The Swashbuckling Physicist's Guide to Complex Numbers

It sounds very unusual that students can take three modules in calculus and yet have never come across complex numbers. They're a very difficult thing to avoid. In fact, it is essentially impossible to avoid them in any introductory course on linear algebra, as they _will_ appear when discussing eigenvalues(unless ALL of your matrices are both symmetric AND positive definite, which really couldn't be the case unless you were cheating).

I would recommend introducing Eulers formula via a Taylor series. They should be familiar with those. It takes about three or four lines and is fairly digestible.

It sounds very unusual that students can take three modules in calculus and yet have never come across complex numbers. They're a very difficult thing to avoid. In fact, it is essentially impossible to avoid them in any introductory course on linear algebra, as they _will_ appear when discussing eigenvalues(unless ALL of your matrices are both symmetric AND positive definite, which really couldn't be the case unless you were cheating).

I would recommend introducing Eulers formula via a Taylor series. They should be familiar with those. It takes about three or four lines and is fairly digestible.

People don't get this in high school? I remember it being a pretty big chunk of precalc....

Sounds like the high school and university level math + physics curriculum been watered down significantly, or at least rearranged in a very uncoordinated erratic manner.

What onymous and JC said.

Complex numbers were covered in my Algebra 2 class. This is the math class that typical college-bound students took in 11th grade and smarter-than-average students (the ones who went on to take calculus while still in high school) took in 10th grade. I could see teaching calculus without complex numbers, but you would definitely need them again by the time you reach differential equations (is that part of Union's calculus sequence? At my undergrad school, it was considered a separate course which was still required for physics and most engineering majors.) or linear algebra.

But then, I'm just an old guy--back when I was in high school, Ronald Reagan was an actual man (who happened to be in the White House at the time) and not a myth.

They don't teach it in Calculus I-III, Vector Calculus, ODE 1 or Linear Algebra I-II at my university, and we follow the regular semester schedule

They don't teach it in Calculus I-III, Vector Calculus, ODE 1 or Linear Algebra I-II at my university, and we follow the regular semester schedule

It is quite possible to have seen complex numbers without having seen complex exponents. I think I ran into complex numbers in HS freshman year when we did quadratic equations, and then again in HS senior year where we studied complex multiplication as rotations in the complex plane. Neither covered complex exponents.

Complex exponents would have been covered in a junior-year university course called, I believe, Complex Analysis. But it was optional even for most math majors.

I love this web page about the Imaginary Unit and some games it plays:

http://mathworld.wolfram.com/i.html

The mathematician's son referred to in the joke at the end (to which ex-astrophysicist Eric W. Weisstein added a follow-up punchline) was my son...

I think it's highly variable. I had a wonderful high school teacher who introduced us to complex numbers, reducing matrices by hand, and many other nifty tricks. However, she clearly wasn't the norm, as when I took a group theory course the junior?senior? math major grading my homework balked at my answer for the 5th roots of 1, which I quickly wrote down as e^2*pi*i/5, e^4*pi*i/5, ... In fact, when I later took a complex analysis course, the first week was spent introducing all the math majors to complex numbers. Very disconcerting for the few physicists in the class.

Sounds like the high school and university level math + physics curriculum been watered down significantly, or at least rearranged in a very uncoordinated erratic manner.

In my school about 20 years ago, complex numbers showed up in 9th grade if you were on course to take calculus in 12th (on that path, 9th grade math class was algebra, 10th geometry and trig, 11th more advanced algebra, probably what's called "pre-calculus" these days, 12th calculus). So even people who never planned to take calculus (or even the more advanced algebra class) would probably have encountered them. They first showed up in the context of quadratic equations, as you might guess, and so they might have even shown up in my 8th grade math class (though I don't remember for sure).

I will say that just because I saw complex numbers reasonably early on doesn't mean I understood them at the time. I think I had to go through a year or so of fairly challenging physics and math classes in college to understand why people would use them in practice, rather than merely a sort of convenient fiction for making roots of equations work.

I've done the math department a disservice by saying that they don't teach about complex numbers when I meant to say that they don't teach about complex exponentials. That ought to teach me not to blog at seven in the morning.

There is not all that much uniformity to high school math curricula. My recollection is that we talked about complex exponentials in "12th-grade" math (I took it in 11th grade, before AP calc in my senior year), but didn't do much with them beyond asserting that the complex exponential was cosine plus i sine (which was abbreviated "cis," for no useful reason). We certainly didn't talk about complex numbers as vectors in a complex plane, or anything like that, which is the key element that I need for teaching quantum.

Other people I've talked to about this had different experiences. And every time I talked about complex exponentials without doing this lecture, I got nothing but blank stares from at least half of the class.

In your notes, you have both c-c*=2a, and c+c*=2a ...

Shouldn't c-c*=2ib ?

The XKCD comic is a nice touch.

Complex numbers simply aren't in many (if any) freshmen level calculus texts. They're not considered part of the standard curriculum.

Generally speaking, there's way too much material to cover in the freshmen calculus sequence as it is, without throwing complex numbers in. I don't think the "fault" can be put on the math departments who just don't have enough time to cover everything that everybody (in every major) needs.

One should also keep in mind that calculus in the complex plane is much more complicated than the calculus of real numbers, starting with the definition of a limit. Most freshmen aren't mathematically mature enough to face complex variables "done right", especially one a first pass through calculus.

complex analysis is only fun until someone loses an i.

While on the subect, an homage to Hokusai and Zelazny: 24 Views of the Complex Exponential Function

First, nice work Chad. My only criticism, in addition to not using the Taylor series to show how the complex function is an analytic continuation of the real one, is omitting its greatest benefit: it makes trig identities, particularly for angle addition, completely unnecessary. It replaces those identities with algebra. You give the key result used in AC circuit problems by showing how signs become phases (bottom of page 3), but didn't apply it on page 4 or show how general it is.

Second, what #3, #4, and #5 said ...
I learned about the complex plane and e^{ix} in my one-semester HS trig class (second half of algebra 2 and trig) several decades ago. Our one-semester college trig class omits the topic completely. So, yes, trig in college has been watered down. (Compare French, where one year in HS translates into one semester in college.) But before I blame it all on them, they are dealing with students who don't know any geometry. Their HS geometry classes are a joke (few proofs), probably thinned out so they can save a semester or two to get to calculus. And many have never even had a class in geometry, and we don't have one in the pre calculus sequence. They learn most of their geometry in physics 2 and calc 3.

Our calculus classes don't appear to use complex numbers, and even a differential equations course buries them. Typically they use them for a nanosecond to derive the characteristic equation, then just use that equation without i (or j) appearing again. Physicists tend to do the opposite, just using the complex equation for everything.

By the way, I think my students gain a huge advantage when they get to engineering school because of the day I spend on that subject, plus repetitive use of it in AC circuit problems. Yours will come back to thank you later.

CCPhysicist

When was the high school geometry course eliminated?

I think there is a serious disconnect between what a lot of physicists think is or should be taught and what actually gets taught, in pretty much any other department on campus. Paul in comment #15 described it aptly. This was so for both U State Podunk City and R1 I did my grad work at: diff eq is a separate class taken after one completes the cal sequence. YMMV for linear and/or matrix algebra, but they still typically have cal as a prereq.

This seems to be especially true for older physicists. My advisor once shanghaid me into driving for a short trip, and said person was flumoxed that I'd never had a class in fluid dynamics, nor heard of modeling traffic flow as an incompressible fluid (i.e., traffic jam == shock wave). The dismay redoubled when I pointed out that fluid dynamics is typically taught in mechanical engineering, not physics departments, because who has time to take extra classes when the quantum/electrodynamics/stat mech/classical dynamics sequences loom quickly and fill up all available time on top a year and a half of state required core classes.

Also, the Dimensions video on the complex plane a commenter mentioned in the course report post is really good.

agm,

Several of my older profs mentioned back when they were undergrads (ie. in the 1950's and 1960's), they didn't even have any undergraduate modern physics and quantum mechanics courses back then. Back then, the undergraduate physics curriculum they went through had courses like fluid mechanics + hydrodynamics, mechanics of materials + elasticity theory, continuum mechanics, classical thermodynamics + heat/mass transfer, calculus of variations, etc ...

To answer #19, I think high school geometry was eliminated somewhere between 1985 - 2000... roughly the period between my graduation from high school and my first year as a professor.

I remember my university physics course, how relieved my friends and I were to have reached geometric optics after surviving magnetic fields and Faraday's Law. Out of the whole semester, geometric optics problems were the simplest to work, because they only involved a couple physics principles and everything else was geometry.

Flash forward to my first year teaching the university physics course, and geometric optics wins the "lowest average exam score" award. I was shocked.

Now when I reach that topic, I know to give a brief lecture on simple geometry rules... how to identify equal angles in a skew line crossing two parallel line, and what the sum of angles in a triangle equals. And I know to require students to map out the path of light through material, providing either the physics or geometry reason for each step.

I asked the chair of our math dept. why students didn't know this basic geometry. He told me that, in high school, all the Euclid theorem-proof stuff has been replaced with something akin to "determine the area of a rectangle by measuring two sides".

Please fix the error on page four, in the equation that intends to show the second order partial derivative with respect to time. You are having the first order derivative on the left hand side, make it the second order derivative.

Also, standard notation for second order derivative is d2f/dx2, not df2/dx2.

Thanks

BrianT

Heh. I never would have thought that geometric optics would have become a "hard" part of the course! (Perhaps this may have to do with me not teaching the optics section of freshman physics for a long time. For many years, I was mainly teaching the mechanics section of freshman physics.)

Googling around for high school textbook publishers online, it appears that "integrated math" textbooks are very popular. In many of these "integrated math" textbook series, it appears that euclidean geometry has been given the short shrift by relegating it to two (or three) full chapters out of 30 chapters (encompassing three books). Some of these textbooks don't appear to even cover the conic sections anymore either. Complex numbers are only mentioned in the context of solving quadratic equations with negative discriminant, and hardly anywhere else.

The bankrupt State of California (today is the first of the alternate Fridays when almost all State offices are closed and state employees start their roughly 9% unpaid furlough) does still have High School Geometry Standards. Those, plus the ones for Algebra and Chemistry and Physics and Earth Science are all here:

[PDF]
Mathematics Content Standards - Content Standards

I have formally complained about the Framework for Mathematics, created by a different committee, which is purported to be the foundations from which these standards emerge. My complaint was handled by a Math professor that I knew, who went to Sacramento and spoke with several of those committee members about my complaint that the first definition, which defined Mathematics, was god-awfully wrong, would include Music Notation, and confused applied Math for Science with pure Math inextricably. They blew him off. He then asked: "Have any of you ever actually TAUGHT Mathematics or Science in any middle school or High School?"

"No," they each said, one way or another. "But that's irrelevant."

I was wondering the uses of complex exponentials. I have never ran across them. I have seen complex number especially in circuits . Could somenone describe the uses of complex exp.

Blake: It's primarily a notational convenience. You get to do all of the neat mathematical tricks you can do with exponentials of real numbers, and you don't have to carry sine and cosine terms around separately. This comes in handy when you do things like deriving Fourier transforms, or you play around with plane waves, or various other problems that pop up in advanced physics. You could do any or all of these things with sines and cosines, but the bookkeeping is frequently harder.

The notation also makes it easier, pedagogically speaking, to introduce certain other special functions such as the hyperbolic functions (which are, in effect, trigonometric functions with imaginary arguments) and the Bessel functions (which often pop up in problems involving cylindrical symmetry).

Something I've long wondered: what happens if you take zero to the power of complex numbers? I know, 0^(positive real) = 0, 0^(negative real) = "infinity" and 0^0 = "undefined" - but what about 0^i etc?

0^i = e^(i ln(0)) = e^(-i infinity), of course. So, um, not terribly well-defined, because e^(-i x) oscillates as x->infinity.

@ #19:
I don't know when, but my answer is the same is that of Brian @ #23. All I know is that they never did any proofs, have no clue at all about the surface area of solids, have to be taught about alternate interior angles, etc, and apparently first see conic sections in our pre-calc class.

They only see complex numbers in the context of the quadratic equation, and most of our math instructors have no idea that complex numbers are ever used for anything real at all until I tell them.

@ #27:
Complex exponentials are where the complex numbers in circuits come from! You could come to my AC classes next month, but in the meantime just think about V - R I - L dI/dt = 0 from a loop equation with I = Imax e^{i omega t}. You magically get V = (R + i*omega*L)I, and there it all is: V(t) = |Z|e^{i phi}*I(t) once you do a bit of algebra in the complex plane. You still need to solve for I(t) in terms of V, but that is just division. You get the phase really easily, and also automatically get the - sign when a capacitance is present.

Compare that derivation to the use of angle addition identities to figure out what the phase is! No contest.

Phasors are just complex numbers in disguise.

Thanks onymous, I should have put more thought into getting that far. Now I recall Feynman talking about the oscillation/undefined nature of e^(i infinity). Hmmm, how far can we apply about zero in reverse, to find any way to answer 0^x = 5 or etc? (Looks like no solution set, but in the past, we found interesting numbers by extending solutions to previously {"absurd" types of answers....) That brings up further issues, like how to deal with complex infinity in general. Do we have for example, 5 + (infinity)i, but then for polar we need r = infinity but theta can be various values. The two ways of "split infinity" aren't commensurable, since "theta" for the a +/- (infinity)i always limits out to zero or pi, and for +/-infinity + bi we have theta = pi/2 or 3pi/2. I know, infinity "isn't a number" but we can use limits to define it, so has anyone done good work on complex infinity issues? I have the hardest time finding any.

Jonathon -
Thanks for the link to the CA standards. Those are the things we did when I was in HS, but my students do not do proofs. I'll have to remember to ask them if they can bisect an angle with a compass. I still know, many decades later.

I think it doesn't get taught because proofs can't be put on the "standard" exam.

Neil, you are getting into some fairly deep territory here.

On your first question: think about it. If 0^x = 5, then x ln 0 = ln 5, and as has been previously pointed out ln 0 = -infinity. Therefore x cannot be a finite number. The same argument applies for any finite and nonzero complex number you put on the right hand side. This is why 0^0 is considered undefined.

As for your question about infinity: The way to think about f(z) (where z is a generic complex number) as z -> infinity is to do the variable substitution w = 1/z and see what happens to f(w) as w -> 0. As long as f(z) is an analytic function,* you can do this. It turns out that exp(z) is well defined for any finite z but the limit to exp(w) as w -> 0 cannot be made independent of the direction from which you approach the limit (the technical term for this scenario is "essential singularity"). By contrast, although the argument depends on the direction of approach, 1/z^n will always go to infinity as z -> 0 (this is known as a "pole of order n"), and conversely z^n will have a pole of order n at infinity.

*An analytic function is a function which satisfies certain conditions of continuity and differentiability. I would have to re-read my complex analysis textbook for the exact definition, but one of the key points is that the limit as you approach any point in the complex plane (other than isolated singularities) of the function and its derivatives must be independent of the direction of approach. Most if not all functions which model actual physical systems are analytic: powers of z, exp(z), and most of the other functions described in physics texts and in references like Abramowitz and Stegun. However, some useful functions are not analytic: f(z) = z*, for instance, is not an analytic function. Note that analyticity is a stronger constraint on functions of a complex variable than when you are confined to the real numbers.

Speaking of things I should have though of ... Sorry onymous but we have a problem with your derivation. Note that according to the same logic, however acceptable for non-zero numbers: 0^5 = e^(5 ln(0)) = e^(-5 infinity) which just does not work since we accept (especially from direct series multiplication) that 0^5 etc. = 0. Er, now what? This looks like a paradox...

OK maybe not so bad, since e^(-5 infinity) tends to zero in the limit. In any case, we need to use "infinity" in math and can't just blow it off.

Ah yes, the differences between math in the math department and math in the physics department. At the university of washington, I've found our two introductory courses on mathematical physics cover more than 6 quarters worth of material from the math department (diff. eqs., linear algebra, real analysis, complex analysis, etc). As such i've found it particularly useless to take any further math courses...
i found it particularly funny when i was trying to take the math department's differential equations course (required by my department, but already covered in the math. physics course) and they refused to use complex notation. everything was always a linear combination of sines and cosines, never the real part of a complex exp. or the im. part.

Re # 33 | CCPhysicist | February 6, 2009 3:15 PM

"Jonathon - Thanks for the link to the CA standards."

You're welcome. Technically, I spell my name "Jonathan" but you used the default error, and I don't mind. Isaac Asimov used to complain when someone spelled "Issac."

Standards writer? The Standardetti? They're well-intentioned, and you can usually see why they are what they are (locally). But, globally, they a patchwork of a lattice of sieves of misunderstandings. Professionals in any field, given any of the individual standards, always react by saying: "But..." and listing the crucial exceptions and alternatives and clarifications and complexifications.

State and Federal regulations require the teacher to "teach to the standards." That is one step better than "teaching to the test." But barely.

In one case Sep-Dec 2008 I got better results by distributing to a few students the standards in one subject SPANISH LANGUAGE VERSION. I don't speak Spanish (a stupid position to have been in in New York City, and equally stupid for me resident in Southern California).

But I wonder what the native Language was for the committee that wrote, say, the Physics standards. I doubt that they were actual laboratory Physicists. I likewise doubt that most of the people who wrote the Math Standards think well enough in Math to have published refereed papers in math (as opposed to refereed papers in the Pedagogy of Math).

I am, as of last month, a card Carrying member of the National Council on the Teaching of Mathematics. That is very different from being a member of the American Mathematical Society.

When my wife published in The Physics Teacher, it got her promoted, even though the virtually illiterate fool who's Chair of her Department complained in writing that it was not good enough for Phys Rev Let (where once and only once he was the junior of 3 co-authors, the other two of whom were very prolific and probably didn't consider it worth the hassle to kick him off the title page). He also sneered, in writing, that one of our refereed papers read "as if by Asimov." Which, of course, we took as a compliment.

Neil: We can make the discussion more rigorous by considering 0^a (for some real number a) as the limit of z^a as z -> 0. Define z = r * exp(i*theta)--modulus and argument notation (r and theta are both real; r > 0). Then z^a = r^a * exp(i*a*theta). If a > 0, then r^a -> 0 regardless of what theta is. If a < 0, then r^a -> infinity, again regardless of theta. The case a = 0 runs into trouble because, unlike with nonzero a, it matters what order you take the limits. You might argue that fixing a = 0 first leads to an unambiguous statement of z^a =1, but if you take the z -> 0 limit first it matters whether a -> 0 from the positive or negative direction.

As for the case of 0^i, let's see what happens to the above argument when a is allowed to be purely imaginary (we can derive the case of a complex by multiplying this result with the real case above). In this case r^a is oscillatory with period approaching 0 as r -> 0, so the limit does not make sense.

So why, in the case of nonzero real a, does the order not matter? Because even if you take the limit of a from off the real axis, the oscillatory part goes to a definite fixed value (because the oscillation period goes to infinity). Thus the limit exists even if you let z -> first and then fix a.

Chad,

Nice notes, but from a logical point of you definitely cheated. If you define complex conjugation by

(a + bi)* = a - bi

that does not necessarily mean

(e^ix)* = e^(-ix)

It's also not fair to go from

e^ix = cos(t) + i sin(t)

to asserting

t=x,

when really all you can say is

t=f(x)

Nice notes, but from a logical point of you definitely cheated.

Absolutely.
That's the "swashbuckling physicist" part of the lecture...

As I put it in class, these are more like plausibility arguments than anything that a mathematician would accept as a proof, but they get the basic ideas that we need for QM across, and none of the actual statements are untrue.

Re #41:

Feynman said:

"A great deal more is known than has been proved."
[Quoted in The Music of the Primes : Searching to Solve the Greatest Mystery of Mathematics (2003) by Marcus du Sautoy]

and

"I think I can safely say that nobody understands quantum mechanics."

[The Character of Physical Law (1965) Ch. 6; also quoted in The New Quantum Universe (2003) by Tony Hey and Patrick Walters]