Complex Numbers
Core ideas
Represent complex numbers in Cartesian (a + bi) and polar/modulus-argument forms. Perform addition, multiplication, division, and conjugation; use De Moivre's theorem for powers and roots; and interpret loci in the Argand diagram (circles, perpendicular bisectors, half-lines). Polynomial equations with real coefficients have conjugate root pairs.
Traps
- Argument must be in the correct quadrant; use atan2 logic, not just arctan(b/a).
- When taking nth roots, there are n distinct roots spaced by 2π/n; forgetting some loses marks.
- Conjugate root theorem applies only to polynomials with real coefficients.
- For division, multiply numerator and denominator by the conjugate of the denominator.
- Loci like |z - a| = |z - b| describe a perpendicular bisector, not a circle.
Rule of thumb
For powers and roots, convert to modulus-argument form: z = r(cos θ + i sin θ), so z^n = r^n(cos nθ + i sin nθ). For roots, add 2kπ to θ before dividing by n.
Worked example
Find the cube roots of 8i. Write 8i = 8(cos π/2 + i sin π/2). Then z = 2(cos((π/2 + 2kπ)/3) + i sin((π/2 + 2kπ)/3)) for k = 0, 1, 2. This gives arguments π/6, 5π/6, 3π/2, so roots are 2(cos π/6 + i sin π/6), 2(cos 5π/6 + i sin 5π/6), and 2(cos 3π/2 + i sin 3π/2).