As always with this author, I followed a link to a math explanation video expecting to point out how it is only a superficial treatment that breaks down under r

As always with this author, I followed a link to a math explanation video expecting to point out how it is only a superficial treatment that breaks down under rigorous analysis, only to be dissapointed in my inability to find any such flaws. Seriously, this guy is awesome. If you need to learn math, see if he has a video about it. I do have 1 gripe with this video though, and that is his handwaving around "natural". More specifically, as far as this particular case goes, there is nothing natural about the choice of e^x, or the significance of pi. As he identifies, we are interested in some function that maps adders into multipliers with the property f(x+y)=f(x)f(x). To uniquely identify such a function, we need to add an additional constraint. He chose to add f(pi) = -1. He justifies this by arguing that pi is the length you would travel along the unit circle to arrive at -1. This is true (and the underlying reason why pi and e end up being natural), however using this argument seems to break the abstraction for me. Under this construction, in the equation f(i * pi)=-1, "i * pi" is an object in the set of multipliers, and "-1" is an object in the set of adders. Specifically, "i * pi" is a function which takes a plane (or perhaps a point) and rotates it, while "-1" is a function which takes a plane (or point) and slides it. He then invokes an unstated mapping, g, to convert the multiplier [0] "i * pi" into the real number "pi". He than insists that g(x) gives the distance a point would travel along the unit circle when the multiplier x is applied to it. At this point, because arriving at the point "-1" [1] from the point "1" through rotation requires traveling pi distance, it makes sense that f(i * pi) = f(g^-1(pi)) = -1 I am still not convinced that (within this construction alone), this is a more natural choice that saying that f(i) = 1, but would agree that it is one of the two natural choices. To get to f(x) = e^x being the natural choice requires showing that it comes up in all sorts of unrelated parts of math, so it is probably more natural. [0] It might be better to speak of "rotational multipliers" here, as I am not sure how to natural define g(x) for multipliers that stretch the plane, instead of or in addition to simply rotating it. [1] This "1" is again distinct from the adder "1" and multiplier "1", but plays a central role in defining them, so I do not object to its usage.