@Anonymous: That’s a great insight! This site has a cool diagram showing the spiral interpretation: http://www.songho.ca/math/euler/euler.html.
I love that. any number can be written in terms of e ‘growing’ at a certain rate for a certian amount of time! brilliant!
[…] the "restoring force" like "positive or negative interest". This makes the sine/e connection in Euler's formula easier to understand. Sine is like e, except sometimes it earns negative interest. There's more to […]
Kalid, you are my new bestfriend. 
@Loui: Math brings everyone together!
thanks!
Second paragraph under “But Shouldn’t We Spin Faster and Faster?”: mustn’t it be “if our growth rate was twice that (2ln(2)), it would look the same as growing twice as fast (2x vs x)" instead of "if our growth rate was twice that (2ln(2)), it would look the same as growing for twice as long (2x vs x)”? Thank you.
Really helpful for the project that my group is doing in Math class! Thanks, I understand this so much more clearly now!
the comment above is from me, i forgot to write my name but i just wanted all of you math lovers to know that i LOVE math!
math is a wonderful thing
math is a really cool thing
so get off your ass, lets do some math
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@Paul: Great question. I clarified the sentence a little – e^rt has the cool property that increasing the growth rate (2*ln(2) vs ln(2)) would have the same effect as increasing the amount of time spent (2x vs x).
@Sam: Awesome, glad it helped!
@Jamie: Math rocks :).
This article is amazing! Reading it makes you feel like a genius.
In the section “what does i as an exponent do” it seems like you are saying that 3^4 is equivalent to ln(3) * 4. Obviously, this is wrong.
Nevermind I see what I was missing.
3^4 = (e^ln(3))^4
3^i = (e^ln(3))^i
Kalid, the intuitions your articles provide are of unmatched quality. Thanks, and keep writing!
I really like the physics interpretation wererogue mentioned (angular and uniform acceleration), which is an excellent metaphor. But for me (who thought about this topic like 19^th century Benjamin), the most important inside you provided is that (-1)^n becomes a less awkward special case if you have complex numbers. The e^(i pi n) model now feels more elegant and obviously superior 
I need to read more (all) of your articles, but in case you haven’t yet: Would you please demystify residues and contour integrals ?
Doh, two! wrong words in the analogy. Although the idea of viewing the imaginary part as kind of centripetal force seems still nice, the real part is of course not uniform but proportional to the position. So x’’ = x works perfectly, but the analogy (with one acceleration growing exponentially, and the other one staying constant, if I got it right this time) is maybe not that nice after all.
@Rick: Thanks! I love sharing those aha moments with people
@Peter: Thanks for the comment – I’ve updated the article to be more clear in that section.
@Benedikt: Wow, appreciate the note! I’d love to do those topics as I learn more about them!
For the analogy, yes, you have to
@Rick: Thanks! I love sharing those aha moments with people
@Peter: Thanks for the comment – I’ve updated the article to be more clear in that section.
@Benedikt: Wow, appreciate the note! I’d love to do those topics as I learn more about them!
For the analogy, yes, you have to sort of pick the portions where it works best. There’s clearly some issues with scaling and rotating at once – I prefer to split them up.
So, if your exponent is e^(a+bi) then I see that as a combination of real exponential growth and imaginary exponential growth (rotation). It’s really e^a * e^bi and you get the best of both :). I’m not sure if this answers your question? If it doesn’t, let me know.
I’m still trying to grasp the concepts, and have some basic ideas.
One analogy to this movement is like a pen moving in the x direction and writing on a rotating flat disk below the pen (the disks diameter is parallel to the x axis). A (2-dimensional) spiral will be created. The spiral will be perpendicular to the pen then. This is just a basic analogy, perhaps a rotating cylinder centered on the x axis would be another analogy. The drawn figure would be a 3-dimensional spiral.
If anyone knows a similar type of analogy, let us know.