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=10
^{4}fixing the MSSM’s current naturalness problem, and N=10^{16}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.

Surprised to hear Euler’s formula is absent from introductory texts. After discovering it as a youth, I used it just like this, and found it a wonderful alternative to the pain that my classmates were suffering without it. But I always assumed its absence from our curriculum was merely one more manifestation of our crap teachers and textbooks. Kudos for re-introducing it to the world.

Peter,

You have a typo in your exponential notes on page 9.

Where you are evaluating the integral of cos(ax)cos(bx), on the second line, the second term in the integrand should be exp[i(b-a)x].

See, I’m looking out for you.

What’s up, Peter. Nice post, but I was expecting something about the latest hype as well:

http://www.theguardian.com/science/life-and-physics/2015/sep/14/straws-in-the-lhc-wind-lepton-universality-and-an-update-on-that-bump

As far as the category theory is concerned, the way to obtain some interesting results is to avoid generalizing too much. For example, the Turaeev-Viro 3-manifold invariants were obtained by using special tensor categories (associated to representations of quantum groups at roots of unity) while the invariants of 4-manifolds can be associated

with 2-categories. Moreover, the 2-categories also appear as 2-groups or crossed modules, which generalize our notion of symmetry (which is based on the notion of a group). This is useful for constructing new TQFT’s and quantum gravity theories.

Isn’t Euler’s Formula and Trigonometry taught at grade 12?

“An engineer / scientist … starts out knowing a great deal about very few things but over time learns more and more about less and less until eventually s/he knows everything about almost nothing”.

“A philosopher / architect … starts out knowing a little about a large number of things but over time learns less and less about more and more until eventually s/he knows almost nothing about everything”

Can you give a few more details about N-Naturalness? I cant find any of Arkani-Hamed’s slides are on the web, but I’ve heard about his idea anecdotally from a few sources.

Thank you for your pointer to my website. I should clarify that the kind of unification that I pursue in my work is not through (Lawvere-style) category-theoretic generalization/axiomatization, but rather through transfer of suitable invariants. While category theory can be regarded as a unifying language since its concepts are sufficiently general to specialize to many important ‘concrete’ notions in specific fields of Mathematics, Grothendieck toposes provide a substantially different way for relating different mathematical theories with each other, which is based on the possibility of transferring invariants across their multiple representations. The kind of insight that these novel topos-theoretic methods can bring is therefore much deeper than the “static unification” through generalization achieved by category theory (the distinction between these two different kinds of unification is explained at this page): they allow, for instance, to translate a difficult-to-check property in the context of one mathematical theory into a completely different-looking one which can be much easier to tackle in another theory, and to generate concrete results in different mathematical areas which might be hard to prove using alternative methods (a list of examples is available on my website).

“By the end of this course, we will see that the exponential function can be represented as a power series”

Peter, why leave them ‘dangling in expectation’ to the end of the course? The power series follows in two lines – satisfies the stated equation and meets the initial condition, both only requiring calculus 1if I am not mistaken?

You would then have a 100% ‘do it all yourself’ physicist’s exposition!

Ru,

In the US Euler’s formula is not part of the standard secondary school curriculum. Even worse, it’s not part of the calculus curriculum at the college level. I hope this is not true in most other countries.

One fundamental problem (besides that of not having a crucial, powerful tool) is that without using Euler’s formula, there’s no sensible way of understanding why the addition formulas (and thus most trig identities) are true. As a result, students are led to believe that mathematics is about some list of formulas you should memorize without understanding why they are true.

Another way of getting the same thing (cos + isin\theta is a rotation by \theta), while avoiding complex numbers would have been to use two by two rotation matrices, but that requires introducing matrix multiplication.

I don’t really understand why the US math curriculum makes this choice not to use Euler’s formula. It seems to be that the logic is “to understand Euler’s formula you need to understand complex functions of a complex argument, and we can’t talk about that until a complex variables course”

sm,

The difference between a math and a physics course is that in the math course you’re supposed to pay attention to whether the power series converges, and whether what it converges to is the function you have in mind. Much of the later part of the course is devoted to understanding series and convergence issues, so, yes, I’m leaving them dangling…

Andrew,

Sorry, it’s a slightly complicated picture, and I don’t think I’d be doing anything very helpful by explaining badly what I understood of it here. He says a paper is underway, collaborators are Cohen, D’Angelo, Hook, Kim, Pinner, maybe you can find out more from one of them.

Verissimo,

The Butterworth article isn’t hype at all, it does an excellent job of explaining the story.

It is unfortunate to see promising young researchers potentially destroying their own careers by being so clueless with regards to social skills, as Olivia Caramello seems so bent on doing right now. I hope someone would give her better advice, and that she would follow it.

anon,

I don’t know her field and I didn’t look that carefully at all the details of her exchanges with other mathematicians. But some of them did seem to be giving her good advice, and I hope she’ll make use of it. The issue of “folklore theorems” in mathematics is a very tricky one for a lot of reasons. Her choice to make all this debate quite public is very unusual. To the extent it makes the situation with the mathematics and its history clearer, that may be a good thing. On the other hand, to the extent it gets people worked up and annoyed about priority claims, not a good thing at all.

mitch,

Thanks. Fixed.

I think Dr. Caramello comes off well in the exchange.

Besides the accusers admitting to writing letter of reference that insulted her, one accuser defends himself with this statement.

“You’ll recall that, when you first told me about the `de Morganization’ construction (the largest dense de Morgan subtopos of an arbitrary topos) I was very surprised — not so much by the fact that this construction existed, but by the fact that no-one had spotted it before.

However, a little later I discovered that I had found the construction myself,…”

that is just absurd.

Alex Heller, who taught me topology my first year in graduate school, was a student of Eilenberg in the 40s and around while category theory was being developed. Used to drive me crazy in class, every proof was “the diagram commutes.” That said, though Alex was a category theorist in some meaningful sense, even he used to call it “general abstract nonsense” which I believe was a quote from Eilenberg. It is possible to get to general. Then again, what Caramello is doing is somewhat different, Grothendieck had a very interesting take on things. Not that I’m an expert on that sort of stuff. I have crossed paths with Grothendieck in his work on dessins des enfants.

As to “… Calculus II … textbooks … refusal to use Euler’s formula …”

years ago when I was at Georgia Tech a physics PhD candidate friend of mine

was teaching a comparable calculus course using a book with no Euler’s Formula

and

he took it upon himself to show how Euler’s Formula made calculations easy and natural.

The students rebelled (it is not in the book – we don’t want to know about complex numbers – complex number theory is not a prerequisite – …. etc …)

and complained to the head of the department.

My friend was admonished and told to apologize to the students

and to avoid all use of the devil’s tool of complex numbers in the course.

Maybe that was not the isolated incident that I thought it was

and

maybe that is why publishers even today keep Euler’s Formula out of such textbooks.

Tony

Tony,

Hopefully that won’t be the reaction of the students this time. The odd thing is that complex numbers are a topic in high school math, so is trigonometry. But explaining to students the relation between the two subjects is somehow off limits.

I was first introduced to Euler’s formula in pre-calc in high school. It’s about the only thing I remember clearly from that class, actually, because I thought my teacher had found God or something. Very few things that I can recall floored me like that did. It was also the first time my lack of innate skill in mathematics made me a little depressed.

I think that for more adventures students, specially if they have a computer software background, an introduction to the Chebyshev polynomials and their relationship to the trigonometry and de Moivre’s formula may be interesting (this is actually how computers calculate the trigonometric and many other functions).

My high school (in the late ’70s in the rural Midwest) had only trig as the highest math, and we barely got into complex numbers. Then in college (engineering school) I did math through PDEs, but we didn’t talk much about complex numbers; I got more about complex numbers in my control systems class (Laplace transforms) than the math classes. But being something of a math and science nerd, some years ago I found Paul Nagin’s book “An Imaginary Tale: The Story of sqrt(-1)”, from Princeton Science Library Press; it is ostensibly a history of complex numbers, and quite entertaining at that, but it also has a lot of semi-rigorous derivations of things like Euler’s formula, de Moivre’s formula, and a lot more. Might be worth recommending to your students as a resource.

“Euler’s Formula and Trigonometry” notes that e^(

iπ) = −1 and then says that the equation “relates three fundamental constants of mathematics”.I like the discussion in the Wikipedia article Euler’s identity, which is based on the equation e^(

iπ) + 1 = 0.I got the formula from Eric Temple Bell’s Men of Mathematics, who rhapsodized over its astonishing combination of all the fundamental constants in math. But I never was exposed to it as something useful.

The question of exposure to the complex numbers is an interesting one, from what I can tell they have been largely disappeared from the curriculum, even for math majors. At my university, there isn’t complex variable course, and realistically you could get through a whole major without really seeing them, except peripherally. I just checked, and Stewart’s precalc book (a standard) does mention them, about 5 pages pretty early, then a few more when it does polar coordinates. It mentions DeMoivre. So students will probably see them then, for a minute, and might not again until grad school. I saw them first in high school, not because we got taught them, but because all the math nerds did it on their own. And when I took Calc in high school, we did do Euler. I also of course did complex as an undergrad (out of a very interesting book by Polya).

Huh. I don’t know about Euler’s formula, but we certainly learned about complex numbers, basic operations with them, and the idea of the complex plane in high school. Has all of that disappeared from the curriculum?

Academic Lurker,

Complex numbers are taught in high school, generally in the context of roots of polynomials. Euler’s formula is about the exponential function, and essentially says that you can extend it from the real numbers to the complex numbers, preserving the property that exponents add when you multiply. The remarkable, beautiful fact is that when you do this you get trig functions: the exponential of an imaginary number is a complex number with real part the cosine, imaginary part the sine.

Where I went to school, i was all about factoring, as I recall, and that was all we were tested on. I found the exercises chugging through complex conjugates, etc., very tedious. Euler’s formula was just an aside, alas. We were wrapping up the complex numbers unit, and he pulled it out of his hat, so to speak, during the last lecture. We never had time to learn what it was “good for”. I think we were just meant to be amazed at how raising e to the i*pi power gives you -1, of all things, and he walked us through the proof to hammer it home. I was duly amazed.

Nice notes on Euler’s formula. There’s another typo at bottom of page 9, in the last point where you’re deriving the orthogonality of the exponential functions: you need minus signs in various places. Either you want n -> -n in the exponentials or in the conditionals and the text below.

MB,

Thanks! Fixed.

Peter,

You say “One fundamental problem (besides that of not having a crucial, powerful tool) is that without using Euler’s formula, there’s no sensible way of understanding why the addition formulas (and thus most trig identities) are true.”

What’s wrong with drawing the picture? You can quickly derive both addition formulas by drawing a couple of right triangles and using the definitions of cosine and sine.

https://en.wikipedia.org/wiki/Proofs_of_trigonometric_identities

In my experience teaching college math and physics, the students are already struggling to remember the definitions of basic trig functions (hence SOH-CAH-TOA). Introducing complex numbers, rules for multiplying and conjugating them, and another formula (Euler’s) for memorization is probably going to overwhelm them.

Then again, the better students in the class will probably love it. I know I was amazed when I first learned about Euler’s formula. It was an episode of the Simpsons. It flew by Homer after he crawled through a portal to the “hypothetical” third dimension.

@Peter: I meant that while I remember complex numbers being covered in high school, I can’t recall whether Euler’s formula specifically was covered. I suspect that, as LMMI mentions, it might have been mentioned as an aside and so didn’t stick.

My first clear memory of Euler’s formula being presented in a way that made clear how useful and cool it is was in the Vibrations and Waves section of one of my early university physics courses.

G.S.,

Would any mathematician who needed to derive the addition formula do it using that triangle argument, or would they use Euler’s formula (as the page you link to also does)? Why use a non-obvious argument about triangles when there’s an obvious one using exponentials?

As Academic Lurker points out, the other reason students should know Euler’s formula is that it’s crucial for solving ODEs, including the basic ones used in physics to describe wave motion. If it’s needed for a second year physics course, it should be taught early on in the math curriculum.

I was wondering how our ODE textbook deals with this and just took a look at the book for our basic math ODE class. They use Euler’s formula, don’t assume students know it, put the explanation for why it’s true in a footnote. The idea that that’s the way math students should encounter this subject is kind of appalling.

So I specifically went to the bookshop to take a look at the book for 12th grade (“première enseignment secondaire classique“):

The formula

It’s a bit short, in a “just learn this” kind of way, but it’s there. I do remember that our

professeur~30 years ago enthusiastically showed us that it was indeed an extension of the exponential over the reals and we went on from there.Well, really, the best way to teach the various trig identities is to use the Taylor series 🙂

Peter is right, it’s depressing that this has been dropped from the math curriculum (in large part), especially since calc books have added so much useless garbage. I learned calculus first out of Sherwood Taylor, my dad’s book from City College in the early 50’s, it was 5 x 9 and maybe an inch and a half thick, missing most of what’s in current calculus books, but somehow they included stuff like Euler, and actual proofs.

I’m a little surprised by this whole discussion about Euler’s formula. The way I learned math in the first-year calculus course at my University (as a physicist) is more or less the way math students learned it as well — one introduces the exp function in a complex plane via its power series, prove convergence, basic identities etc, and then go on to introduce both trigonometric and hyperbolic functions (also over the full complex plane) by splitting exp into real and imaginary part, and evaluating the corresponding series on the real line, imaginary line, etc. The Euler formula is then actually a part of the definition of sin and cos.

The interesting question then is what do these functions and their power series have to do with triangles and planar geometry. This connection is then being made in a nontrivial manner, by studying the relationship between the complex plane and the real plane, and the map between U(1) and SO(2)…

But I was generally under the impression that everyone learns math in this way in a university course. The “elementary” trigonometry was something one learns in high school, but that any serious calculus course (university level) would introduce all elementary functions more rigorously, as a suitable power series in a complex plane, etc. So I am surprised that this is not the curriculum at every university out there.

Also, the Euler formula, despite its elegance, is not universally powerful in proving identities in trigonometry. For example, the famous identity

A sin x + B cos x = \sqrt(A^2 + B^2) sin (x + arctan B/A),

is a bit more tough nut to crack using Euler formula. 🙂

HTH, 🙂

Marko

vmarko:

Depends what you mean by “using Euler’s formula”. Simply by considering (Asin(x)+Bcos(x))/sqrt(A^2+B^2) we have (as the mathematicians say) reduced to problem to the case(s) already proved by Euler’s formula.

I did the same first year calc in college that all the other science majors had to take, including the physics and math majors (if they didn’t place out, which was rare, of course). Complex numbers barely came into it. Neither Euler’s formula nor the famous Identity were even mentioned, if I recall (and I would have, since the lattered bordered on mystical to me).

Well, to me the argument for including it early is simply that it’s amazing. I know, I know, it’s also terribly useful. But is it gauche to simply marvel at it? Generate a little more excitement for a subject that many find intimidating without having to resort to gimmicks, like they do in my niece’s high school? Sorry, OT probably. I’ll shut up.

Peter,

Every mathematician before Euler used arguments about triangles to prove those trig identities. While it’s true that I use Euler’s identity today to quickly derive trig identities and perform integrals involving trig functions, that method is only “obvious” to me because I’ve spent the last 20 years using complex numbers. I agree that they should learn it, but I question whether introducing it in Calc 2 is wise. I think you’ll end up confusing more students that you end up helping.

This whole thing actually reminds me of my own experience TA-ing a lower level physics class in grad school. I was always appalled that physics books simply state the solution to the simple harmonic oscillator differential equation as a linear combination of sine and cosine (or as complex exponentials). You can prove that it works, but there’s never an explanation for how that solution was guessed in the first place. It’s usually chalked up to “experience” that the students don’t have.

The only method they had seen up until then for solving a differential equation was to try a power series. So, away I went. I spent a discussion session showing that the power series solution that you get ends up being a linear combination of the power series for sine and cosine (which they had memorized from a previous class). Then I showed that this can be equivalently written as a linear combination of the power series for complex exponentials with opposite sign (by proving Euler’s identity from the power series).

The A students liked it, but most of the B and below students (the ones who could most use the discussion session) stopped attending after that.

G.S.,

I’ve taught Calculus II several times before, and often tried saying something about complex numbers and Euler’s formula. Some students get it, some don’t. The big problem is not that this is too hard for them, but that it’s not in the book, so they don’t have a good source outside of class to learn from. This year I tried to do a bit better on that with the notes I wrote, but that can’t replace the book.

The problem with saying “don’t teach this in Calculus II” is that there’s no where else we do teach this (I checked, it’s not explained in the ODE book). So, many students either leave here with several years of university mathematics education, but not knowing this, or they have to pick it up in the gutter (their physics class on wave phenomena….)

Quick application and mastering Euler’s formula is taught to every sophmore electrical engineer, as the practical method to represent AC currents, and more generally Narrowband Stochastic processes Every cellular modem works on the “I and Q” representataion of the received Signal + Noise.

It is so common like using a calculator, that probably nobody remembers that Euler invented it more than 300 years ago…..

sm,

The standard way to prove the formula I quoted is to use addition formula for the sine on the right-hand side, and then use the identities expressing sin and cos in terms of tan, reaching the left-hand side after a little algebra. That is the standard trigonometric proof.

My point was that there is no simpler way to prove it, say by using Euler formula but without invoking addition formula and sin-to-tan, cos-to-tan identities. So Euler formula is not an all-powerful magic bullet for everything in trigonometry. You can use it to simplify proofs of *some* theorems, but by no means *all* of them. Some things are just equally hard, whether you use Euler formula or not. 🙂

That said, I’m still completely perplexed to hear that it is not the main part of the standard curriculum at universities. I learned it as a central piece of complex analysis — second year undergraduate calculus course (both for math and physics majors). Somehow it seems odd that the University of Belgrade, Serbia, has a stronger undergraduate calculus course than Columbia University, New York. If that’s really the case, I’ll never suggest to any future student to ever go study at CUNY, since its math curriculum is just sub-par.

Best, 🙂

Marko

Math 1a is the basic mathematics course for all Caltech freshmen.

The curriculum is here:

http://www.math.caltech.edu/~2014-15/1term/ma001a/

Lecture 23 is complex numbers (lecture notes PDF: http://math.caltech.edu/~nets/lecture23.pdf )

There was a major revision in undergraduate mathematics requirements in 2012, the detailed outline is here (PDF): http://cue.caltech.edu/documents/22-core_with_tables_appendix.pdf

e.g.,

You may want to read through the document, e.g., later on:

The specific problems have been identified and are made up with a at-home online course: Transition to Mathematical Proof (TMP) course, for incoming students.

The TMP outcome is supposed to be:

Perhaps a good way to get a handle on what has happened as far as student’s preparation in high school is to consider Spivak’s calculus book. As I understand it (one of my professors in college was good friends with Spivak at Princeton, so I got various stories) he wrote it basically from his teaching notes for honors calc at Brandeis in the mid 60s. You would have trouble using it in a junior level analysis course now. He proves e is transcendental! It’s one of the prettiest books ever written, but no one is close to ready for it, certainly not as freshmen. Hell, in Calculus on Manifolds Spivak says all a student needs is one year of calc and one semester of linear algebra. In 22 years I’ve had one student who could handle that book as an undergraduate, and she’s a math professor at Carleton now.

I learned about Complex number in grade 12.

We used something called cis(x) = cos(x) + isin(x)

you get the idea. We just didn’t call it Euler’s formula.

It made things much easier.

Dear Peter,

The idea of your book seems very similar (although, of course, completely different in style) to this book http://www.springer.com/us/book/9783319192000, also published by Springer. Have you already read it? If yes, would you recommend reading it before or after reading your book (or not at all)?

vmmarko:

“My point was that there is no simpler way to prove it, say by using Euler formula but without invoking addition formula and sin-to-tan, cos-to-tan identities.”

And my point was that it is just a rewriting of the addition formula, which has already been elegantly proved by Euler.

Are you complaining that it is cheating to use cos(arctan(B/A))=A/sqrt(A^2+B^2), which follows using scaling to set sqrt(A^2+B^2)=1 and Peter’s unit circle – no more really than was already put into Euler (from the geometrical point of view) in the first place?

Long live Euler!

Dear Peter,

Longtime lurker —

Thanks for the interesting notes on Euler’s formula — I will share with my children, who are in geom/trig and calculus in school. One topic that this always raises for me, though, is WHY e and pi are transcendental. There seems to be little material available for interested laypersons. As I understand it, pi is transcendental because e is, and they are both transcendental because they have something to do with what we mean about having multidimensional space with directions orthogonal and how it is we rotate. Can you suggest good reading about 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?

they have to pick it up in the gutter (their physics class on wave phenomena….)It wasn’t easy learning Euler’s formula on the mean streets, but it toughened me up.

I almost mentioned this earlier, but I don’t think it invokes Euler’s formula. My copy (second edition) is around here somewhere, but I was lazy and just looked what was online.

And yes, we used it as freshmen (’84). My high school didn’t offer calculus, so I took it by correspondence course from the University of Wisconsin Extension. This was enough put me in the honors calc tier and do it all over from scratch. (Nonetheless, I didn’t qualify for honors analysis with a certain eyepatch-wearing fellow the next year; the substitute was not pretty. I’m not sure what honors analysis actually used, but I tried working my way through Apostol on my own.)