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This explanation strikes me as a little too aggressive in throwing out the notation with the bathwater, only reaching its result by redefining the terms we alre

This explanation strikes me as a little too aggressive in throwing out the notation with the bathwater, only reaching its result by redefining the terms we already have intuition for into space-stretching operations that don't work like arithmetic does in my head. In that sense I think I have the same problem with this proof that I do with the standard one, where you add the Maclaurin series of cos(θ) and i * sin(θ) and match term-by-term with the series for e^(i * θ). The problem is, at the point you can actually show equality, the things on one side aren't obviously a rotation and the things on the other side aren't obviously an exponential. I'm not just hear to yell at clouds. I was given a proof that I truly love by a professor I adore, which I think really does give insight into what all these operators are doing. The best video I can find with it is here: https://www.youtube.com/watch?v=-dhHrg-KbJ0 https://www.youtube.com/watch?v=-dhHrg-KbJ0 (Skip to 7:30 if you're already comfortable with the limit definition of e^x) The basic summary is: 1) e^iθ is equal to (1 + iθ/n) ^ n for large n 2) That base, (1 + iθ/n), plotted as a complex number, has length approaching 1, angle approaching θ/n 3) The base squared, (1 + iθ/n)^2, by de moivre's theorem, forms another point as if the transformation from (0, 1) were repeated twice — that is, the length stays one, and another tiny angle is added for a total of 2θ/n 4) The full result is therefore n transformations, taking the path along the unit circle, traveling θ and arriving at cos(θ) + i * sin(θ)

mistermumble 10y ago
Thanks for the pointer to the video ("E to the Pi for Dummies" by Mathologer). It is definitely worth watching. Much more understandable, imho, than the one referenced at the top of this thread. Your text summary of the video is accurate and likely makes sense to some people but not so much for others (namely, me). Tour words are hard to grok until you see the visual depiction --- until you see the sequence of n transformations become a spiral arrangement of triangles that ends up approximating the (-1, 0i) point in the complex plane.
jacobolus 10y ago
Your (1) is just one of many possible (technically equivalent) definitions for the exponential function. And arguably not the most basic/natural/intuitive one. The primary motivation for the exponential function is to be the inverse of the logarithm function. And the motivation of logarithms is to convert multiplication problems to addition problems, so they could be solved with table lookups (later performed on a slide rule) instead of difficult arithmetic. That is, log ab = log a + log b. Which is to say, exp(c + d) = (exp c)(exp d). Just setting this constraint along with the derivative exp’ 0 = 1 is enough to characterize the exponential function. Or you can get to this function in many other ways, e.g. by solving the differential equation d/dx exp x = exp x; by the series 1 + x + x^2/2 + x^3/6 + ...; or by defining the logarithm as the definite integral of 1/x starting at 1, and then taking the exponential function to be its inverse. I like defining the (complex) exponential as a conformal mapping between the cylinder and the plane minus a point.
- unholiness 10y ago
  Of course! I think there's a good reason (1) is the first definition of e^x we get in school, though. I don't think it's the most "natural" definition, but I think it's the most grok-able. It's easy for high-schoolers to learn precisely _because_ each part of it is so intimately familiar that it's clear without hand-waving precisely what everyone's talking about. When we then step into viewing real exponentials as "The thing that gets bigger at a rate proportional to where we're at", it's un-abstract, which is honestly important for intuition. Of course you're going to want to then view it with the calculus definition, or as a matrix, or as an operator... But I wouldn't start there. So I feel the same about the complex version. There's another nice proof in this thread that starts with the calculus definition and follows a similar track. It's a much more beautiful proof! But I worry some will lose intuition when we start the conversation with "a complex exponential's rate of change is perpendicular to its position"... That said, if I were to ever teach a math class on complex numbers, I think fleshed-out versions of these two 4-step proofs are the ones that together best build intuition for what "rotation in complex exponentials" really means. That rotational aspect is the foundation of much of the handwaving done in control theory, DSP, ASP, signals and systems, and really anything involving fourier and laplace transforms. So, it's important to grok it well.
  - jacobolus 10y ago
    Anything which repeats on some periodic interval (e.g. a signal which repeats in time) can be represented geometrically as being defined with respect to uniform motion around a circle (sometimes there are physical circles involved, and sometimes it’s just an abstract circle). Complex numbers (a two-part complex of a scalar part and a bivector part) and the complex logarithm/exponential are the natural formalism to use for describing uniform circular motion, and are therefore the natural formalism for any kind of periodic signal. The way to teach this is to start with vectors in the plane, and then teach about geometric products/quotients of vectors (this is a subject called “geometric algebra” or “Clifford algebra”, and the basics are plenty accessible to high school students). All of the mystery is removed from complex numbers when they are taught this way. http://www.shapeoperator.com/2016/12/12/sunset-geometry/ http://www.shapeoperator.com/2016/12/12/sunset-geometry/
hasbroslasher 10y ago
I agree, the suggestion that anyone could/would be able to just throw out everything they know about arithmetic is sketchy at best, and even it's sketchier to make it seem as though a result like Euler's formula could purely be derived from geometric intuition. Sure, it "makes sense" in the video, but I don't think it's possible to truly grok and appreciate the result without having done any calculus. The real joy of math isn't in knowing, it's in proving. Anyway: as for proof methods, I like the Maclaurin series trick with this one. Too bad the internet hasn't come up with a universal/nice way of writing math notation in a readable format yet, I'd like to sketch that proof here...
- unholiness 10y ago
  The Maclaurin trick is what I'd call the "standard proof", and I think it's the source of a lot of the lack of intuition surrounding e^i... It's exactly what you call it, a "trick", where you prove two rabbits are identical by transforming them both into an infinite number of hats. On the one hand, it's all sound logic and algebra, but on the other hand, I certainly don't blame Randall Munroe for seeing that proof in high school and going on to say "I have never been totally satisfied by the explanations for why e to the ix gives a sinusoidal wave."[0] [0] https://xkcd.com/179/ https://xkcd.com/179/
  - hasbroslasher 10y ago
    Love that xkcd, but I found the Maclaurin "trick" to be extremely common-sensical... If you're already finding the area of infinitely many small boxes under a curve then Taylor series don't seem that weird. Once you are convinced that Taylor series are O.K. then this proof is really cool. I also think it's weird to show high-schoolers this, as about 1% of high school seniors have the mathematical knowledge needed to grok this level of math. Anyways - people have differing levels of magic tolerance, and some people don't have a lot of background in calc, so I don't blame them for looking for other explanations.
DiThi 10y ago
I came here to post that video by Mathloger, it's very well explained.

mistermumble 10y ago

Thanks for the pointer to the video ("E to the Pi for Dummies" by Mathologer). It is definitely worth watching. Much more understandable, imho, than the one referenced at the top of this thread. Your text summary of the video is accurate and likely makes sense to some people but not so much for others (namely, me). Tour words are hard to grok until you see the visual depiction --- until you see the sequence of n transformations become a spiral arrangement of triangles that ends up approximating the (-1, 0i) point in the complex plane.

jacobolus 10y ago

Your (1) is just one of many possible (technically equivalent) definitions for the exponential function. And arguably not the most basic/natural/intuitive one. The primary motivation for the exponential function is to be the inverse of the logarithm function. And the motivation of logarithms is to convert multiplication problems to addition problems, so they could be solved with table lookups (later performed on a slide rule) instead of difficult arithmetic. That is, log ab = log a + log b. Which is to say, exp(c + d) = (exp c)(exp d). Just setting this constraint along with the derivative exp’ 0 = 1 is enough to characterize the exponential function. Or you can get to this function in many other ways, e.g. by solving the differential equation d/dx exp x = exp x; by the series 1 + x + x^2/2 + x^3/6 + ...; or by defining the logarithm as the definite integral of 1/x starting at 1, and then taking the exponential function to be its inverse. I like defining the (complex) exponential as a conformal mapping between the cylinder and the plane minus a point.

unholiness 10y ago
Of course! I think there's a good reason (1) is the first definition of e^x we get in school, though. I don't think it's the most "natural" definition, but I think it's the most grok-able. It's easy for high-schoolers to learn precisely _because_ each part of it is so intimately familiar that it's clear without hand-waving precisely what everyone's talking about. When we then step into viewing real exponentials as "The thing that gets bigger at a rate proportional to where we're at", it's un-abstract, which is honestly important for intuition. Of course you're going to want to then view it with the calculus definition, or as a matrix, or as an operator... But I wouldn't start there. So I feel the same about the complex version. There's another nice proof in this thread that starts with the calculus definition and follows a similar track. It's a much more beautiful proof! But I worry some will lose intuition when we start the conversation with "a complex exponential's rate of change is perpendicular to its position"... That said, if I were to ever teach a math class on complex numbers, I think fleshed-out versions of these two 4-step proofs are the ones that together best build intuition for what "rotation in complex exponentials" really means. That rotational aspect is the foundation of much of the handwaving done in control theory, DSP, ASP, signals and systems, and really anything involving fourier and laplace transforms. So, it's important to grok it well.
- jacobolus 10y ago
  Anything which repeats on some periodic interval (e.g. a signal which repeats in time) can be represented geometrically as being defined with respect to uniform motion around a circle (sometimes there are physical circles involved, and sometimes it’s just an abstract circle). Complex numbers (a two-part complex of a scalar part and a bivector part) and the complex logarithm/exponential are the natural formalism to use for describing uniform circular motion, and are therefore the natural formalism for any kind of periodic signal. The way to teach this is to start with vectors in the plane, and then teach about geometric products/quotients of vectors (this is a subject called “geometric algebra” or “Clifford algebra”, and the basics are plenty accessible to high school students). All of the mystery is removed from complex numbers when they are taught this way. http://www.shapeoperator.com/2016/12/12/sunset-geometry/ http://www.shapeoperator.com/2016/12/12/sunset-geometry/

hasbroslasher 10y ago

I agree, the suggestion that anyone could/would be able to just throw out everything they know about arithmetic is sketchy at best, and even it's sketchier to make it seem as though a result like Euler's formula could purely be derived from geometric intuition. Sure, it "makes sense" in the video, but I don't think it's possible to truly grok and appreciate the result without having done any calculus. The real joy of math isn't in knowing, it's in proving. Anyway: as for proof methods, I like the Maclaurin series trick with this one. Too bad the internet hasn't come up with a universal/nice way of writing math notation in a readable format yet, I'd like to sketch that proof here...

unholiness 10y ago
The Maclaurin trick is what I'd call the "standard proof", and I think it's the source of a lot of the lack of intuition surrounding e^i... It's exactly what you call it, a "trick", where you prove two rabbits are identical by transforming them both into an infinite number of hats. On the one hand, it's all sound logic and algebra, but on the other hand, I certainly don't blame Randall Munroe for seeing that proof in high school and going on to say "I have never been totally satisfied by the explanations for why e to the ix gives a sinusoidal wave."[0] [0] https://xkcd.com/179/ https://xkcd.com/179/
- hasbroslasher 10y ago
  Love that xkcd, but I found the Maclaurin "trick" to be extremely common-sensical... If you're already finding the area of infinitely many small boxes under a curve then Taylor series don't seem that weird. Once you are convinced that Taylor series are O.K. then this proof is really cool. I also think it's weird to show high-schoolers this, as about 1% of high school seniors have the mathematical knowledge needed to grok this level of math. Anyways - people have differing levels of magic tolerance, and some people don't have a lot of background in calc, so I don't blame them for looking for other explanations.

DiThi 10y ago

I came here to post that video by Mathloger, it's very well explained.