Complex Numbers-Definition, Formula, Properties

Complex numbers are an extension of the real numbers, allowing for the inclusion of solutions to equations that do not have real solutions.

Here’s a detailed explanation:

Definition

A complex number is of the form:

z=a+bi

• Where: a and b are real numbers.
• i is the imaginary unit, defined by the property i²=−1.

Components

(1) Real Part: The a in z=a+bi is called the real part and is denoted as ℜ(z).
(2) Imaginary Part: The b in z=a+bi is called the imaginary part and is denoted as ℑ(z).

Representation

Complex numbers can be represented in different forms:

Cartesian Form

The standard form a+bi, where a and b are real numbers.

Polar Form

A complex number can also be represented in polar coordinates as: z=r(cosθ+isinθ)

where:

• r=∣z∣=✓a²+b² is the magnitude (or modulus) of z.
• θ=arg(z)=tan^-1(b/a) is the argument (or angle) of z.

This can also be written using Euler’s formula:
z=reiθ

Operations

Addition and Subtraction

(a+bi)+(c+di)=(a+c)+(b+d)i

(a+bi)−(c+di)=(a−c)+(b−d)i

Multiplication

(a+bi)(c+di)=(ac−bd)+(ad+bc)i

Division

(c+di)/(a+bi)=(c+di)(c−di)/(a+bi)(c−di)
​={(ac+bd)+(bc−ad)i}/c²+d²

Conjugate

The conjugate of z=a+bi is z¯=a−bi.

Modulus and Argument

Modulus (Magnitude): The distance from the origin to the point (a,b) in the complex plane.
∣z∣=✓a²+b²

Argument (Angle): The angle θ between the positive real axis and the line representing the complex number.

θ=arg(z)=tan^-1(b/a)

Properties

Complex Conjugate:

The complex conjugate z¯ of z=a+bi is z¯=a−bi.

Multiplication by Conjugate:

z⋅ z¯=(a+bi)(a−bi)=a²+b²=∣z∣²

Euler’s Formula:

eiθ=cosθ+isinθ

De Moivre’s Theorem:

For any integer n

(reiθ)n=rⁿeinθ

Applications

Complex numbers are used in many fields, including:

Engineering: Especially in signal processing and control theory.
Physics: Quantum mechanics and electrical engineering.
Mathematics: Solving polynomial equations, fractals, and complex analysis.
Computer Graphics: Representing rotations and transformations.

Visualization

Complex numbers can be visualized on the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part. Each complex number corresponds to a unique point on this plane.

Summary

Complex numbers provide a powerful extension to the real numbers, enabling the solution of equations and problems that cannot be addressed within the realm of real numbers alone. They have a wide range of applications in various scientific and engineering disciplines.