The semester here is finally underway, and I’m getting back to work on my quantum mechanics and mathematics book (latest version available here). Current plan is to have a final version by next spring, with publication by Springer late next year. This semester I’m teaching Calculus II, a subject where there’s only one thing I really dislike about pretty much all textbooks, their refusal to use Euler’s formula. Since I couldn’t find an online source I was completely happy with, I spent some of the last couple days writing up some notes for the students on Euler’s Formula and Trigonometry, which maybe someone else will find useful. In other news:
- Nima Arkani-Hamed was here today, giving a talk on a new model he calls “NNaturalness”. The basic idea is to consider something like N copies of the Standard Model, with N a large number. Large N fixes the technical naturalness problem, with something like N=104 fixing the MSSM’s current naturalness problem, and N=1016 fixing the non-supersymmetric problem. He makes clear that he’s well aware that this is a pretty contrived thing to do, but argues that it’s interesting one can do this while evading dramatic disagreement with experiment, and coming up with potential CMB signatures soon observable (e.g. the effective number of relativistic degrees of freedom).
He has a nice description of the naturalness problem as “in any theory where we can compute the mass of the Higgs it has a fine-tuning problem”. Probably there are people out there who think they have a way to compute the Higgs mass who would disagree with him. To me the problem is that the theories he’s talking about (GUTs, string landscape) don’t actually explain anything about the underlying physics of electroweak symmetry breaking (where does the Higgs field come from and why does it have those couplings?). Given this, it’s unclear why one should worry about the fine-tuning.
He describes the landscape and the multiverse as “like democracy, the worst idea except for everything else”, and gives a defensive argument for why one should study alternatives like “NNaturalness”, even if they’re not as good as the multiverse (which he finds “simple and deep”). To him it’s worth thinking about alternatives to the multiverse (as a “foil”) not because the multiverse is untestable pseudo-science, but because maybe one shouldn’t just give up. So, it seems that at this point he’s not quite signing up with the intellectual suicide of multiverse mania, although he sees it as the most attractive path available.
In other Arkani-Hamed news, the IAS has an article about his activities promoting a next generation collider here.
- The KITP has a newsletter here, including a description by Graham Farmelo of his visit there. Oddly, no matter what he writes about, Farmelo almost always includes an unconvincing defense of string theory and/or the current activities of string theorists (for examples, see here, here, here and here). In this case he assures us that the KITP theorists are not given to “mathematical adventurism”. I think he’s right, but that’s the problem…
- Someone pointed me recently to Olivia Caramello’s web-page on Unifying theory and her arguments with fellow category theorists. I had a youthful infatuation with category theory, but ultimately came to the conclusion that there’s a real danger in that kind of “unification” of going too far in the direction of saying less and less about more and more. Many of the ideas involved are powerful and attractive, but the remarkable thing about mathematics is that, even for the lover of grand ideas, less generality is sometimes even more so.
Update: One more. If you’re in the Bay Area next week, you might want to head up to MSRI for a series of elementary talks on the Langlands program by Edward Frenkel.
re-“why pi must be transcendental and why it is that it has the value of about 3, (order 1) rather than 0.1 or 1000?”
the value of about 3 comes easily from the definition of pi as the ratio of the circumference to the diameter of a circle. The perimeter of a square circumscribing the circle is 4 x diameter and the perimeter of an inscribed square is 2 x square root of 2
x diameter. The circumference of the circle is greater than 2.83 and less than 4 times
the diameter which puts pi somewhere near 3. (if you look at inscribed and circumscri bed hexagons you get pi between 3 and 3.46).
why pi must be transcendental is not so easy.
maybe already said : of the “Euler’s Formula and Trigonometry, ” pdf : page 5 on top, 6th line (3th line of the expansion) is wrong : you forgot a pair of brackets ;
Nice PDF, thanks for that
Yes, one can define “cis” that way and show that it has all the properties of an extension of the exponential to pure imaginary arguments, and that’s a common strategy. Still, once one has done that, I don’t see why one shouldn’t just drop the “cis” notation and use the exponential notation, which will be much simpler to work with.
Thanks I hadn’t seen that book. It does have some similar goals to what I’m writing, and there’s a significant amount of overlap. I’d describe my version as going for more mathematical depth, which is why it’s about twice as long. Also, that book sticks to the fairly conventional Lagrangian symmetry story of most physics textbooks, while I’m trying to emphasize parts of the story that aren’t in such books, including paying much more attention to the Hamiltonian point of view.
As somebody who was accepted to Caltech as an undergraduate from overseas in the 1980s, this sounds incredible. At the time I could scarcely believe the type of remedial-level mathematics (and physics, and chemistry, for that matter) on which the best and brightest from the finest US high schools wasted a year and more of tertiary education. The Euler Formula that Peter Woit’s students are encountering for the first time should surely have been year 9 or 10 high school material for any student destined for a tertiary education in any quantitative field.
That the situation might be even worse today is both hard to understand and dispiriting to consider.
Are things as bad outside the USA? Are “four year” tertiary degrees also becoming two year programs on top of two years (or worse) of remedial secondary school material elsewhere in the industrialized world?
The situation with the level of math and science instruction in US secondary schools is bad, but I don’t think any worse now than in the 1980s, and in some ways better. As far as I can tell, the US high school math curriculum right now is more or less identical to what I experienced forty years ago in the early 1970s (and so is the set curriculum for calculus classes, whether offered at the end of high school, or beginning of college). On the positive side, many more US high school students are taking calculus classes in high school.
But, with Euler’s formula not even in the calculus textbooks, US students are still highly unlikely to see that in high school.
I looked at my own freshman math book, from 1962 “Thomas”, which apparently still has nominal successors in print. Euler’s formula is there, quite prominently, though near the end. There is an online version of a near-recent successor, and its still there.
I taught for many years junior level quantum mechanics to chemists, and we used that formula a lot; the students were not bothered by it. What they were bothered by was the idea that they had to generate even an obvious three line derivation or proof. We never of course asked for mathematical rigor.
It’s great that you passed on your Euler notes; that kind of commitment to teaching is too often missing!
I teach at a university somewhere outside the US, and I always work in either Euler’s formula (almost exactly as you present it!) or the 2×2 matrix version or both as soon as possible. The matrix version allows me to make the linear transformation/matrix link. This is in Calc 1,2,or 3: because it is guaranteed they haven’t seen it either in high school or with any earlier prof.
I use the power series definition for exp, but heuristically (and if they are interested,
I will explain the convergence). I hand out a little resume of the material to them since no book has it. I also draw a graph of the function e^(it): \R\to\C
and explain that looking at this “Slinky” from the side gives you sin and cos.
And to draw sin, I set my arm in “harmonic motion” and walk along with time. And: there is always some mention of music synthesis, as a motivation.
This is just one example of similar missing material. As a student I had many moments of frustration/fury when I realized some obvious explanation had been left out. almost as if the secrets were being kept within the guild.
Somebody mentioned Spivak and old editions of Thomas. Both I have warm memories of for completely different reasons. Spivak is simply beautiful and is a great backup for the prof, but is only good for that rare student who wants to really know what is going on and is ready to learn on their own. Old Thomas has some great word problems as I recall, although it is too dry for self-study.
Back to Euler, when I teach power series (calc level) I use that to treat Fourier series
-proving orthonormality, and writing the formulas, is so much easier than with sin/cos-and then I make the Fourier series- Taylor/Laurent series link. This is done not too rigorously: mainly just to blow their minds- as it did for me when I realized that–in grad school!!!–In other words, it gives them a good advertisement for what is to come, and a taste of why math is at once so cool and so powerful and so simple.
Does anyone have links to talks on the Nima’s N-naturalness? Am curious to understands its implications for the thermal history of the early universe.
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Maor suggests that the idea of the exponential function first arose from the concept of compound interest and the question of what would happen if the compounding became instantaneous.
I was a young kid when the banks started decreasing their compounding periods back in the ’60s, and I remember hoping that I would get incredibly rich as they reduced their compounding periods to zero!
Of course, my hopes were dashed because the limit of (1 + x/n) to the nth is just exp (n), which is, alas, a finite limit for any real x.
You can in fact easily derive the power series for exp(x) from this by just taking the binomial expansion of (1 + x/n) to the nth and taking the limit n -> infinity. This easily shows that the limit is finite.
If you then plug in an imaginary number, you get the limit as n goes to infinity of (1 + i x/n). For n very large, it is easy to see that one is taking a small angle close to x/n and “compounding” it n times to get to the angle x. (A bit of thought shows that in the limit n goes to infinity, the radius remains at 1: e.g., look at the binomial expansion for (1 + i x /n) times its complex conjugate all to the nth..) Using the binomial expansion to get the power series of course works in exactly the same way as in the real case.
All this does not meet the mathematician’s standards of rigor (although I suspect it can be “cleaned up” as easily as any other approach). However, if Maor is right, it may be close to the historical development. And, multiplying (1 + i x/n) by itself n times,. is a somewhat intuitive way of reaching the angle x (in the limit n goes to infinity of course) on the unit circle.
This is a comment to your notes on Euler’s formula.
Do you know the book `Numbers’ (multiple authors, edited by J.H. Ewing), Springer GTM 123? Among many other things, it has a very nice treatment of complex numbers, exponential function, trigonometric functions and, of course, pi. One might even find that treatment a bit too comprehensive.
I know the book, hadn’t looked at it recently. It is quite wonderful, and the chapter on “pi” has an excellent treatment of the issues.
I’ve just taught Euler’s theorem and De Moivre’s theorem to my A-level Maths class and I wanted one of my brightest students to do some independent study on how to use it to integrate and differentiate products and powers of trigonometric functions (as extra reading) – I’m going to share your document with her. Thanks 🙂 @Olier – do you know of any resources / books which describe how to use Eulers formula for Fourier Series? That might be good extension material too!