argument

The argument of a complex number is the direction it points, measured as an angle from the positive real axis. If the modulus tells you how far the point sits from the origin, the argument tells you which way to face — turning counterclockwise from the positive real direction until your arrow lines up with the number.

For z = a + bi the argument is written arg(z), and it satisfies tan(arg(z)) = b/a, with the quadrant of (a, b) deciding the actual angle. The argument is usually measured in radians. Multiplying two complex numbers adds their arguments, which is the secret behind why complex multiplication rotates.

An honest subtlety: the argument is not unique, because adding a full turn of 2π gives the same direction. To pin down a single value, one often uses the principal argument, conventionally taken in the range from -π to π. Also note arg(0) is undefined — the origin has no direction.