3 ms·

Of course! I think there's a good reason (1) is the first definition of e^x we get in school, though. I don't think it's the most "natural" definition, but I th

Of course! I think there's a good reason (1) is the first definition of e^x we get in school, though. I don't think it's the most "natural" definition, but I think it's the most grok-able. It's easy for high-schoolers to learn precisely _because_ each part of it is so intimately familiar that it's clear without hand-waving precisely what everyone's talking about. When we then step into viewing real exponentials as "The thing that gets bigger at a rate proportional to where we're at", it's un-abstract, which is honestly important for intuition. Of course you're going to want to then view it with the calculus definition, or as a matrix, or as an operator... But I wouldn't start there. So I feel the same about the complex version. There's another nice proof in this thread that starts with the calculus definition and follows a similar track. It's a much more beautiful proof! But I worry some will lose intuition when we start the conversation with "a complex exponential's rate of change is perpendicular to its position"... That said, if I were to ever teach a math class on complex numbers, I think fleshed-out versions of these two 4-step proofs are the ones that together best build intuition for what "rotation in complex exponentials" really means. That rotational aspect is the foundation of much of the handwaving done in control theory, DSP, ASP, signals and systems, and really anything involving fourier and laplace transforms. So, it's important to grok it well.

jacobolus 10y ago
Anything which repeats on some periodic interval (e.g. a signal which repeats in time) can be represented geometrically as being defined with respect to uniform motion around a circle (sometimes there are physical circles involved, and sometimes it’s just an abstract circle). Complex numbers (a two-part complex of a scalar part and a bivector part) and the complex logarithm/exponential are the natural formalism to use for describing uniform circular motion, and are therefore the natural formalism for any kind of periodic signal. The way to teach this is to start with vectors in the plane, and then teach about geometric products/quotients of vectors (this is a subject called “geometric algebra” or “Clifford algebra”, and the basics are plenty accessible to high school students). All of the mystery is removed from complex numbers when they are taught this way. http://www.shapeoperator.com/2016/12/12/sunset-geometry/ http://www.shapeoperator.com/2016/12/12/sunset-geometry/

jacobolus 10y ago

Anything which repeats on some periodic interval (e.g. a signal which repeats in time) can be represented geometrically as being defined with respect to uniform motion around a circle (sometimes there are physical circles involved, and sometimes it’s just an abstract circle). Complex numbers (a two-part complex of a scalar part and a bivector part) and the complex logarithm/exponential are the natural formalism to use for describing uniform circular motion, and are therefore the natural formalism for any kind of periodic signal. The way to teach this is to start with vectors in the plane, and then teach about geometric products/quotients of vectors (this is a subject called “geometric algebra” or “Clifford algebra”, and the basics are plenty accessible to high school students). All of the mystery is removed from complex numbers when they are taught this way. http://www.shapeoperator.com/2016/12/12/sunset-geometry/ http://www.shapeoperator.com/2016/12/12/sunset-geometry/