Complex numbers are numbers that have a real part and an imaginary part. They are usually denoted as a + bj, where a is the real part and bj is the imaginary part. The imaginary part is always multiplied by j, which is the square root of -1. Complex numbers can be added, subtracted, multiplied, and divided like real numbers, but have additional properties such as conjugation, modulus, and argument. Complex numbers are used to solve polynomial equations that have no solution in the set of real numbers. Complex numbers are used in many applications in physics, engineering, and computer science. They can be used to describe phenomena such as electromagnetic waves, oscillating systems and electronic circuits.

Given a complex: \(z=a+jb\) where \(a,b \in \mathbb{R}\) \(\rightarrow\) algebraic notation
Real part: \(Re(z)=a\)
Imaginary part: \(Im(z)=b\) \(\rightarrow\) if \(a=0\) then \(z\) is a pure imaginary
If \(b=0\) then \(z\) is a real number
Conjugate: \(\overline{z} = a-jb\), \(z\) and \(\overline{z}\) have the same real part \(Re\), but the imaginary part \(Im \) different

Operations

Sum: \(z+z’=(a+a’)+j(b+b’)\)
Product: \(z.z’=aa’-bb’+j(ab’+a’b)\)
Reverse & ; quotient: $$\frac{1}{z} = \frac{1}{a+jb} = \frac{a-jb}{a^2+b^2} = \frac{a}{a^2+b ^2} +j\frac{-b}{a^2+b^2} = \frac{1}{a+jb} *\frac{\overline{z}}{\overline{z}}$$

Operations and conjugates

$$\overline{z+z’}=\overline{z}+\overline{z’}$$ $$z+\overline{z’}= 2a = 2* Re(z)$$ $$\overline{z*z’}=\overline{z}*\overline{z’}$$ $$z-\overline{z’}= 2jb = 2j* Im(z)$$ $$z*\overline{z’}=a^2 + b^2$$ $$\overline{\frac{z}{z’}}= \frac{\overline{z}}{\overline{z’}}$$

Trigonometric writing

Module of z:\( \vert z \vert = \sqrt{a^2 + b^2}= \sqrt{z*\overline{z’}}=r\)
Argument of z: \( arg(z)= \theta + k*2\pi\) where \( k \in \mathbb{R}\)
$$cos(\theta)=\frac{a}{r}$$ $$sin(\theta)=\frac{b}{r}$$ Cartesian coordinates : \(M(a,b)\text{ }z=a+jb=r*(cos⁡(\theta)+j.sin⁡(\theta))\)
Polar coordinates: \( M[(r,\theta)]\)

Operations on Modules & ; Argument

Module

$$ \vert z \vert +\vert z’ \vert \geq \vert z+z’ \vert$$ $$ \vert z \vert =\vert \overline{z} \vert$$ $$\vert z \vert *\vert \overline{z} \vert = \vert z*z’ \vert$$ $$ n \in \mathbb{R}, \vert z^n \vert ={\vert z \vert}^n$$ $$ z \neq 0, \vert \frac{1}{z} \vert =\frac{1}{\vert z \vert} \rightarrow \vert \frac{z}{z’} \vert = \frac {\vert z \vert}{\vert z’ \vert}$$

Argument

$$z \neq 0\text{ and }z’\neq0 $$ $$arg⁡(z.z’)=arg⁡(z)+arg⁡(z’) [2π]$$
[2π] means: modulo 2π
$$n \in \mathbb{N}, arg⁡(z^n)=n*arg(z) [2π]$$ $$arg⁡(1/z)=-arg⁡(z) [2π] \rightarrow arg⁡(\frac{z}{z’}) = arg⁡(z) – arg⁡(z’) [2π] $$ $$arg⁡(\overline{z})=-arg⁡(z) [2π]$$

Exponential writing

$$e^{j\theta}= cos(\theta)+j.sin(\theta)$$ $$\vert e^{j\theta} \vert= 1$$ $$arg(e^{j\theta})= \theta + k*2\pi \text{ where } k \in \mathbb{Z}$$ $$ z = r.e^{j\theta}$$ Properties:
$$e^{j\theta}*e^{j\theta’}= e^{j(\theta+\theta’)}$$ $$\frac{1}{e^{j\theta}}= e^{-j\theta}= \overline{e^{j\theta}}$$ $$\frac{e^{j\theta}}{e^{j\theta’}}= e^{j(\theta-\theta’)}$$ $$(e^{j.\theta})^n= e^{j.n.\theta}$$

Euler formula

$$\begin{array}{cc} e^{j\theta}= cos(\theta)+j.sin(\theta) \\ e^{-j\theta}= cos(\theta)-j.sin(\theta ) \end{array}$$ $$\rightarrow cos(\theta)=\frac{e^{j\theta}+e^{-j\theta}}{2}$$ $$\rightarrow sin(\theta)=\frac{e^{j\theta}-e^{-j\theta}}{2j}$$ $$cos^3(\theta)=(\frac{e^{j\theta}+e^{-j\theta}}{2})^3 = \frac{1}{2}*(e^ {3j\theta}+e^{-3j\theta}+3e^{j\theta}+3e^{-j\theta}) =\frac{1}{4}*(cos(3\theta)+ 3cos(\theta))$$ $$\rightarrow (cos(\theta)+j.sin(\theta))^n = cos(n\theta)+j.sin(n\theta)$$ $$cos(a+b)+j.sin(a+b)= (cos(a)+j.sin(a))*(cos(b)+j.sin(b)) = cos(a).cos(b)-sin(a). sin(b)+j[cos(a).sin(b)+sin(a).cos(b)]$$ $$cos(a+b) = cos(a).cos(b)-sin(a).sin(b) \rightarrow \text{Real part}$$ $$sin(a+b) = cos(a).sin(b)+sin(a).cos(b) \rightarrow \text{Imaginary part}$$

Tagged: complex numbers, real part, imaginary part, imaginary unit “i”, polynomial equations, physics, engineering, computer science