Level 8: Complex numbers and beyond

Numbers · level 8 of 8 · rough age 17 and beyond (a guide for parents; anyone can start anywhere)

In level 7 you met the complex numbers as pairs \(a + bi\) and as points on a plane. This level gives them their second face: a length and an angle. With it, multiplying becomes "multiply the lengths, add the angles", powers and roots become easy, and the five most famous numbers in mathematics meet in one line, \(e^{i\pi} + 1 = 0\). The level ends at the edge of school mathematics: the theorem that the complex numbers are enough for every polynomial, and the strange four-dimensional numbers that come after them.

Lesson	You will be able to
Polar form	Find the modulus and argument of any complex number, switch between \(a + bi\) and \(r(\cos\theta + i\sin\theta)\), and prove that multiplying multiplies lengths and adds angles
De Moivre's theorem	Raise complex numbers to high powers by hand, prove the theorem by induction, and derive the double- and triple-angle formulas
Roots of unity	Solve \(z^n = 1\) and \(z^n = w\), draw the roots as a regular polygon, and prove that the roots of unity add to zero
Euler's formula	Explain where \(e^{i\theta} = \cos\theta + i\sin\theta\) comes from using power series, use it to calculate, and say what \(e^{i\pi} + 1 = 0\) does and does not mean
The fundamental theorem of algebra	State the theorem, count roots with multiplicity, factor polynomials such as \(x^4 + 1\) completely, and explain why real polynomials split into linear and quadratic factors
Beyond the complex numbers	Calculate with quaternions, use them to rotate a point in space, and explain what is lost at each step up the ladder of number systems

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