Complex Numbers and Algebra
Forms and operations
z=a+bi, i^2=-1, conjugate z̄=a-bi. Modulus |z|=√(a^2+b^2); argument arg z measured from positive real axis. Polar form z=r(cosθ+isinθ)=r cisθ. Then |z1z2|=|z1||z2|, arg(z1z2)=arg z1+arg z2; division subtracts arguments.
De Moivre and roots
(cosθ+isinθ)^n=cos nθ+isin nθ. Roots of z^n=r cisθ are z=r^{1/n} cis((θ+2kπ)/n), k=0,...,n-1, equally spaced around a circle. Polynomials with real coefficients have non-real roots in conjugate pairs.
Worked example
Solve z^3=8i. Write 8i=8 cis(π/2). Then z=2 cis((π/2+2kπ)/3), k=0,1,2: 2 cis(π/6), 2 cis(5π/6), 2 cis(3π/2). In rectangular form these are √3+i, -√3+i, -2i. Check: each has modulus 2 and cube 8 cis(3θ)=8 cis(π/2+2kπ)=8i.
Loci
|z-z0|=r is a circle centre z0 radius r; |z-z0|=|z-z1| is the perpendicular bisector of the segment joining z0,z1; arg(z-z0)=α is a ray from z0 at angle α.
Traps
Wrong quadrant for arg; forgetting principal argument range (-π,π]; treating i as a real variable; missing roots; applying conjugate-root theorem when coefficients are not real; not excluding the point z0 from a locus; degree/radian mode in cis.