4 ms·

I agree with you that e is not arbitrary. I say that the period of 2π for e^ix is arbitrary, because we've arbitrarily defined periods of sin and cos as 2π. If

I agree with you that e is not arbitrary. I say that the period of 2π for e^ix is arbitrary, because we've arbitrarily defined periods of sin and cos as 2π. If we defined a function sin to take not an angle in radians, but in degrees (with a period of 360.0), and used that definition of sin in our math, then our complex e^ix would have a period of exactly 360, and the entire complex math would still work — for example, Euler's formula below would still hold: e^ix = cos x + i sin x And people in comments would rave about how magic number 360 is, and its magic properties were discovered by Romans two thousand years ago.

codeflo 3y ago
You seem to think that the 2pi is injected into the definition of e^ix somewhere, but actually it's the other way round, 2pi comes out as a theorem. I'll give the rough outline. exp(x) for complex x is simply defined to be the infinite sum from k = 0 to infinity of x^k/k!. That is, exp(x) = 1 + x + x^2/2 + x^3/6 + x^4/24 + x^5/120 ... (BTW, the motivation for this definition is that exp'(x) = exp(x), which shouldn't be too hard to see because it's already a Tailor series.) Purely from this you can prove that exp(ix) with real x is periodic with period 6.28... It just so happens that this number is also the circumference of the unit circle.
- quickthrower2 3y ago
  I had never spotted that before, each term of the series is the integral of the previous. That is pleasing! The Pi thing feels now less of a coincidence than the fact that exp is a power. That probably falls out of expanding the polynomials but it so ingrained as taken for granted that it is wonderous when you think about it.
- nlitened 3y ago
  Hey, you're right! So Pi is special :)
quickthrower2 3y ago
I guess radians are “magic” in that sin and cos can be defined by infinite series that look nice (and feel canonical). You have to manipulate those series to get 360 or even just revolutions.
- nlitened 3y ago
  As codeflo showed in a sibling comment, actually I am wrong, and Pi's special magic also becomes from Taylor's series expansion. So turns out that Pi is the real magic number rather than just our arbitrary choice!

codeflo 3y ago

You seem to think that the 2pi is injected into the definition of e^ix somewhere, but actually it's the other way round, 2pi comes out as a theorem. I'll give the rough outline. exp(x) for complex x is simply defined to be the infinite sum from k = 0 to infinity of x^k/k!. That is, exp(x) = 1 + x + x^2/2 + x^3/6 + x^4/24 + x^5/120 ... (BTW, the motivation for this definition is that exp'(x) = exp(x), which shouldn't be too hard to see because it's already a Tailor series.) Purely from this you can prove that exp(ix) with real x is periodic with period 6.28... It just so happens that this number is also the circumference of the unit circle.

quickthrower2 3y ago
I had never spotted that before, each term of the series is the integral of the previous. That is pleasing! The Pi thing feels now less of a coincidence than the fact that exp is a power. That probably falls out of expanding the polynomials but it so ingrained as taken for granted that it is wonderous when you think about it.
nlitened 3y ago
Hey, you're right! So Pi is special :)

quickthrower2 3y ago

I guess radians are “magic” in that sin and cos can be defined by infinite series that look nice (and feel canonical). You have to manipulate those series to get 360 or even just revolutions.

nlitened 3y ago
As codeflo showed in a sibling comment, actually I am wrong, and Pi's special magic also becomes from Taylor's series expansion. So turns out that Pi is the real magic number rather than just our arbitrary choice!