Term 1 review: binomial expansions, complex numbers, and vectors, lines and planes

Weeks 4 to 7 build on one picture: a complex number x + yi is the point (x, y), with distance |z| = r and angle θ. First find r, then the reference angle α and the quadrant to get the principal amplitude, and write z = r(cos θ + i sin θ), or z = re^(iθ). In this form multiplication multiplies the moduli and adds the amplitudes, division divides and subtracts, and the power zⁿ = rⁿ(cos nθ + i sin nθ) is De Moivre’s theorem. Every amplitude must be brought back into ]−π, π] by adding or subtracting 2π. Roots: Zⁿ = a has n roots, with modulus r^(1/n) and amplitudes (θ + 360°k)/n; on the Argand plane they are the corners of a regular n-sided polygon. A square root in the form x + yi is found from x² − y² = a, 2xy = b and x² + y² = |a + bi|. The cube roots of 1 are 1, ω, ω², and ω lets you simplify long expressions: reduce powers by ω³ = 1 and replace 1 + ω + ω² by 0. When you finish, check one value in the original equation.