Multiply or divide the moduli, while adding or subtracting the arguments, respectively. For an integer power, raise the modulus to that power and multiply the argument by the power.
This is HL-only content from AHL 1.13, Complex numbers continued. It may be assessed on any IB Mathematics: Applications and Interpretation HL paper, and a GDC is available on every paper.
Write each non-zero complex number in polar form:
where is the modulus and is an argument. Geometrically, the modulus represents a stretch and the argument represents a rotation from the positive real axis.
| Operation | Rule |
|---|---|
| Multiplication | |
| Division | , where |
| Integer power |
The integer-power rule is De Moivre's theorem. Multiplication combines stretches by multiplying moduli and combines rotations by adding arguments; division reverses these operations.
For example, let
For multiplication,
so .
Similarly,
A useful check is to interpret each result geometrically. In the product, the stretch factors two and three combine to give six, while the rotations combine to give five pi over twelve. In the quotient, dividing by z two removes its stretch and rotation, producing modulus two thirds and angle negative pi over twelve.
Before using the rules, make sure both numbers are written in the same angular unit. If the answer must be Cartesian, convert only after completing the polar calculation. This avoids expansion and reduces sign errors, especially when the argument lies in another quadrant.
Arguments differing by , where , represent the same complex number. If a principal argument is requested, adjust the result to the required interval.
Show the modulus calculation and argument calculation separately before stating the final polar or exponential form. The common misconception is multiplying arguments during multiplication: arguments are added, while only moduli are multiplied. Use radian mode unless degree symbols are explicitly given, and retain exact multiples of when possible.
