Converting Complex Number from rectangular form into Polar form & Euler's form or Exponential form

Описание: Definition:
A complex number is a number that can be expressed in the form:
z = a + bi
\where:
1. a is the real part
2. b is the imaginary part
3. i is the imaginary unit, satisfying i^2 = -1
Components:
1. Real part (a): The real component of the complex number.
2. Imaginary part (b): The imaginary component of the complex number.

Operations:
1. Addition: (a + bi) + (c + di) = (a + c) + (b + d)i
2. Subtraction: (a + bi) - (c + di) = (a - c) + (b - d)i
3. Multiplication: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
4. Division: (a + bi) / (c + di) = ((ac + bd) / (c^2 + d^2)) + ((bc - ad) / (c^2 + d^2))i

Applications:
1. Algebra: Solving equations with complex roots.
2. Geometry: Representing points in the complex plane.
3. Calculus: Analyzing functions with complex variables.
4. Engineering: Applications in electrical engineering, signal processing, and control systems.

Key Concepts:
1. Complex conjugate: The complex conjugate of z = a + bi is z̄ = a - bi.
2. Modulus: The modulus of z = a + bi is |z| = √(a^2 + b^2).
3. Argument: The argument of z = a + bi is the angle θ = arctan(b/a).

Ways of representing Complex Numbers:
1.Rectangular Form or cartesian form
2.Polar form or circular form
3.Euler's form or exponential form

A complex number z can be expressed in rectangular form as:
z = a + bi
where:
1. a is the real part
2. b is the imaginary part
3. i is the imaginary unit, satisfying i^2 = -1

Polar Form:
A complex number z can also be expressed in polar form as:
z = r(cosθ + isinθ)
where:
1. r is the modulus (magnitude) of z
2. θ is the argument (angle) of z

Exponential or Euler's Form:
Using Euler's formula, e^(iθ) = cosθ + isinθ, complex numbers can be expressed in exponential form as:
z = re^(iθ)


Applications:
Complex numbers have numerous applications in:
1. Electrical engineering: AC circuits, signal processing
2. Control systems: Stability analysis, controller design
3. Signal processing: Filter design, modulation analysis
4. Mathematics: Solving equations, algebraic manipulations

Key Concepts:
1. Complex conjugate: z̄ = a - bi
2. Modulus: |z| = √(a^2 + b^2)
3. Argument: θ = arctan(b/a)