> 10^(iπ) is some horrible complex number Sure, you want to use a convenient coordinate system. But you already have 2 horrible numbers in the conventional coo

> 10^(iπ) is some horrible complex number Sure, you want to use a convenient coordinate system. But you already have 2 horrible numbers in the conventional coordinate system: the π and e. In the conventional (“natural”) coordinate system for logarithms, the coordinates represent powers of e and rotations by 1/2π turns, respectively. You have just decided that you’ll use those specific horrible numbers in the logarithmic coordinates, because e and π are what you get out if you choose a coordinate system for the logarithm of quotients-of-vectors where the derivative of log(x) at the point x = 1 is 1. This turns out to be convenient for simplifying differential equations and related formulas. Personally I like using a coordinate system where the “scale” axis of the logarithm uses units of doublings, and the “rotation” axis of the logarithm uses units of full turns. If I use that one when making a picture, I can easily tell you exactly what the value is (as a rational number) at every grid intersection. e.g. here is a plot of a Möbius transformation, https://raw.githubusercontent.com/jrus/images-for-observable/master/v.-*(z%2B1)%7C(z-1).b.jpg https://raw.githubusercontent.com/jrus/images-for-observable... Every contour at the most prominent level is either a doubling or 1/12 of a turn. (You’ll notice that when we zoom on any part of the picture, we see little rectangles rather than little squares; the “natural” coordinate system is a square one, which is also often convenient.)