Complex numbers

Builds on

Required prerequisites, all the way down: 29

You will learn to

• Add, multiply and divide complex numbers and represent them in the complex plane.
• Convert between Cartesian and polar forms and interpret multiplication as rotation combined with scaling.
• Use De Moivre's theorem to find the n-th roots of unity, and solve quadratics with negative discriminant.

Definition

• A complex number is z = a + bi with a and b real and i² = −1; ℂ denotes the set of complex numbers.
• The modulus of z = a + bi is |z| = √(a² + b²), and its conjugate is z̄ = a − bi.
• Polar form: z = r(cos θ + i sin θ), where r = |z| and θ is an argument of z.

Theorem

• Multiplying complex numbers multiplies their moduli and adds their arguments.
• De Moivre: (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ for every integer n.
• There are exactly n complex n-th roots of unity, and they are the vertices of a regular n-gon.
• Every non-constant polynomial with complex coefficients has a complex root (the fundamental theorem of algebra, proved later).

Common misconceptions

• Thinking complex numbers are 'imaginary' in the sense of illegitimate.
• Using √(ab) = √a · √b for negative a and b: √(−1) · √(−1) = −1, not √1 = 1.
• Believing ℂ can be ordered in a way compatible with its arithmetic, as ℝ can.

Leads to

Optional discoveries

• Fields and Galois theory ℂ can be constructed as the quotient ring ℝ[x]/(x² + 1).

Further reading

• Tristan Needham, Visual Complex Analysis (1997)