Introduction to complex numbers

Today we will talk about complex numbers – what they mean, how they are represented, why they are needed – and will introduce some of their properties.

Complex numbers

Everyone knows different sets of numbers that they studied back at school, such as natural, whole, rational, irrational, and real numbers. The largest one (in a sense that it includes all the others) among the aforementioned is the set of real numbers. But is there a set of numbers that would allow all arithmetic properties (addition, subtraction, multiplication, and division) while including the set of real numbers? The answer to that question is yes: it is the set of complex numbers.

A complex number has two parts – real and imaginary. In algebraic form, it is represented the following way:

$z = a + b \cdot i$

The first part of the sum is the real part, while the second one that includes multiplication by $i$ is the imaginary one. $i$ is called the imaginary unit. The imaginary unit is defined by: $i = \sqrt{-1}$. Therefore $i^2 = -1$.

Complex numbers properties

1. Complex plane
   Complex numbers can be represented on a cartesian plane (and radius-vector, connecting the origin with that point): the number $z = x + y \cdot i$ has a corresponding dot on the plane with coordinates $(x; y)$. So $X$-axis corresponds to the real part of the complex number, while $Y$-axis corresponds to the imaginary part. We’ll show how it looks on the graph:

   

2. Module
   The module (absolute value) of a complex number is the length of the respective line on the complex plane. The module of the complex number $z = x + y \cdot i$ is labeled $|z|$ and is defined by the expression $|z| = \sqrt{x^2 + y^2}$. The module of the complex number is a real number.

3. Conjugated number
   If we have a complex number $z = x + y \cdot i$, then the number $\overline{z} = x - y \cdot i$ is called conjugated (or complex conjugated) to $z$ and is denoted $\overline{z}$. On the complex plane, conjugated numbers transform into each other via plane reflection over $X$-axis. The module of the conjugated number is the same, and their arguments have different signs. At the plane, a complex number and its adjoined number will look this way:

From these properties, you can see how complex numbers are related to the plane.

Conclusion

In this chapter, we discussed complex numbers and some of their properties and definitions, which helped us represent their link to the plane and real numbers. In the future, these seemingly simple properties will help you in different areas of both mathematics and programming. Complex numbers are used across the board in applied areas, such as computer vision.