5.2 Complex Numbers

Any number of the form $a+i b$, where $a$ and $b$ are real numbers, is defined to be a complex number. We often use Z for a complex number.

In the above complex number equation, z becomes:

1. Purely real if b =0; therefore z = a.
2. Purely Imaginary if a = 0; therefore z = ib (Imaginary numbers are the numbers when squared it gives the negative result; for example $\mathrm{i} = \sqrt{-1}$; $\mathrm{i}^2 \text { is }-1]$)
3. complex number if $(a \neq 0, b \neq 0)$

Examples:

Real Numbers Examples : $3,8,-2,0,9,13$
Imaginary Numbers Examples: 3i, 7i, -2i, 8i, $\sqrt{i}$
Complex Numbers Examples: $3+7 \mathrm{i}, 7- 6 \mathrm{i}, 0+20 \mathrm{i}=20 \mathrm{i}, 2+\mathrm{i}$.

In a complex plane (complex number Z as a vector) it looks like this:

What are Real Numbers?

Any number which is present in a number system such as zero, positive, negative, integer, rational, irrational, fractions, etc. are real numbers. It is represented as $\operatorname{Re}()$. For example: $12,-25,0,1 / 3,2.6$, etc., are all real numbers.

What are Imaginary Numbers?

The numbers which are not real are imaginary numbers. When we square an imaginary number, it gives a negative result. It is represented as $\operatorname{Im}()$. Example: $\sqrt{-2}$, $\sqrt{-5}$, $\sqrt{-13}$ are all imaginary numbers.

The complex numbers were introduced to solve the equation $x^2+1=0$. The roots of the equation are of form $x=$ $\pm \sqrt{-1}$ and no real roots exist. Thus, with the introduction of complex numbers, we have Imaginary roots. We denote $\sqrt{-1}$ with the symbol ‘i’, which denotes lota (Imaginary number).

For the complex number $z=a+i b, a$ is called the real part, denoted by $\operatorname{Re} (z)$ and $b$ is called the imaginary part denoted by $\operatorname{Im} (z)$ of the complex number $z$. For example, if $z=2+i 5$, then $\operatorname{Re} (z)=2$ and $\operatorname{Im} (z)=5$.

Two complex numbers $z_1=a+i b$ and $z_2=c+i d$ are equal if $a=c$ and $b=d$.

Graphical representation

The graph below shows the representation of complex numbers along the axes. Here we can see, the $x$-axis represents real part, and $y$ represents the imaginary part.

$\text { Let us plot a graph of complex number } 3+4 i  \text {as shown below}$

Is Zero(0) a complex Number?

As we know, 0 is a real number. And real numbers are part of complex numbers. Therefore, 0 is also a complex number and can be represented as $0+0 i$. 0 can be considered purely real and purely imaginary.