Complex Numbers

Chapter 2 Complex numbers arithmethic
Chapter 2.1 Definition
Complex number consists of parts, real and imaginary
Fig. 2-1
Equivalent complex numbers symbols
Real numbers are positioned on the x-axis. Other words. It requires 1D space. Complex numbers are positioned on the Re x, Im y axies and require 2D space. So the complex number is similar to the vector. Similar but similar only!
Here you are some z1…z2 complex numbers examples
 

Fig. 2-2
z1…z9
complex numbers examples
Note:
z6, z1 
and z2 have the real part only. So there are the real numbers -1,0,+1

Chapter 2.2 The most “absurd” complex numbers assumption
Get your attention at the quotation marks.  Maybe this absurd isn’t so big?

Rys. 2-3
j imaginary numbers definitions
This “absurd” assumption facilitates calculations in the real world! You will see in a while that complex numbers are:
added as for vectors
rotated when multiplicated
The a/m facilitate operations on the sinusoids. Sinusoids are very important in the electricity!
The parameters R-resistance, L-inductance and C-capacity:
– change the current amplitude sinusoid
– 
move the current phase sinusoid
acc. to the input sinusoid voltage
The amplitude and the phase may be calculated with the aid of the common trigonometry. But the formulas are very complicated then, compared to the complex numbers operations.

Chapter 2.3 Four principal complex numbers operations
Chapter 2.3.1 Addition
Real components part  sum = real components sum
Imaginary
components part  sum =Inaginary components sum
 

Fig. 2-4
z3=z1+z2
z3=(-5+4j)+(7+5j)=-5+4j+7+5j=2+9j
Note:
Complex numbers are  as vectors when added

Chapter 2.3.2 Subtraction
Real components part  difference = real components  difference
Imaginary
components part  difference =Inaginary components  difference
Or
Subtraction is the opposite number (-z) addition

Fig. 2-5
z3=z1-z2
z3=(-5+4j)-(7+5j)=-5+4j-7-5j=-12-1j

Chapter 2.3.3 Multiplication
The same as 2 binomials multiplication
(1+x)(2+y)=2+y+2x+2xy
But remember the “absurd” assumption j*j=-1.
An example
z1=0.4+1.6j
z2=1=0.8j
z3=z1*z2=(0.4+1.6j)*(1-0.8j)=0.4+0.32j+1.6j-1.28j*j=1.68+1.28j
You will see that multiplication means rotation!
But to be convinced about it, you have to treat the complex number like vector with the length |z| and the with the angle α.
Other words- complex numbers- modulus argument form.

Fig. 2-6
Complex
number as modulus |z|and argument α
here
z=4+6j –>tgα=6/4–>α=arctan(6/4)=56.31º
Two definitions:
–>z=4+6j
–>modulus |z|and argument α
point this same number z
the first is more convenient once, the second is more convenient once too.

Fig. 2-7
The z3=z1*z2 product is a number
with modulus |z3| and argument α3
|z3|=|z1|*|z2|
α3=α1+α2
We will not prove it, but this is a common trigonometry.
I hope that you see the rotation α3=α1+α2=75º+(-38.66º)=37.30º.

There are the consecutive number +1 mutiplications by number
j*1=j<–Start
j*j=-1
j*(-1)=-j
j*(-j)=1<–End

Conclusion
j*z–>number rotation about +90º!

Fig. 2-8
Consecutive number +1 rotations
For example
number -1 is a number +1 after rotations
-1=j*j*(+1)

Chapter 2.3.4 Division
The division is a some sort of the multiplication. Let’s do a trick and multiply the numerator and the denominator by the same number. The fraction will be not changed of course.
This number is so called conjugate number. The conjugates numbers product is real always!
An example

Fig. 2-9
What’s the z1/z2 division? (or fraction)
No comments.
By the way:
Multiplication 2 z numbers means means
|z3|=|z1|*|z2|
α3=α1+α2
see Fig. 2-7
Division 2 z numbers means means
|z3|=|z1|/|z2|
α3=α1-α2

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