For tedious reasons, I find myself faced with giving what will basically be a pure math lecture next Friday. I need to introduce a bunch of mathematical apparatus that we will need in the coming weeks, and I know that the Math department doesn't cover these topics in any of the classes that these students have taken. If I want them to be able to use this stuff, I need to teach it myself.
Formal mathematics is probably my least favorite part of teaching physics. I'm very much inclined toward the "swashbuckling physicist" approach to math, in which we cavalierly assume that all sorts of picky details will resolve themselves when we eventually have to compare our calculations to experiment. In the fields I deal with, it's usually obvious when a potential solution is un-physical, and we just discard those. As a result, I've always had a very casual approach to dealing with math, which occasionally puts me in a bad place when I have to introduce math to students.
In this particular case, the main idea that they need to get is the Euler relationship between complex exponentials and the trigonometric functions, usually expressed as:
ei x = cos(x) + i sin(x)
It says that when you take the exponential of an imaginary number, the result you get is the sum of a couple of trig functions. It's a wonderfully elegant result, that has all sorts of nice properties when you start dealing with the physics of waves and oscillations. And it's absolutely indispensible when you talk about quantum mechanics, as wavefunctions are necessarily complex quantities.
the problem is, I have absolutely no idea how to introduce it to these particular students...
Part of the problem is that I have a deep loathing of just stating anything as a fact to be memorized. That's the sort of behavior that gives physics a bad reputation among students of other disciplines, and on top of that, it's not terribly effective. Without some context, it becomes just a bit of trivia taken on faith, and students are prone to forget it before the end of the class period, let alone the final exam.
When I introduce physics formulae, I always make a point to provide some justification. Either I derive the results from first principles, or I appeal to physical intuition and common sense. I think that works better than just stating results, both in terms of student retention of the important facts, and also for giving a more complete picture of how the field works. There are a few places where I'm forced to punt-- the Schrödinger equation being the obvious example-- but I try to at least make plausibility arguments for everything.
I'm sort of stumped on the Euler relationship, though, because the only way I know to justify it relies on mathematics the students haven't had yet. You can make a swashbuckling-physicist argument that it obviously has to be true based on the series expansions of ex, cos(x) and sin(x), but they haven't seen series expansions yet, either, and I don't have time to introduce that as well.
I'm open to suggestions, here. Does anybody know a good way to introduce the Euler relationship without reference to series expansions? Or am I just stuck with Proof by Blatant Assertion?
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The most intuitive explanation for me has always been starting the unit circle in the complex plane, as defined by e^ix; I find it easy to imagine traveling around that circle and looking at the projection on the real plane.
Of course, you're still starting with defining the unit circle on the complex plane as e^ix, the truth of which is less than obvious. But it has a nice graphical element to it that the series expansion lacks, making it a bit more intuitive to me.
How about demanding that
1) d/dx exp(i*x) = i*exp(i*x)
Knowing that first-order linear differential equations have only one lineary independent solution, and using the boundary condition
exp(i*0) = 1
You show that cos(x) + i*sin(x) obeys (1) and (2), and hence it's the solution.
I don't have specific advice, but I could suggest a few places you could go for inspiration. The Feynman Lectures has a chapter on algebra (vol. 1, lecture 19), in which I believe he discusses this formula. I don't remember how he introduces it, but being Feynman, it might be good.
Also, you could try Visual Complex Analysis by Tristan Needham. It's amazing at giving an intuitive geometric picture of complex analysis that's well connected to the mathematics, and could potentially include a nice way of visually relating the rotational properties of pure phases to the exponential function. (I think some of its main approach is based on the derivative properties of the exponential function, as mentioned by 'Someone', but with more connection to geometric insight.)
I've always found the relationship was most understandable/believable through the power series expansions of the functions, but I'm guessing that if these students haven't seen the Euler equation, they don't know the Taylor series expansions of e^x, sin, and cosine, either.
Will they need power series eventually too? You could do that first! I do it with my non-mathy kids with graphing programs and handwaving. Graph this... now add this term... now add this term, now add this term... hey that kinda looks like the sine function!
Anyway... if you do the Taylor series expansion of e^ix and then square and cube and fouth power all of the i's, and then group the terms WITH i's and the terms WITHOUT i's you get (the series expansion of cosine) + i (the series expansion of sine).
Not exactly elegant, but it always impresses ME when it works out.
Well, phooey, somehow I got all excited and didn't read the last paragraph. But I still say if you're willing to devote some handwaving time to power series... it's the only CONVINCING derivation I've ever seen.
I distinctly remember that my high-school precalculus class covered this, along with de Moivre's theorem and taking n^th roots of complex numbers. On the other hand I don't recall how it was introduced. I'm pretty sure we hadn't discussed series expansions for exp, cos, and sin, and we certainly hadn't discussed Taylor series in general, or ODEs, since it was a precalculus class.
So I think maybe the way it was done was to show the complex plane, to discuss how multiplying two complex numbers gives a third number elsewhere in the plane, and then to see that if you multiply two numbers on the unit circle, you get another number on the unit circle and that the angles from the real axis add. So that's pretty simple algebra to do. But that motivates that the angles behave under multiplication in the way that arguments of exponentials do, except of course that they're periodic.
If you do that, then you can at least motivate that you're going to *define* e^(i theta) for theta real to be cos(theta)+i sin(theta), and show that this is consistent with all the usual properties of exponentials and logarithms that they know. I definitely remember that in this class there was a lot of discussion of how multiplying by "i" is rotating the plane by "pi/2", so that's probably how it was done. It might not be the most satisfactory treatment, but it has the advantages of requiring little background and it lends itself to some nice visual demonstrations of how things work.
sines and cosines are useful for oscillating things
Euler combines them both
Cos(x) = Re(e^{ix}), take derivatives and integrals in complex space which reduces life to algebra, take real part in end
e^{i*pi}+1=0 is beautiful!
Maybe they've seen the limit definition exp(x) = lim (1+x/n)^n/ Works just as well for complex arguments...
For showing that e^ix is the unit circle, the place that Evan Murdock suggests you start, I'd use the nth roots of 1, which obviously have to be of unit magnitude (because otherwise you get away from the unit circle, which can't be because 1n has to be one). Start with square roots, then cube, then *maybe* 4th, then generalize. (Or maybe generalize after 3rd).
This also assumes that they know that sin2x + cos2 = 1, if they don't know that, um...Pythagorean theorem time, I guess.
Oh, if they know polar coordinates already, that should help a lot.
Feynman has a chapter dedicated to exactly this, but his approach may not meet your needs.
Think about what the students understand by the symbol "e^x". Presumably they think of it as a number multiplied by itself repeatedly. Euler's formula cannot be derived from this definition, even heuristically. I suspect that not making this clear may only result in further confusion.
Some other property of e^x must be used. There are two that spring to mind. Either e^x*e^y=e^(x+y) or d(e^kx)/dx=ke^x.
Wikipedia contains three proofs in addition to the Taylor expansion:
http://en.wikipedia.org/wiki/Euler%27s_formula
Any of them could be used.
If they can differentiate sin, cos, e^x and quotients then the proof "using calculus" is short and sweet.
The proof using differential equations is a little longer, but if you are teaching waves it might fit in very nicely with the rest of your course.
The "heuristic argument" makes use of the property e^x*e^y=e^(x+y). It needs more background in trig idenities and polar form of complex numbers, but it avoids calculus except implicitly in identifying the base as e by behaviour near x=0.
Ah, I have strong feelings about this one. I first saw this junior year of high school, when the calculus teacher we were going to have the next year asked a couple of us if we knew what e^(i*pi) was. He told us the answer, and promised us that by the time we were done with his class we'd understand. A classmate and I spent a lot of time arguing conceptutally about why this should be so, what exponents mean, etc. When we actually took calc, I felt let down by the formal definition using series expansions (which I still didn't really grasp), with no intuitive explanation of the meaning.
The one I eventually contructed for myself is something like... sin(x) and cos(x) have these cyclical derivatives (which has to do with the fact that they're defined in relation to a circle.) Second derivative of either is just itself with a negative sign out front, and the fourth derivative of either is just itself.
Meanwhile, e^x is a function which equals its derivative at all points (it took me a while to understand that this is the definition of e, that the derivatives of 2^x and 3^x are tantalizingly close to the original function, but e^x is the value where they're actually equal.) So anyway, it is similar to sin(x) and cos(x) in that its fourth derivative and eighth derivative and 12th derivative also give back the original function. It's just that sin(x) and cos(x) have this thing with negative signs...
And also meanwhile, i has sort of cyclical properties too. i^0=1, i^1=i, i^2=-1, i^3=-i, i^4=1, i^5=i. There's this every fourth power thing going on, again...
When you note all of these facts, it almost seems like there has to be a way to relate these functions.
Fiddle around and you find e^i*x has all the same properties as cos(x)+i*sin(x). All the derivatives match. What's more, they have the same "initial value" at x=0 (the only place where we can actually evaluate e^ix in that form.)
Obviously this doesn't rise to the level of a proof, but it is a motivation...
Finally, I found it much easier to think about e^ix in problems after my quantum mechanics professor graphed it. He said z=e^ix. I want to plot z vs. x. Of course, z is a complex number, so I need two axes to plot it on.
He drew three axes, Re(z), Im(z), and x. Then he drew a sort of helix, starting at x=0, Re(z)=1, Im(z)=0, and spiraling around the x axis, showing that the magnitude was always one, but the real and imaginary components (which behaved just like components of a vector) varied between -1 and 1. This made it much easier to vizualize the dependence on sine and cosine, and the relationship of this function to a circle. This professor always referred to e^ix as the "rotini function." I still think of momentum eigenstates and so on as "rotini function solutions".
(He was my favorite professor.)
This is an approach that I've found works all right with bright high school students who have seen the formula printed somewhere and want to know what it means. I'll write E(theta) as short hand for cos theta+i sin theta.
Show that E(theta)*E(phi)=E(theta+phi), this just uses standard trigonometric formulas for the sum of angles. Therefore, your students should believe that E(theta)=E(1)^{theta}.
There is no way that you can prove that E(1)=e^{i} in a rigorous sense, because your students don't have any definition of raising numbers to a complex power to start with. But here is a plausible argument. Suppose that there was some complex number a such that E(1)=e^a. Then E(theta)=cos theta+i sin theta=e^{a*theta}. For small theta, the left hand side is 1+ i*theta and the right hand side is 1+a*theta, plus terms of order theta^2. The only way that they can match is if a is i.
Here are some pictures I drew the last time I thought about this: 24 Views of the Complex Exponential Function. If you can start by picturing the shape of the complex exponential function in four dimensions, the rest is easy.
I don't see that you've got a problem here. All of this stuff is simply true by definition: the sine is the rise over the hypotenuse, the cosine is the run over the hypotenuse, and these are where you get to when you rotate the hypotenuse around the origin. Q.E.D.
The supposedly mysterious fact that these "quantities" turn up when we talk about ocillations, electrical currents, and so on is no mystery at all: we've defined our terms in those disciplines to be the same again. It's as if a wave should turn out to be wavy: right, that's the word we use for it, onl these are math words.
The absolute value of e^{ix} is 1, so it can be drawn as a point on the unit circle in the complex plane. The projection onto the real and imaginary axes are then cos(x) and sin(x). All those concepts (absolute value, projections...) are geometrical, and a nice picture is always easier to remember than an arbitrary looking formula.
Alternatively, how did Euler discover the formula? likely to still be the easiest explanation.
It can be tricky
I fail to see why this discussion went on beyond comment #2 by 'someone'.
How about demanding that
1) d/dx exp(i*x) = i*exp(i*x)
That's clever, but differential equations are even scarier than series expansions... I do like it, though, and may use it in another context.
The most intuitive explanation for me has always been starting the unit circle in the complex plane, as defined by e^ix; I find it easy to imagine traveling around that circle and looking at the projection on the real plane.
Of course, you're still starting with defining the unit circle on the complex plane as e^ix, the truth of which is less than obvious. But it has a nice graphical element to it that the series expansion lacks, making it a bit more intuitive to me.
This is probably what I'll end up doing. Or, rather a combination between that and Moshe's comment about projections onto the axes. I think I can finesse the magnitude thing, because I have to define the magnitude anyway, and once you show that A*A is the length of a vector in the complex plane, then it's trivial to show that e^ix has magnitude one, which puts it on the unit circle.
Feynman has a couple of nice tricks for this, and a really clever way of showing the oscillatory nature of e^ix, but it's one of those successive-approximation things that you need to be Feynman to pull off. It's also something that was clearly written before cheap electronic calculators...
One of the things that my maths lecturer did, was show that cos x + i sin x (which I'll refer to as "cis x") obeys the usual laws that you'd expect of a^x where a is a number to be determined. For example, a^x * a^y = a^(xy).
Another approach is that if they already know trig identities, you could show how exp(ix) makes them easier to understand.
As an example, if they know the identities:
This makes more sense if you expand:
and match up the real and imaginary parts.
But personally, I think the best way to understand this is geometrically. Take the differential equation, for example. If you think of x as a point in the Argand plane, then ix is a tangent vector to the origin-centered circle that passes through x. If the point "follows" the tangent, it'll "obviously" move in a circle.
Another thing which may help: The hyperbolic trig functions have a similar identity. Most high school students, if they remember it, learned the hyperbolic trig functions as:
If j is a number such that j^2 = 1, then:
You can trivially check that one by setting j=1 or j=-1.
This identity, of course, turns up in the geometric algebra view of Minkowski space.
Oh, and yet another thing.
The reason that the Euler equation works is because i^2 = -1, not because you're working specifically in complex numbers.
Complex numbers have a simple matrix representation, where:
In this representation, I think the differential equation representation is much more obvious. When you work it out, you get:
The right hand side is the standard 2D rotation matrix.
But, of course, to define the exponential of a matrix, you need the diff-eq or power series stuff. I'm with Lieven -- #2 is the way to go. They can understand it if they know how to take a derivative assuming you postulate uniqueness.
Aaron: Yup, you do.
The reason why I bring up the matrix representation is that some people might find it easier to believe. The Argand plane has some geometric properties that some people find hard to believe. Why would multiplication by i be a rotation? Using an explicit rotation matrix instead makes things a bit easier to believe, especially if you've seen rotation matrices before.
I think it was Charles Stevens' The Six Core Theories of Modern Physics which pointed out that special relativity isn't hard to understand, it's just hard to believe. I think that's true of a lot of things.
Differential equations (#19) ?? Postulating uniqueness (#22) ?? The only fact a student needs to know to get through comment #2 is that df/dx=0 implies that f is a constant function.
I didn't say the uniqueness in this case was hard :).
It's probably too long to help you next Friday, but I recommend you take a look at Where Mathematics Comes From by Lakoff and Núñez. The last part of the book is all about ways of explaining the Euler relation.