Complex Numbers

Introduction to the basics of Complex Number Calculation

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

$z_{1} \cdot z_{2} = (x_{1} + y_{1} i) \cdot (x_{2} + y_{2} i)$

$= x_{1} \cdot x_{2} - y_{1} \cdot y_{2} + i (x_{1} \cdot y_{2} + y_{1} \cdot x_{2})$

Example

$(1 + 2 i) \cdot (4 + 3 i) = (1 \cdot 4 - 2 \cdot 3) + i \cdot (1 \cdot 3 + 2 \cdot 4) = - 2 + 11 i$

The following picture shows a multiplication as graphic

$(2 + 4 i) \cdot (5 + 2 i) = 2 +24i$ .

Conjugate a complex number

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

The conjugate to $z = a + b i$ is written $z = a + b i$

Property of the operation $\bar{z_{1} + z_{2}} = \bar{z_{1}} + \bar{z_{2}}$ and $\bar{z_{1} \cdot z_{2}} = \bar{z_{1}} \cdot \bar{z_{2}}$

In the following example we search the sum of $z_{1} = 1 - 2 i$ and $z_{2} = 6 + 4 i$ , and the conjugate $\bar{z_{1} + z_{2}}$

Sum:

$z_{1} + z_{2} = (1 - 2 i) + (6 - 4 i) = 7 + 2 i$

conjugated:

$\bar{z_{1} + z_{2}} = \bar{7 + 2 i} = 7 - 2 i$

Division of complex numbers

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

$z_{1} / z_{2} = (z_{1} / z_{2}) \cdot (\bar{z_{1}} / \bar{z_{2}})$

Example for calculating the quotient

$\frac{3 - 2 i}{4 + 5 i} = \frac{3 - 2 i}{4 + 5 i} \cdot \frac{3 - 2 i}{4 - 5 i} = \frac{12 - 15 i - 8 i + 10 i^{2}}{16 - 25 i^{2}} =$ $\frac{12 - 10 - 23 i}{16 + 25} = \frac{2 - 23 i}{41} = \frac{2}{41} - i \cdot \frac{23}{41}$

The real part is

Re (\frac{3 - 2 i}{4 + 5 i}) = \frac{2}{41}

The imaginary part is

Im (\frac{3 - 2 i}{4 + 5 i}) = - \frac{23}{41}