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Complex numbers — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Multiplying two complex numbers. In the complex plane, multiplying adds the two angles and multiplies the two lengths.

    Multiplying is turning

    Complex numbers are introduced as an algebraic dodge for square roots of negatives. They are better understood as the arithmetic of rotation, at which point every rule stops needing to be remembered.

    part 1 · algebra
  2. The Mandelbrot set. Points of the complex plane shaded by how long the iteration takes to escape, with the set itself the innermost region.

    The shape in every picture of itself

    One line of arithmetic, repeated, with a single complex number as its only input. Sort the numbers by whether the result stays bounded and the boundary between the two answers is the most complicated object anyone draws from a rule this short.

    part 2 · dynamics
  3. A Julia set. Points of the complex plane shaded by how long the iteration takes to escape, with the set itself the innermost region.

    One c, one picture

    The same iteration, with the parameter held still and the starting point varied instead. Every complex number gets its own picture, and moving the parameter a hair can shatter it into dust.

    part 3 · dynamics

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