Using complex numbers to represent regions on an Argand diagram

Understanding Complex Numbers

Complex numbers are numbers that consist of a real part and an imaginary part.
They are traditionally written in the form a + bi, where ‘a’ is the real part, ‘b’ is the imaginary part, and ‘i’ is the square root of -1.
When ‘b’ is 0, the complex number is simply a real number. But when ‘a’ is 0 and ‘b’ isn’t, the complex number is referred to accurately as a ‘pure imaginary number’.
Complex numbers can also be expressed in polar form, r(cos θ + i sin θ) or r cis θ which is useful on an Argand diagram.

The Argand Diagram

An Argand Diagram provides a graphical way to represent complex numbers.
The horizontal axis (real axis) represents the real part of a complex number, while the vertical axis (imaginary axis) represents the imaginary part.
Thus any complex number ‘a + bi’ can be represented as a point on the Argand diagram at the coordinates (a, b).

Representing Regions on an Argand Diagram

To represent a region in the complex plane (on an Argand diagram), simply shade the appropriate area.
For example, all complex numbers with a real part greater than 2 would represent a shading to the right of the line ‘Re(z) = 2’ on the Argand diagram.
The set of all complex numbers with an absolute value, or modulus, less than 1 represents a disk of radius 1 centered at the origin of the complex plane.

Application Tips with Complex Numbers and Argand Diagrams

Look at the inequality you’re aiming to describe with your region. The inequality could make reference to the real part, the imaginary part, or the modulus.
If the inequality refers to the modulus or absolute value of complex numbers, it’s usually referring to points a certain ‘distance’ from the origin.
Conversely, if the inequality only mentions the real or imaginary parts, your region will be half the complex plane, upwards or downwards (for imaginary part inequalities), or rightwards or leftwards (for real part inequalities).
Practice plotting complex numbers and regions frequently to understand and visualize them better.
Working with previous example problems can bolster your skills in analysis, understanding and application on this topic.

Remember to apply the principles gradually, starting with simpler numbers and regions and advancing to more complex ones. Mastery of basic concepts and terms in complex numbers and the Argand diagram is essential in mathematics at this level.