Loci and regions on the Argand diagram

A locus (plural loci) is the set of all points that obey a rule. "All the points 3 cm from this dot" is a locus: it's a circle. In this lesson the points are complex numbers $z$ on an Argand diagram, and the rules are written with modulus and argument.

Take two points on the Argand diagram, $a$ and $z$. The complex number $z - a$ is the journey from $a$ to $z$: to get from $a$ to $z$ you have to add $z - a$.

For example, if $a = 1 + i$ and $z = 4 + 5i$, then $z - a = 3 + 4i$: go 3 across and 4 up.

So the two things you already know about a complex number now mean:

• $\lvert z - a \rvert$ is the length of that journey: the distance between $z$ and $a$. (Here $\lvert 3 + 4i \rvert = 5$.)
• $\arg(z - a)$ is the direction of that journey: the angle you travel at to get from $a$ to $z$, measured from the positive real direction.

Read every locus out loud using these words and the shape becomes obvious.

Watch the signs. The rule always has the form $z$ minus the fixed point. So $\lvert z + 1 - 2i \rvert$ must be rewritten as $\lvert z - (-1 + 2i) \rvert$: it is the distance from the point $-1 + 2i$, which is $(-1, 2)$.

$\lvert z - a \rvert = r$ says: the distance from $z$ to $a$ is always $r$. Every point a fixed distance from a centre lies on a circle, so this is the circle with centre $a$ and radius $r$.

Inequalities give the inside or outside:

• $\lvert z - a \rvert < r$: distance less than $r$, so the inside of the circle.
• $\lvert z - a \rvert > r$: the outside of the circle.

Boundaries. If the boundary itself is allowed ($\leqslant$ or $\geqslant$), draw it as a solid line. If it is not allowed ($<$ or $>$), draw it dashed.

$\lvert z - a \rvert = \lvert z - b \rvert$ says: $z$ is the same distance from $a$ as from $b$.

Which points are equally far from two given points? Picture standing exactly halfway between two friends, then stepping sideways at right angles to the line joining them: you stay equally far from both. The locus is the straight line through the midpoint of $a$ and $b$, at right angles to the line joining them. It is called the perpendicular bisector ("perpendicular" means at right angles, "bisector" means it cuts the segment in half).

For an inequality, ask which side is closer: $\lvert z - a \rvert < \lvert z - b \rvert$ is the side of the line nearer to $a$.

$\arg(z - a) = \theta$ says: to get from $a$ to $z$ you always travel in the direction $\theta$. Starting at $a$ and heading off at angle $\theta$ traces out a straight line that starts at $a$ and goes on forever in one direction. This is a half-line (or ray).

• The angle is measured from the positive real direction (pointing right), anticlockwise positive, just like an ordinary argument.
• The starting point $a$ itself is not included, because when $z = a$ there is no journey and $\arg 0$ has no value. Mark it with an open circle.
• It goes in one direction only. The opposite direction would be $\theta \pm \pi$, a different argument.