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 (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)
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 (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)