Re:The exponential function is a miracle
by jacobolus on Friday September 13, 2019 @03:09PM
(#20965323)
> Eulers fornulae^ix = cos x + i sin xwas discovered in 1748
Discovered by Roger Cotes in 1714, in IMO its most natural form, ix = log(cos x + i sin x)
> actual important identity you care about is that you can add angles by multiplying
Personally I think the multiplicative concept of rotations is the natural one, with rotations associated to points on a circle embedded in the plane, rather than associated to arclengths. The amazing thing is that we can compose rotations (naturally multiplicative) by first taking the logarithm of the rotations (a.k.a. angle measure) adding them, and then taking the inverse logarithm. The logarithm is a tool which turns multiplication into addition.
The protractor and the slide rule turn out to be more or less the same concept, which is why we can substitute the former for the latter in https://en.wikipedia.org/wiki/Prosthaphaeresis
Discovered by Roger Cotes in 1714, in IMO its most natural form, ix = log(cos x + i sin x)
> actual important identity you care about is that you can add angles by multiplying
Personally I think the multiplicative concept of rotations is the natural one, with rotations associated to points on a circle embedded in the plane, rather than associated to arclengths. The amazing thing is that we can compose rotations (naturally multiplicative) by first taking the logarithm of the rotations (a.k.a. angle measure) adding them, and then taking the inverse logarithm. The logarithm is a tool which turns multiplication into addition.
The protractor and the slide rule turn out to be more or less the same concept, which is why we can substitute the former for the latter in https://en.wikipedia.org/wiki/Prosthaphaeresis