Complex Numbers-Addition, Multiplication, division, Conjugate, Power, Absolute Value,solve equations

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In this video we will learn about complex numbers and their properties, we learn what is the conjugate of a complex number. How to add, subtract, multiply and divide the complex numbers.
we learn about how to raise to a power.
We will learn about absolute value or modulus or magnitude of a complex number.
we will learn about solving equations that their roots are complex numbers and much more.
And how to graphically represent a complex number.

Complex numbers are numbers that extend the concept of the one-dimensional number line to the two-dimensional complex plane. They consist of a real part and an imaginary part, and they are used in mathematics, physics, engineering, and other fields to solve problems that cannot be addressed using only real numbers.
Definition:

A complex number is written in the form:
z=a+bi,

where:
a: The real part of the complex number (Re(z)=a).
b: The imaginary part of the complex number (Im(z)=b).
i: The imaginary unit, defined as i=sqrt(-1)

, with the property that i^2=-1

Complex Plane:

Complex numbers are represented graphically in the complex plane, where:

The horizontal axis (real axis) represents the real part (a).
The vertical axis (imaginary axis) represents the imaginary part (b).

Each complex number corresponds to a unique point in the plane.

Properties and Applications:

Roots of Equations: Complex numbers allow for solutions to equations like x^2+1=0 which have no real solutions.

Algebraic Closure: The set of complex numbers (CC) is algebraically closed, meaning every polynomial equation has solutions in CC.

Applications:
Electrical Engineering: Used to analyze AC circuits using phasors.
Quantum Mechanics: Represent quantum states and wave functions.
Signal Processing: Simplify the representation of sinusoidal signals.
Geometry: Model rotations and transformations.

Fractals and Chaos: Complex numbers are fundamental in generating fractals like the Mandelbrot set.