(Replying to myself since we hit the nesting depth.) 1. Here's a deeper intuition: any circle is just the unit circle, scaled up or down. Any number is just 1.

(Replying to myself since we hit the nesting depth.) 1. Here's a deeper intuition: any circle is just the unit circle, scaled up or down. Any number is just 1.0, scaled (and rotated, if complex) by the exponential function that was run for some rate and for some amount of time. e^rt is a rocket ship of constant change, we just decide how long to stay on for, and we can get to any number. In other words, for any number a: a = 1.0 * e^ln(a) This formulation is useful if we know we're going to be taking exponents on our number a, i.e. we really want a^b. (If we know we'll be rotating our number, maybe we write it in polar coordinates, etc.) So, the intuition is: "I know I'm going to be taking my number to various powers, so let's get the base settings for e^rt dialed in. pi/2 is the setting for how long we'd ride e^i for in order to get to 90 degrees. I should expect pi/2 somewhere in the answer as I take it to various powers." 2) I haven't taken complex analysis, so my understanding isn't nuanced enough here either. Technically, i^i can be multi-valued, for this graphical analogy let's settle on the principal root (https://www.math.hmc.edu/funfacts/ffiles/20013.3.shtml https://www.math.hmc.edu/funfacts/ffiles/20013.3.shtml). > (e^(2 i pi))^(1/2) is not e^(i pi), it's e^0. (e^(2 i pi))^(1/2) is asking for the square root of 1, which is both 1 [e^0] and -1 [e^(i pi)]. Again there may be a subtlety here, but I'm not sure how the above statement is incorrect (barring a technicality like "we always mean the positive root"). For the purposes of an intuition it makes sense, I think.