Complex Numbers
Core concepts
Understand Cartesian form a+bi, modulus, argument, conjugate, and polar form r cis(theta). Be fluent converting between forms, applying De Moivre's theorem, and solving polynomial equations over the complex numbers. Know the conjugate root theorem for real-coefficient polynomials and interpret complex roots geometrically in the Argand plane.
Traps
- Argument ambiguity: use the correct quadrant; principal argument lies in (-pi, pi].
- Forgetting that multiplying by the conjugate rationalises denominators.
- De Moivre applied to (r cis theta)^n requires the modulus raised to n, not left unchanged.
- Missing that non-real roots of real polynomials occur in conjugate pairs.
Worked example
Find the cube roots of 8 cis(pi/3). Write 8 cis(pi/3)=8 cis(pi/3+2k pi). Then z=2 cis((pi/3+2k pi)/3) for k=0,1,2, giving arguments pi/9, 7pi/9, and 13pi/9. The three roots are equally spaced around a circle of radius 2.
Rule of thumb
When raising to a power or extracting roots, always convert to polar form first; the modulus and argument then behave independently.