Complex Numbers

• z = x + iy, with i² = −1. Re z = x, Im z = y, conjugate z* = x − iy.
• Two complex numbers are equal ⇔ their real parts are equal and their imaginary parts are equal.
• Divide by multiplying top and bottom by the conjugate of the denominator.
• Polynomials with real coefficients have non-real roots in conjugate pairs.
• |z| = √(x² + y²); arg z is the angle with the positive real axis, usually −π < θ ≤ π.
• |z₁z₂| = |z₁||z₂| and arg(z₁z₂) = arg z₁ + arg z₂; for division, divide moduli and subtract arguments.
• Loci: |z − a| = k (circle), |z − a| = |z − b| (perpendicular bisector), arg(z − a) = α (half-line).

Calculating with x + iy

Add and subtract real and imaginary parts separately. Multiply by expanding and replacing i² with −1. To divide, multiply numerator and denominator by the conjugate of the denominator, since (a + bi)(a − bi) = a² + b² is real. Show full working for multiplication and division.

If a polynomial has real coefficients and z is a root, so is z*. A quadratic factor with those two roots is z² − 2(Re z)z + |z|².

Lengths, angles and their geometry

On an Argand diagram, x + iy is the point (x, y). Its modulus r is the distance from O and its argument θ is the angle from the positive real axis (anticlockwise positive). Always sketch the point to put θ in the right quadrant.

Geometric effects: z* is the reflection of z in the real axis. z₁ + z₂ follows the parallelogram rule, and z₁ − z₂ is the vector from z₂ to z₁. Multiplying by z₂ scales by |z₂| and rotates by arg z₂ (multiplying by i rotates through π/2).

Equating real and imaginary parts

Let the square root be x + iy. Squaring gives x² − y² + 2xyi; equate real and imaginary parts, eliminate y, and solve the resulting quadratic in x², keeping only the positive value of x² (x is real). There are always two roots, negatives of each other.