Complex Numbers
Core concepts
Work confidently with Cartesian form (z=a+bi), modulus (|z|=\sqrt{a^2+b^2}), argument, conjugate, and polar form (z=r(\cos\theta+i\sin\theta)=re^{i\theta}). Know De Moivre's theorem for powers and roots, and interpret multiplication as rotation-and-scaling in the Argand plane. Understand conjugates as reflections, polynomial roots occurring in conjugate pairs for real coefficients, and locus problems such as (|z-z_0|=r) (a circle) and (|z-a|=|z-b|) (a perpendicular bisector).
Traps
Argument ambiguity is the classic trap: use the correct quadrant and a consistent principal range, often ((-\pi,\pi]). Remember (i^2=-1), rationalise denominators using the conjugate, and do not assume (\arg(z_1z_2)=\arg z_1+\arg z_2) without adjusting for the principal range. When finding (n)th roots, produce all (n) distinct roots spaced by (2\pi/n). Watch modulus equations that yield lines rather than circles.
Worked example
Solve (z^3=8i). Write (8i=8e^{i\pi/2}). Then (z=2e^{i(\pi/6+2k\pi/3)}) for (k=0,1,2), giving arguments (\pi/6), (5\pi/6), (3\pi/2). Rule of thumb: convert to polar first, then apply De Moivre and list every root.
Exam strategy
Sketch the Argand plane, state modulus and argument explicitly, and verify roots by substitution or symmetry.