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... what? I guess if it makes sense to you, ok, but "continuous growth in a circle" to explain the complex exponential sounds like cloud-gazing to me: the huma
by jordigh 10y ago
... what?
I guess if it makes sense to you, ok, but "continuous growth in a circle" to explain the complex exponential sounds like cloud-gazing to me: the human mind making up patterns where there are none.
The only way it makes sense to me is by looking at the power series and seeing how the exponential power series is almost the same as the sine and cosine power series (alternatively and equivalently: all of the exponential, hyperbolic, and circular functions satisfy almost the same differential equation).
- jerf 10y ago"Continuous growth in a circle" may indeed be a personal visualization, but "complex numbers are deeply connected to rotation" is a mainstream point of view in mathematics.
- jordigh 10y agoSure, complex multiplication is rotation, no problem with that part. To multiply two complex numbers you add their angles and multiply their lengths.
- ittekimasu 10y agoParent is referring to the ODE \dot{z} = i z.
- ThrustVectoring 10y agoI think I was a bit loose in the wording in order to try to get the phrasing as compact as I could. The exponential function is continuous proportional growth - the limit of adding a part of a vector to itself proportional to the current vector and the "step size" as you take the step size to zero. (Real numbers are just the one-dimensional special case of vectors here). Anyway, my point is that Euler's identity generalizes to any number that encapsulates "sideways" in Euclidean geometry. Any quaternion "x" that is a unit of rotation will satisfy "e^pi*x = -1". You can't get that easily from the power series understanding.
- abecedarius 10y agoNo, it makes sense. exp(i epsilon) = 1 + i epsilon, to first order, i.e. when epsilon is near 0. You can think of 1 + i epsilon as a tiny rotation, when you multiply by it. exp(i t) = the product of a lot of tiny rotations, just as in real numbers exp(x) = the product of a lot of tiny scalings. The product of rotations equals one rotation by the sum of their (individually tiny) angles. As t increases continuously, exp(i t) rotates around from 1 continuously. The unchanging speed of rotation corresponds to how in e.g. bacterial growth, exp(t), the population gets some constant factor bigger in every equal interval. (I first got this picture from the Feynman lecture on algebra, http://www.feynmanlectures.caltech.edu/I_22.html http://www.feynmanlectures.caltech.edu/I_22.html, which doesn't seem to ever explain it that way explicitly. It's funny how memory works.)
- jordigh 10y agoOkay, this makes a little more sense to me, since it's just the power series again. But I don't think that calling a rotation a growth makes sense. Also, to me, the approximation e^(i epsilon) ~ 1 + i epsilon kind of begs the question.
- ThrustVectoring 10y agoe^(epsilon) ~ 1 + epsilon in general, because of the definition of the derivative and the fact that d/dx (e^x) = e^x
- jordigh 10y agoRight, the latter fact is equivalent to considering exp the solution of a differential equation, which is a much bigger fact. Treating the approximation as an innocent brute fact is what I considered begging the question, because why should this differential equation be valid for complex values?
- abecedarius 10y agoI was trying to make a geometric intuition accessible to high schoolers who haven't mastered calculus -- though I haven't tried it on an actual teenager. Of course there's also value in looking deeper into the foundations.