Regions in Argand diagrams

An Argand diagram is a graphical representation of complex numbers. It’s a two-dimensional diagram in which the horizontal axis represents the real part of a complex number and the vertical axis represents the imaginary part.
In an Argand diagram, a complex number is represented by a point in the complex plane. For example, the complex number z = a + bi is represented by the point (a, b).
Every point in the plane corresponds to a unique complex number. Therefore, a region in the Argand diagram represents a set of complex numbers.

The modulus of a complex number z = a + bi is the distance from the origin to the point (a, b) in the Argand diagram. It is given by

= sqrt(a^2 + b^2).

The argument of a complex number is the angle it makes with the positive real axis. If the angle is measured counterclockwise, it is positive, and if it’s measured clockwise, it’s negative.
A complex number can be written in polar form as z = r(cos θ + i sin θ), where r is the modulus and θ is the argument. This is also known as the cis form.
Euler’s formula, e^(iθ) = cos θ + i sin θ, gives a relationship between exponential and trigonometric functions, which simplifies many calculations in complex number theory.
Understanding how to manipulate complex numbers graphically using Argand diagrams is a key skill. This involves adding, subtracting, multiplying, and dividing complex numbers, as well as finding their moduli and arguments.
When translating algebraic operations to geometric operations on the Argand diagram, remember that addition of complex numbers corresponds to a translation, multiplication corresponds to a scaling and rotation, and conjugation corresponds to a reflection in the real axis.

Inequalities involving complex numbers often define regions in the Argand diagram. For instance, the inequality

z - z1

< r defines the region inside a circle center z1 and radius r in the complex plane.