Complex Numbers
What you must understand
You must handle complex numbers in Cartesian, polar (modulus–argument) and exponential forms, converting fluently. Core skills: arithmetic, conjugates, modulus and argument properties, De Moivre's theorem for powers and roots, sketching loci and regions in the Argand plane, and using complex numbers to factor real polynomials and prove trigonometric identities.
Traps
- Argument ambiguity: arguments are defined modulo (2\pi); for roots or when locating a point in the second or third quadrant, adjust by (\pi) so the argument matches the actual quadrant.
- Modulus–argument errors in products: (|z_1 z_2| = |z_1||z_2|) but (\arg(z_1 z_2) = \arg z_1 + \arg z_2) only up to multiples of (2\pi).
- Loci confusion: (|z - z_0| = r) is a circle; (|z - z_1| = |z - z_2|) is a perpendicular bisector; (\arg(z - z_0) = \theta) is a ray (not a full line), excluding the point (z_0).
- Conjugate roots: real coefficients force non-real roots to occur in conjugate pairs - use this to factor quartic and higher polynomials.
- Principal value: (\sqrt{-1}) conventions and branch choices can silently change an answer.
Rule of thumb
When a problem involves powers or roots, switch to polar form; when it involves sums, distances or conjugates, stay in Cartesian form.
Worked example
Solve (z^3 = 8i). Write (8i = 8,\text{cis},\frac{\pi}{2}). Then [ z = 2,\text{cis}\left(\frac{\pi/2 + 2k\pi}{3}\right),\ k=0,1,2, ] giving arguments (\frac{\pi}{6}, \frac{5\pi}{6}, \frac{3\pi}{2}). The three roots are equally spaced around a circle of radius 2.