Seen via +Malin Christersson: A spectacular-looking website explains complex numbers and some of the things they're good for, including a remarkable visualization of how a Julia-set fractal is made, and another of how complex numbers are used to analyze wave phenomena in physics and engineering.

This is stuff that it took me years to learn how to visualize intuitively, displayed all in one place with animations. I also like the remarks at the beginning about the problems with how we present mathematics to students, and turn beautiful things into unpleasant experiences.

My only criticisms are that (1) the site is technically demanding, probably requires a recent browser and might push yours pretty hard (my laptop's fan was running the whole time I was looking at it), and (2) it was initially not obvious to me how to start reading the article (just scroll down as usual).
An experiment in math publishing, using MathBox, explaining the often misunderstood subject of Complex Numbers.
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