The formula
(a + bi)(c + di) = (ac − bd) + (ad + bc)i
What it means
Complex arithmetic follows ordinary algebra with one extra rule: i² = −1. Addition and subtraction work part by part; multiplication expands like binomials, and the i² term flips sign; division multiplies top and bottom by the conjugate to clear the imaginary part from the denominator.
How to do it by hand
- For addition, add the real parts and the imaginary parts separately
- For multiplication, expand as binomials and replace i² with −1
- For division, multiply top and bottom by the conjugate of the denominator
- The modulus is √(a² + b²) and the argument is the arctangent of b over a
What is worth knowing
Polar form is where multiplication becomes easy: multiplying two complex numbers multiplies their moduli and adds their arguments, which turns a messy binomial expansion into two simple operations. That is the whole basis of De Moivre's theorem and of how roots of complex numbers are found. Complex numbers were resisted for two centuries — Descartes coined 'imaginary' as an insult — but they turn out to be unavoidable: they are why every polynomial of degree n has exactly n roots, and they are the natural language of alternating current, quantum mechanics and signal processing.