Complex Numbers
Complex Numbers tests algebraic, geometric and polar forms. You need fluency with conjugates, modulus-argument form, De Moivre’s theorem, loci and polynomial roots.
Core rules
- z=a+bi has |z|=√(a²+b²), arg z=atan2(b,a), and z=r(cosθ+i sinθ)=re^{iθ}.
- Multiplication multiplies moduli and adds arguments; division divides moduli and subtracts arguments.
- De Moivre: (re^{iθ})^n=r^n e^{inθ}. The nth roots are equally spaced around a circle, separated by 2π/n.
- For real-coefficient polynomials, non-real roots occur in conjugate pairs.
- Loci such as |z-a|=r describe circles; arg(z-a)=α describes a ray from a, excluding the point itself.
Worked example
Compute (1+i)^8. |1+i|=√2 and arg(1+i)=π/4. By De Moivre, (1+i)^8=(√2)^8 e^{i8π/4}=16e^{i2π}=16. Using degrees instead would give 16e^{i360°}, which is the same value but must be converted consistently.
Traps
Mixing degrees and radians; choosing the wrong principal argument; forgetting all nth roots; assuming conjugate roots for complex coefficients; losing the modulus factor when raising to a power; sketching a locus as a full line when it is only a ray.