Complex numbers

📚The mathematics

Cartesian form

Define \( \mathrm{i} \) by \( \mathrm{i}^{2} = -1 \). A complex number is \( z = x + \mathrm{i}y \), with real part \( x \) and imaginary part \( y \) — note the imaginary part is \(y\), a real number, not \( \mathrm{i}y \).

Addition and subtraction work componentwise. Multiplication expands as normal, then uses \( \mathrm{i}^{2} = -1 \):

The conjugate \( z^{*} = x - \mathrm{i}y \) flips the sign of the imaginary part. Its key property is that

which is always real. That is what makes division work: multiply numerator and denominator by the conjugate of the denominator, exactly as you would rationalise a surd.

Modulus and argument

The argument is conventionally given in the range \( -\pi < \theta \le \pi \), and it is measured from the positive real axis.

Polar form and multiplication

Multiplication and division are far simpler in polar form:

Geometrically, multiplying by a complex number is a rotation combined with a stretch: rotate by its argument, scale by its modulus. Multiplying by \( \mathrm{i} \), which has modulus 1 and argument \( \tfrac{\pi}{2} \), is therefore a pure rotation through a right angle — a fact worth carrying, because it explains a lot of Argand diagram geometry.

Adding complex numbers, by contrast, follows the parallelogram rule, exactly like adding vectors.

Roots of polynomials

If a polynomial has real coefficients, its non-real roots occur in conjugate pairs. So being told that \( 2 + 3\mathrm{i} \) is a root immediately gives you \( 2 - 3\mathrm{i} \) as well, and those two combine into a real quadratic factor:

Rather than expanding brackets with \( \mathrm{i} \) in them, build the quadratic from the sum and product of the pair: the sum of the roots is 4 and the product is \( 4 + 9 = 13 \), giving \( z^{2} - (\text{sum})z + (\text{product}) \). It is faster and there is nothing to sign-error.

Then divide the original polynomial by that quadratic to find the remaining roots. The condition matters: if the coefficients are not all real, roots need not come in conjugate pairs and this method does not apply.

Loci in the Argand diagram

Three standard loci, each of which is a geometric statement about distance or direction:

• \( |z - a| = r \) — the points at distance \(r\) from \(a\): a circle, centre \(a\), radius \(r\).
• \( |z - a| = |z - b| \) — the points equidistant from \(a\) and \(b\): the perpendicular bisector of the segment joining them.
• \( \arg(z - a) = \theta \) — the points in a fixed direction from \(a\): a half-line (a ray) starting at \(a\), at angle \( \theta \) to the positive real direction.

Read \( |z - a| \) as “the distance from \(z\) to \(a\)” and all three become obvious. Note that the third is a half-line, not a full line — the opposite direction has argument \( \theta + \pi \), not \( \theta \) — and that the starting point \(a\) is itself excluded, since \( \arg 0 \) is undefined. Mark that with an open circle.

Inequalities give regions: \( |z - a| \le r \) is the disc including its boundary, and \( |z - a| \ge |z - b| \) is the half-plane on the \(b\) side of the bisector. Shade, and state whether boundaries are included.