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I was reading the Wikipedia entry on Euler's identity ($e^{i\pi}+1=0$) and I came across this statement:

"The mathematician Carl Friedrich Gauss was reported to have commented that if this formula was not immediately apparent to a student upon being told it, that student would never become a first-class mathematician."

I know that this isn't too important and shouldn't be taken seriously, but I was just wondering what led him to say this. Is it really obvious? Was he assuming that the student should already know the series expansion for $e^x$? Etc.

Note: I am interested in Gauss's statement in particular, not a proof of Euler's identity.

All views on this will be appreciated, thanks.

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  • $\begingroup$ See math.stackexchange.com/questions/675139/…, math.stackexchange.com/questions/326150/…, math.stackexchange.com/questions/341214/proof-that-ei-pi-1, ... $\endgroup$ – Dietrich Burde Sep 22 '14 at 12:14
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    $\begingroup$ Could this just be a statement of the student's understanding of the abstraction of math? That this should not be such a strange relationship because 1 is no stranger than $ i$ which no stranger than calling something 2 or 3. $\endgroup$ – dylan7 Sep 22 '14 at 12:39
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    $\begingroup$ Gauss published the first acceptable proof of the Fundamental Theorem of Algebra as his doctoral dissertation at age 21. He was smarter than everyone. $\endgroup$ – user4894 Sep 22 '14 at 21:04
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    $\begingroup$ In the spirit of Gauss' quotes, I would have replied "The result was apparent to me immediately: but I do not yet know how I am to arrive at them." $\endgroup$ – Isomorphism Sep 23 '14 at 2:28
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    $\begingroup$ I don't think the series expansion would occur at all in the simplest and most intuitive way to see that this identity is true. ${}\qquad{}$ $\endgroup$ – Michael Hardy Oct 5 '14 at 19:29
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The Wikipedia entry, and a few other places that mention this, give as reference the popular book of John Derbyshire on the Riemann hypothesis. The book itself mentions this in the spirit of an apocryphal tale, with the passing remark that "I wouldn't put it past him", and gives no reference at all.

It's probably best to treat this story with utmost suspicion until better evidence turns up. This is the kind of tale that proliferates in mathematics because on the one hand it adds to the legend of the people we admire, whom we suspend critical judgement for in favour of awe and wonder, and on the other hand it plays on the insecurities of the average aspiring mathematician.

For people with such pretense of belief in reason, mathematicians are a little too often purveyors of rumours and hearsay themselves.

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    $\begingroup$ Since John Derbyshire's book has been mentioned, let me take the opportunity to say I think this is an excellent book for the interested amateur (i.e. me :-). It's the first non-professional mathematician book I've read that really explained Riemann's work in this area. $\endgroup$ – John Rennie Sep 24 '14 at 6:10
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If you look at the formula geometrically (and draw the unit circle), you'll see immediately that $e^{\pi i}=-1$, since it's half a circle from the origin.

I don't see any other way for this to be all too apparent.

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    $\begingroup$ That assumes the nontrivial fact that $e^{ix}=\cos x+i\sin x$. (You don't need calculus to derive that, though - only De Moivre and the fact that $\sin(x)\approx x$ and $\cos(x)\approx1$ when $x$ is small, plus the definition of $e$ ($e\approx (1+x)^{1/x}$ when $x$ is small). The approximations for sine and cosine can be seen fairly easily geometrically.) $\endgroup$ – Akiva Weinberger Sep 23 '14 at 3:03
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    $\begingroup$ If you want to fuss about every "nontrivial fact" that was implicitly assumed and used by 18th century mathematicians, you're in for a long evening. $\endgroup$ – Asaf Karagila♦ Sep 23 '14 at 3:06
  • $\begingroup$ P.S. Too tired to write up the proof I was thinking of, so don't ask. $\endgroup$ – Akiva Weinberger Sep 23 '14 at 3:07
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    $\begingroup$ @AsafKaragila but the quote in the question concerns a student, not a (fully trained) mathematician. Would a student of the time have implicitly assumed and used $e^{ix} = \cos x+i\sin x$? $\endgroup$ – David Z Sep 23 '14 at 7:05
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    $\begingroup$ Even more specifically, it concerns a student whom Gauss considers will become a first-class mathematician. That is, it concerns Gauss ;-) But clearly, if the student is "told the formula" without first being told the meanings of $e$, $i$, and $\pi$, then it could not be obvious no matter their calibre. So we must consider what Gauss expects a student to already know, which is either going to be some variant of $e^{i \theta} = \cos \theta + i \sin \theta$, or "geometric" reasoning in the complex plane, or both. What I don't know, is which variant. $\endgroup$ – Steve Jessop Sep 23 '14 at 15:25
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I have to agree with Prometheus, as this is probably bunk. However, I think there is an interesting interpretation to be gleaned from this Gauss-attributed comment.

I've had the distinct pleasure of talking with UChicago's Professor Benson Farb, an esteemed geometer and one of my personal idols, on two occasions.

The first time I talked with him was through a Skype call. The conversation was centered around what I could do, as a budding mathematician, to start heading in the right direction. I don't recall his exact words, but one of his main points was that I should focus on having a strong foundation before getting into advanced topics. I truly agree with this idea; during my sophomore year, I'd work for at least four hours every day just on calculus. Now, I can not only do geometry, but I can do it well because of my stronger background.

The second time I met him was at a conference in Columbus, Ohio. His lecture (link goes to YouTube) was geared toward undergraduate mathematicians. At 45:30, he gives a variation on the same theme: working with "the basics" helped him become successful.

I also had the opportunity to Skype with another UChicago faculty member, Professor Calegari, while I was at an international conference in Canada. Unsurprisingly, he had the same message: foundations are good.

I think we become, as Farb puts it, "enamored with machinery." We tend to focus on the cool things without paying due attention to the smaller gears to the larger machine. Perhaps Gauss, or his impersonator, meant that if we learned that $e^{i\pi}=-1$ before we learned the series expansions for $x\mapsto e^x$, $x\mapsto\sin(x)$, and $x\mapsto\cos(x)$, then we wouldn't be able to function as well, as mathematicians?

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I fail to understand why we are looking at this so hard and not seeing it. We do not anymore have the series expansion of e^x, sin x, and cos x at our immediate reach* but at the time they did. The claim which to us sounds preposterous merely makes him a man of his day.

* It would take us several seconds to recall each one of them.

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I imagine that since Gauss spent most of his life studying complex numbers, it would have been immediately apparent to him that $e^{\pi i}=\cos(\pi)$.

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    $\begingroup$ Gauss did a lot of mathematics that had nothing to do with complex numbers. $\endgroup$ – Gerry Myerson Aug 8 '16 at 9:25
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As an electrical engineer's son, the way I first heard the definition of multiplication of complex numbers was that a multiplication $z\mapsto az$ is consists of rotating and dilating. That immediately tells you that the exponential function of $n$, $\{i^n\}_{n\in\mathbb Z}$ just goes around in circles.

Learning about $e$ is another step along the way. Once you've understood what's "natural" about $e$ and that the graph of that exponential function is a helix, you're almost there.

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