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Comment by ThrustVectoring | Hacker News Reader

Similarly, Euler's Identity (e^i*pi = -1) becomes a lot more intuitive. The exponential function is continuous (edit: proportional) growth, i is turning sideways, sideways continuous growth traces out a circle, and pi is halfway around a circle.

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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 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).

"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.

Sure, complex multiplication is rotation, no problem with that part. To multiply two complex numbers you add their angles and multiply their lengths.

Parent is referring to the ODE \dot{z} = i z.

I 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.

No, 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, which doesn't seem to ever explain it that way explicitly. It's funny how memory works.)

Okay, 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.

e^(epsilon) ~ 1 + epsilon in general, because of the definition of the derivative and the fact that d/dx (e^x) = e^x

Right, 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?

I 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.

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