Complex numbers and the complex plane
Complex-number questions combine algebra with geometry. Expect polar form, modulus and argument, loci, De Moivre's theorem, and polynomial equations over the complex numbers.
Core rules
- For z=a+bi, |z|=√(a^2+b^2) and arg z is the angle from the positive real axis, chosen in the required range. Polar form: z=r(cos θ+i sin θ)=re^{iθ}.
- Multiplication multiplies moduli and adds arguments; division divides moduli and subtracts arguments.
- De Moivre: (r cis θ)^n=r^n cis(nθ). The nth roots of r cis θ are r^{1/n} cis((θ+2kπ)/n), k=0,...,n-1.
- Loci: |z-a|=r is a circle; |z-a|=|z-b| is the perpendicular bisector of ab; arg(z-a)=α is a ray from a, with endpoint conventions to check.
- If a polynomial has real coefficients, non-real roots occur in conjugate pairs.
Worked example
Solve z^3=8i. Write 8i=8 cis(π/2). The modulus of each root is 8^{1/3}=2. The arguments are (π/2+2kπ)/3 for k=0,1,2, namely π/6, 5π/6, and 3π/2. Therefore the roots are 2 cis(π/6), 2 cis(5π/6), and 2 cis(3π/2). In rectangular form the first is √3+i, the second is -√3+i, and the third is -2i. Multiplying any root by a cube root of unity cycles to another root.
Traps
Do not lose the 2kπ term when finding all roots; a degree-n equation has n complex roots counted with multiplicity. Argument choices differing by 2π represent the same direction but different root rotations. For loci, check whether endpoints are included. When conjugating, only the sign of the imaginary part changes.