Polar form of a complex number
Multiplying complex numbers in the form a + bi is slow, and high powers and roots are worse. Is there a way of writing them that makes this easy?
Where you are 3 inside
What this area is about
A nonzero complex number z = a + bi sits at the point (a, b), so it can also be named by its distance from 0 and its direction, just as a point gets polar coordinates. The polar form z = r(\cos\theta + i\sin\theta), with r = |z| > 0, does not help with adding, but it turns multiplying, powers and roots into multiplying lengths and adding angles.
How these fit together
The definition. Modulus, argument and polar form defines the modulus \displaystyle |z| = \sqrt{a^2 + b^2} and an argument \theta, and converts between a + bi and polar form.
A result and its procedure. Multiplying, powers and De Moivre’s theorem shows that a product multiplies moduli and adds arguments, and that \left(r\operatorname{cis}\theta\right)^n = r^n\operatorname{cis}n\theta for every integer n.
A procedure run backwards. Roots of a complex number and roots of unity solves z^n = w for w \ne 0 and a positive integer n: there are exactly n roots, evenly spaced on a circle about 0.
A path through it
Modulus and argument first, then De Moivre’s theorem, then roots, which runs De Moivre’s theorem in reverse.
Afterward, complex eigenvalues use the same picture for rotation-scaling matrices. A later page, the complex plane, revisited (not written yet), is planned to treat complex numbers as points and arrows, with multiplication as rotating and scaling.