Chapter 2 Complex numbers arithmethic
Chapter 2.1 Definition
Complex number consists of 2 parts, real and imaginary
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
z1…z9 complex numbers examples
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?
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 I current amplitude sinusoid
– move the I 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
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
Subtraction is the opposite number (-z) addition
Chapter 2.3.3 Multiplication
The same as 2 binomials multiplication
But remember the “absurd” assumption j*j=-1.
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.
Complex number as modulus |z|and argument α
–>modulus |z|and argument α
point this same number z
the first is more convenient once, the second is more convenient once too.
The z3=z1*z2 product is a number
with modulus |z3| and argument α3
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 j number
j*z–>number z rotation about +90º!
Consecutive number +1 rotations
number -1 is a number +1 after 2 rotations
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!
What’s the z1/z2 division? (or fraction)
By the way:
Multiplication 2 z numbers means means
α3=α1+α2 see Fig. 2-7
Division 2 z numbers means means