Complex numbers

Complex numbers extend the real numbers with i, where i² = −1. Every quadratic has solutions in the complex numbers, and they can be drawn as points on an Argand diagram.

Key facts

  • z = a + bi: real part a, imaginary part b.
  • |z| = √(a² + b²); arg z is the angle from the positive real axis.
  • Complex conjugate: z* = a − bi; zz* = |z|².
  • Divide by multiplying top and bottom by the conjugate.
  • Modulus–argument form: z = r(cos θ + i sin θ).

Worked examples

Example 1

Solve z² + 4z + 13 = 0.

  1. z = (−4 ± √(16 − 52))/2 = (−4 ± √−36)/2
  2. √−36 = 6i z = −2 ± 3i

Answer: z = −2 ± 3i

Example 2

Write (3 + 4i)/(1 − 2i) in the form a + bi.

  1. (3 + 4i)(1 + 2i)/((1 − 2i)(1 + 2i))
  2. Top: 3 + 6i + 4i + 8i² = −5 + 10i. Bottom: 1 + 4 = 5.
  3. −1 + 2i

Answer: −1 + 2i