimal notational standardization.

There are reasons for this though (not necessarily good ones mind you). The students are generally not math majors and aren't going to continue on in math. Many of them have a fairly shaky background in math alltogether, I remember in calcII classes still consistently having students make elementary level errors such as sum of squares is square of sum and even splitting fractions in wrong ways.

All together if you try to teach complex algebra, (and other stuff you mentioned) along with DE you will be doing a disservice to your students. They will be expected to know the things that the department actually wants you to teach them and due to the extra stuff they will no have properly learned the bag of tricks that teachers further along will expect of them.

Complex algebra will be taught later on to those that continue down to math majors or physics majors (and possibly not even those. In some colleges/Universities many math majors require no advanced complex algebra course at all).

In the United States, high school students typically encounter complex arithmetic and algebra for the first time in the context of an Algebra 2 course, and then revisit the topics the following year in Precalc. (For most students this would be either 10th and 11th grade, or 11th and 12th grade.) Complex numbers also appear on the SAT and ACT.

The problem you are having is not that students haven't seen complex numbers before -- it's that they haven't seen them or used them recently. If you are teaching Diff Eq, most of your students are in their second year of college (if not later), which means that it has been roughly two years since they've had to work with or use complex numbers. As you point out, they do not play a role in single-variable Calculus -- but why should they? Calc 1 and Calc 2 are the study of functions of a single real variable; it's hard for me to even think of places where I could make use of complex numbers in those courses, even if I wanted to.

• +1 for they haven't seen them or used them recently Although I just wrote a comment elsewhere about complex numbers coming up when Euler's formula appears in the Taylor series material in Calculus 2, some teachers might quickly pass over this with a brief mention of $e^{\pi i} = -1$ and skip the typical supplementary problems making use of this that textbooks tend to have in the exercise sets. – Dave L Renfro Apr 4 '18 at 14:42
1. Complex roots of the characteristic equation are no big deal. Kids have seen that plenty with the quadratic formula in high school algebra.

2. I worry from your question that you are dragging the course down too much by feeling the need for prereqs and then diverting to quasi teaching them (matrices, partial derivatives, complex algebra, etc.) The vast bulk of ODEs can be taught without getting into that stuff. And the course is so full that you really have a hard time covering everything in a semester anyhow. Even when some tiny aspect of one of these things, say a determinant, is used, and it is not necessary to even cover that in a 15 week ODE course, you should just do the minimum, rather than seeing a need/opportunity to divert the class.

• Yes, students know quadratic formula, but it is kind of useless when you are dealing with higher order differential equations, isn't it? you can't even get past first order differential equations if they don't know chain rule for functions of several variables. – NezPerce Apr 4 '18 at 2:18
• You may know a lot about ODE theory. But you are very new to TEACHING ODEs and have not thought deeply about the purpose of a service course. Instead of trying to invent your own iconoclastic course, teach a standard course and learn something (about teaching, kids, time management, why the standard course was designed the way it was, etc.) before you decide to do it all different. – guest Apr 4 '18 at 3:45
• NP: You are just schmipfing about Americans. How many 4th order ODEs do you really need to deal with. The main one is second order. Spend time on the different cases (under damped, over, etc.) and on forcing functions. That is main ODE needed for control systems, physics, chemistry, etc. Do your kids have that one down? If they do, fine, spend time on teaching them to take the square root of i (personally, I don't remember how to do it either). If they don't have the second order down and you teach them to take fourth roots of negative 1, than you are not prioritizing. – guest Apr 4 '18 at 19:34
• Very cool, Dave. But if NexP hasn't taught the kids how to handle under/over/critical damped equations, then he is misallocating resources. And by "taught the kids", I mean them showing they know the stuff a year later when they are in my control systems class...not saying he covered it...but them knowing it. If he spends time on linear algebra, complex algebra, or calc 3 instead...he is not doing his job and the physics/mechE/chemistry departments have a valid complaint. – guest Apr 4 '18 at 21:06
• @guest That's what I was trying to point to in my answer. Emotionally I'm 100% behind the OP, I hated the way calc or god forbid business calc was taught (students coming in with a background that didn't really let them graph functions were integrating by the end of the semester!) But the point is these are service classes and I was hired to do a job with some description and the whole rest of the university expected the students to leave with a very specific amount of knowledge. I had to do my best to impart that knowledge even if I didn't agree with the methods that had to be employed. – DRF Apr 5 '18 at 5:17