About this topic
Before you begin
Write complex numbers in polar and exponential form, use de Moivre’s theorem for powers, roots and identities, and sketch loci and regions in the Argand diagram.
Learning goals
- Convert between Cartesian, modulus–argument and exponential forms of a complex number
- Multiply and divide complex numbers in polar form and interpret the rotation and scaling
- Use de Moivre’s theorem to find powers and every nth root of a complex number
- Derive multiple-angle and power identities using de Moivre’s theorem
- Represent simple complex equations and inequalities as loci and regions
Syllabus statements covered
- Include: complex numbers expressed in the form r(cos θ + i sin θ), re^(iθ), where r > 0 and −π < θ ⩽ π; multiplication and division of two complex numbers expressed in polar form and their geometrical interpretations; loci of simple equations and inequalities such as |z − c| ⩽ r, |z − a| = |z − b| and arg(z − a) = α (excluding loci of |z − a| = k|z − b|, where k ≠ 1 and arg(z − a) − arg(z − b) = α); use of de Moivre’s theorem to find the powers and nth roots of a complex number, and to derive trigonometric identities.
Choose a lesson
Work through them in order.
- Not startedPolar form and de Moivre’s theoremNextMultiply complex numbers by rotating and scaling, then use de Moivre’s theorem for powers, every nth root and multiple-angle identities.
- Not startedLoci in the Argand diagramNextRead moduli and arguments of differences as distances and directions, then sketch circles, bisectors, half-lines and regions.