Complex Numbers
Complex Numbers tests algebraic manipulation, modulus–argument form, De Moivre's theorem, roots, and locus problems in the Argand plane. The exam often combines a polar-form computation with a geometric interpretation.
Core rules
Write z=x+iy with modulus r=√(x²+y²) and argument θ chosen by quadrant. Polar form is z=r(cosθ+i sinθ). Then z1z2 multiplies moduli and adds arguments; z1/z2 divides moduli and subtracts arguments. De Moivre: [r(cosθ+i sinθ)]^n=r^n(cos nθ+i sin nθ) for integer n. The nth roots of a complex number have equal modulus r^{1/n} and arguments spaced by 2π/n. For the nth roots of unity, the sum is 0 and the product is (−1)^{n−1}. Conjugates satisfy z z̄=|z|², and |z1+z2|≤|z1|+|z2|.
Worked example
Compute (1+i√3)^6. Here r=√(1+3)=2 and θ=π/3 because the point is in the first quadrant. By De Moivre, the power is 2^6[cos(6·π/3)+i sin(6·π/3)] = 64(cos 2π+i sin 2π)=64.
Traps
Choose the argument in the correct quadrant; using θ=tan^{-1}(y/x) blindly puts second- and third-quadrant numbers in the wrong place. De Moivre requires the modulus to be positive and the angle in radians. Do not write (cosθ+i sinθ)^n=cos nθ+i sin nθ and forget the r^n factor. A square root of a complex number has two values, not one.