Personally, I prefer the version with tau (2 times pi) in it rather than the one with pi:
e^(i*tau) = 1
I won't reproduce https://www.tauday.com/tau-manifesto here, but I'll just mention one part of it. I very much prefer doing radian math using tau rather than pi: tau/4 radians is just one-fourth of a "turn", one-fourth of the way around the circle, i.e. 90°. Which is a lot easier to remember than pi/2, and would have made high-school trig so much easier for me. (I never had trouble with radians, and even so I would have had a much easier time grasping them had I been taught them using tau rather than pi as the key value).
Which would be e^(i*tau) - 1 = 0 if you wanted to honor the spirit of the Identity.
Never liked that form of the Euler's formula. I prefer the following:
(-1)ˣ = cos(πx) + i sin(πx)
That's not the point of the Identity. You exponentiated the beauty right out of it.
Beauty is in the eye of the beholder.
Instead shoehorning it into an arbitrary symbol salad by gimping its generality, I prefer the one which makes a statement: "What does it mean to apply inversion partially?"
Personally, I prefer the version with tau (2 times pi) in it rather than the one with pi:
e^(i*tau) = 1
I won't reproduce https://www.tauday.com/tau-manifesto here, but I'll just mention one part of it. I very much prefer doing radian math using tau rather than pi: tau/4 radians is just one-fourth of a "turn", one-fourth of the way around the circle, i.e. 90°. Which is a lot easier to remember than pi/2, and would have made high-school trig so much easier for me. (I never had trouble with radians, and even so I would have had a much easier time grasping them had I been taught them using tau rather than pi as the key value).
Which would be e^(i*tau) - 1 = 0 if you wanted to honor the spirit of the Identity.
Never liked that form of the Euler's formula. I prefer the following:
That's not the point of the Identity. You exponentiated the beauty right out of it.
Beauty is in the eye of the beholder.
Instead shoehorning it into an arbitrary symbol salad by gimping its generality, I prefer the one which makes a statement: "What does it mean to apply inversion partially?"