Complex Numbers

This article gives a short introduction to the basics of Complex Number Calculation. More detailed descriptions can be found in the chapter on complex numbers


Definition of a complex number


With quadratic equations, there is not always a real solution. For example, the equation

\(X^2 + 1=0\) oder eben \(X^2 = -1\)

In order to be able to count on solutions of such equations, the mathematician Leonard Euler introduced a new imaginary number and designated it with the letter \(i\).


A complex number \(z\) consists of a real part \(a\) and an imaginary part \(b\). The imaginary part is marked with the letter \(i\).

\(z=a+bi\)

The imaginary unit \(i\) has the property

\(z^2=-1\)

The value of a complex number corresponds to the length of the vector \(z\) in the Argand plane.


Graphical interpretation of complex numbers


For the graphical interpretation of complex numbers the Argand plane is used. The Argand plane is a special form of a normal Cartesian coordinate system. The difference is in the name of the axles.

The real part of the complex number is displayed on the x-axis of the argand plane. The axis is called the real axis.

The imaginary part of the complex number is displayed on the y-axis of the argand plane. The axis is called the imaginary axis.


The following figure shows a graphical representation of a complex number \(3 + 4i\).
The absolute value \(z\) is \(5\).


Addition and subtraction of complex numbers


The addition and subtraction of complex numbers corresponds to the addition and subtraction of the vectors. The real and imaginary components are added or subtracted

Addition:
\(\normalsize z_{1}+z_2 = x_1+x_2+i(y_1+y_2)\)

Subtraktion: 
\(z_{1}-z_2 = x_1+x_2-i(y_1+y_2)\)

Excamples

\((1+2i)+(4+3i)=(1+4)+i·(2+3)=4+5i\)

\((1+2i)+8i=1+10i\)

\((1+2i)+(4+2i)=5+4i\)

\((1+2i) -(4+2i)=3\)


The following figure shows the graphical representation of the addition \((2+4i)+(5+2i)\).

Multiplication of complex numbers


The multiplication is done by multiplying the parentheses. 

z1·z2 = (x1+y1i)· ( x2+y2i)

= x1·x2 - y1·y2+i (x1·y2+ y1·x2)

Example

( 1+2i)· ( 4+3i)= ( 1·4- 2·3)+i· ( 1·3+ 2·4)= -2+11i


The following picture shows a multiplication as graphic

( 2+4i)· ( 5+2i)= 2+ 24i .

Conjugate a complex number


To divide a complex number, you need the conjugate of a complex number.

The conjugate to   z=a + bi   is written   z=a + bi

Property of the operation   z1+z2 ¯ = z1 ¯ + z2 ¯   and   z1·z2 ¯ = z1 ¯ · z2 ¯

In the following example we search the sum of z1=1-2i   and z2=6+4i , and the conjugate   z1+z2 ¯  

Sum:
z1+z2= (1-2i)+ (6-4i)= 7+2i

conjugated:
z1+z2 ¯ = 7+2i ¯ = 7-2i





Division of complex numbers


Complex numbers are divided by multiplying the numerator and denominator by the complex conjugate of the denominator.

z1 / z2= ( z1 / z2) ( z1¯  /  z2¯ )

Example for calculating the quotient



3-2i 4+5i = 3-2i 4+5i 3-2i 4-5i = 12-15i-8i+10i2 16-25i2 = 12-10-23i 16+25 = 2-23i 41 = 2 41 -i 23 41


The real part is
Re( 3-2i 4+5i )= 241
 

The imaginary part is
Im( 3-2i 4+5i )=- 2341