Complex numbers¶

Until now we have used real numbers. We will turn our attention to complex numbers and use them to understand different concepts. A complex number can be considered as an extension of real numbers. A complex number has real and imaginary part and is represented as: $$ c = a + ib, \quad i = \sqrt{-1} $$

Complex Plane¶

We can represent the complex number $a + ib$ in a two dimensional plane as follows:

Real value $a$ is represented in $x$-axis.
Imaginary value $b$ in $y$-axis.

This is called the complex plane, also called the Argand plane. Every complex number can therefore be associated with a point in the complex plane. In this case: $$ a + ib \equiv (a, b) $$ For example, the point $(3, 2)$ is the complex number $3 + 2i$ which can be represented as:

Note that all real numbers can also be considered as complex numbers with $b=0$, thereby all real numbers fall on the $x$-axis. The length of a complex number $(a, b)$ can be considered as the length of the line joining the origin and the complex number: $$ ||x||^2=a^2+b^2 $$ Complex number can also be represented using the angle $\theta$ with respect to positive $x$-axis and length $r$. $$ c=re^{i\theta}=r({cos\theta + i sin\theta}) $$ Graphically, we can represent this as follows:

Properties¶

Some basic operations and properties on complex numbers.

Addition¶

If $c_1= a_1+ib_1$ and $c_2=a_2+ib_2$, then: $$ c_1+c_2= (a_1+a_2)+i(b_1+b_2) $$

Multiplication¶

Multiplication of complex numbers works just like multiplication of real numbers, just note that $i=\sqrt{-1}$ and do multiplication using distributive property. Then: $$ \begin{aligned} c_1 c_2 &= (a_1+ib_1)(a_2+ib_2)\\ &= (a_1a_2-b_1b_2) +i(a_1b_2+a_2b_1) \end{aligned} $$

Complex conjugate¶

The complex conjugate of $c = a + ib$ is given by: $$ \bar{c} = a - ib $$ We have the following property: $$ \begin{aligned} c \bar{c} &= (a + ib)(a - ib)\\ &= a^2 + b^2\\ &= ||c||^2 \end{aligned} $$

If we look at it in another way: $$ \begin{aligned} \bar{c} &= r(\cos \theta - i \sin \theta)\\ &= re^{-i\theta} \end{aligned} $$