Complex Numbers: Definition and History

Introduction

• Real numbers form the basis for solving mathematical equations.
• Quadratic equations such as x² - 4 = 0 have two real solutions, e.g., x = ±2.
• However, not all quadratic equations have real solutions, x² + 1 = 0 results in x² = -1, which had no real solution until complex numbers.

Italian Mathematician Gerolamo Cardano

• The need for complex numbers arose when solving cubic equations.
• The set of real numbers was found to be insufficient for certain algebraic expressions.

Complex Numbers

• The imaginary unit i was introduced, where i = √-1, allowing mathematicians to solve previously unsolvable equations.
• Carl Friedrich Gauss refined the concept and introduced the term "complex number."
• Complex numbers became essential tools in mathematics, physics, and engineering.

Complex Number Definition

• Complex numbers are expressed as ordered pairs, typically denoted by z, and written as z = a + bi, where a is the real part and b is the imaginary part.
• In engineering, j is used instead of i for the imaginary unit, with the complex number written as z = x + jy, where x is real and y is the imaginary part.

Complex Numbers and Fields

• Complex numbers form an algebraic structure called a field
• Fields use operations such as addition, subtraction, multiplication, and division (except by zero).
• Fields also includes properties like closure, associativity, commutativity, identity, inverses, and distributivity.

Properties of Complex Numbers

Closure Property (Addition and Multiplication)

• For any two complex numbers z₁ and z₂, both the sum z₁ + z₂ and the product z₁z₂ are also complex numbers.

Commutative Property (Addition and Multiplication)

• For any two complex numbers z₁ and z₂, z₁ + z₂ = z₂ + z₁ and z₁z₂ = z₂z₁.

Associative Property (Addition and Multiplication)

• For any three complex numbers z₁, z₂, and z₃, (z₁ + z₂) + z₃ = z₁ + (z₂ + z₃) and (z₁z₂)z₃ = z₁(z₂z₃).

Additive Identity

• There is a complex number 0 = 0 + 0i such that for every complex number z, z + 0 = 0 + z = z.
• The number 0 = 0 + 0i is known as the additive identity.

Additive Inverse

• For every complex number z, there exists a complex number -z such that z + (-z) = (-z) + z = 0.
• -z is called the additive inverse of z.

Multiplicative Identity

• There exists a complex number 1 = 1 + 0i such that, for every complex number z, z * 1 = 1 * z = z.
• The complex number 1 = 1 + 0i is known as the multiplicative identity.

Multiplicative Inverse

• There exists a complex number w, for any non-zero complex number z, such that z * w = w * z = 1.
• w is called the multiplicative inverse of z, and is denoted by z⁻¹.

Distributive Property

• Distributive property dictates that for any three complex numbers z₁, z₂, and z₃, z₁(z₂ + z₃) = z₁z₂ + z₁z₃ and (z₁ + z₂)z₃ = z₁z₃ + z₂z₃.

Equality of Complex Numbers

• Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal.
• If z₁ = x + jy and z₂ = a + jb where x, y, a, and b are real numbers, then z₁ = z₂ if and only if x = a and y = b.

Conjugate Property of Complex Numbers

• The conjugate of a complex number simplifies complex arithmetic, especially with division or finding the modulus.
• For a complex number z = x + jy (where x is the real part and y is the imaginary part), the conjugate, denoted by z̄, is given by z̄ = x - jy.
• The conjugate is made by changing the sign of the imaginary part while keeping the real part the same.

Basic Operations of Complex Numbers

Addition

• Add complex numbers by combining their real and imaginary parts separately.
• Formula: z₁ + z₂ = (x₁ + x₂) + j(y₁ + y₂)
• Example: Given z₁ = 6 + j7 and z₂ = 3 - j5, (6 + j7) + (3 - j5) = (6 + 3) + j(7 - 5) = 9 + j2.

Subtraction

• Subtract complex numbers by subtracting their real and imaginary parts separately.
• Formula: z₁ - z₂ = (x₁ + x₂) - j(y₁ + y₂)
• Example: Given z₁ = 12 + j6 and z₂ = 4 + j5, (12 + j6) - (4 + j5) = (12 - 4) + j(6 - 5) = 8 + j.

Multiplication

• Multiply complex numbers using the distributive property and remembering that j² = -1.
• Formula: z₁z₂ = (x₁x₂ - y₁y₂) + j(x₁y₂ + x₂y₁)
• Example: (2 - j)(3 + j) = 6 + j2 - j3 - j² = 6 + j2 - j3 + 1 = 7 - j.

Multiplying a Complex Number by its Conjugate

• Multiplying a complex number by its conjugate results in a real number
• z * z̄ = |z|²
• If given z = 3 + j2, and z̄ = 3 – j2, formula application: (3 + j2)(3 – j2) = 9 − (−4) = 13

Division

• To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator in order to remove the imaginary part in the denominator.
• z₁/z₂ = ((x₁ + jy₁)(x₂ - jy₂))/(x₂² + y₂²)
• Example: (3 – j)/(4 + j2) = ((3 – j)(4 – j2)) / ((4 + j2)(4 – j2)) = (12 – j6 – j4 + (-1)2) / (16 – (-4)) = (10 – j10) / 20 = 1/2 - j(1/2)

Graphical Representation of Complex Numbers

• Complex numbers can be represented graphically on a complex plane.
• The horizontal axis represents the real part, and the vertical axis represents the imaginary part.
• This representation allows for a visual understanding of complex numbers and their operations.
• The complex number of z = a + bi can be represented as a point P(a,b) in a complex plane.

Plotting Complex Numbers

• For the complex number z = 3 + 4i, the complex number is plotted by moving 3 units to the right on the x-axis, and 4 units up on the y-axis

Standard Form

• Rectangular (Standard) Form: z = x + jy, where x represents the real part of the complex number, y the imaginary part, and j the imaginary unit.

Polar Form

• Represents a complex number based on magnitude (modulus) and angle (argument).
• Given by z = r(cosθ + jsinθ) or z = r∠θ, where r is the modulus (or absolute value) and θ is the argument (or angle).

Modulus (r)

Modulus

• The modulus "r" (or absolute value) measures the distance between the complex number and the origin in the complex plane.
• It is found using the formula r = |z| = √(x² + y²).

Argument (θ)

Argument

• Is the angle θ that the complex number forms with the positive real axis on the complex plane.
• The angle is measured counterclockwise from the positive real axis. The argument of z = x + jy, denoted as arg(z), is calculated as θ = arg(z) = tan⁻¹(y/x).

Sample Problem: Converting Rectangular Form to Polar Form

• Given z = 3 + j4, we find |z| = √(3² + 4²) = 5 and θ = tan⁻¹(4/3) ≈ 0.927 rad, which leads to z = 5(cos 0.927 + jsin 0.927) or z = 5∠0.927.

Exponential Form

• Expessed using Euler's Formula, any complex number z = r(cosθ + jsinθ) can be re-written in exponential form
• z = reʲθ, where r is the modulus and θ is the argument (or angle).

Sample Problems (Exponential Form)

Sample 1

• Converting z = 1 + i to exponential form; |z| = √(1² + 1²) = √2 and θ = arg(z) = tan⁻¹(1/1) = π/4 rad, thus z = √2*e^(jπ/4).

Sample 2

• Converting z = −1 − j√3 to exponential form; |z| = √((-1)² + (- √3)²) = 2 and θ = arg(z) = tan⁻¹((-√3)/-1) = π + π/3 = 4π/3 rad, thus z = 2e^(j4π/3).

Advanced Operations of Complex Numbers

Multiplication using Polar or Exponential Form

• The product of two complex numbers is a complex number whose magnitude is the product of the magnitudes of the individual numbers, and whose angle is the sum of their angles.
• Formaula: z₁z₂ = r₁r₂[cos(θ₁ + θ₂) + j sin(θ₁ + θ₂)] = r₁r₂e^(j(θ₁+θ₂))

Division Using Polar or Exponential Form

• The quotient of two complex numbers is a complex number whose magnitude is the ratio of the magnitudes of the individual numbers, and whose angle is the difference between their angles.
• z₁/z₂ = (r₁/r₂)[cos(θ₁ - θ₂) + j sin(θ₁ - θ₂)] = (r₁/r₂)e^(j(θ₁-θ₂))

Complex Power

Integer Exponents; De Moivre's Theorem

• Allows the easy computation of the power of a complex number in polar or exponential form.
• To raise a complex number to an integer power, raise the magnitude r to the power of n, and multiply the angle θ by n.
• For a complex number z=re^(jθ) = r(cos θ+ jsin θ), then z^n = r^n * e^(jnθ)
• z^n = r^n(cos(θ*)n + jsin θ*n)

Complex Roots

Polar Form

• Formula for finding the nth roots of a complex number:

• zₖ = ⁿ√r [cos((θ + 2πk)/n) + i sin((θ + 2πk)/n)] (Radian Measurement)

Rectangular Form

• zₖ = ⁿ√r [cos((θ + 360° * k)/n) + i sin((θ + 360° * k)/n)] (Degree Measurement)

General

• Note the value obtained when k=0 gets the principal values

Complex Logarithm

• Gives a complex number as defined Formula
• ln(z) = lnr + jθ
• r is the modulus
• the angle is the argument
• the logarithim of a compolex number has multiple values

Primary Values Formula

• ln(z) = lnr+ jө

General

• z = x +jy

• The logarithim has multiple values due to the periodic nature of the arguement

• express using general values is given using *formual for finding the nth roots of a complex number:

• zₖ = ⁿ√r [cos((θ + 2πk)/n) + i sin((θ + 2πk)/n)] (Radian Measurement)

Complex Power

• Also known as, Complex Exponent.
• Operation of taking a complex number by another
• using formula z^w = e^wln z)

Euler's Relation

• States that trigonometric functions have a relationship with exponential functions
• Formula e^jθ = cos θ + j sin θ
• inverse form: e^{-jθ} = cos θ − j sin θ

Trigonometric and Hyperbolic Functions Expressed

• cos θ =(e^{jθ}+ e^{-jθ})/2
• sin θ = (e^{jθ}-e^{-jθ})/2j
• cosh x = (x+ e^{-x}))/2
• sinh x = (e^{x}- e^{-x}))/2

Trigonometric Functions For Complex Numbers

• cos z = cos(x + jy) = cos x cosh y – j sin x sinh y
• sin z = sin(x + jy) = sin x cosh y – j cos x sinh y

Trigonometric Functions For Complex Numbers

• sinh z = sinh (x+jy) = sinhx cosy – j coshx siny
• cosh z = cosh (x+jy) = coshx cosy – j sinh x sin y

Inverse

• Inverse trigonometric funnctions can be extended to complex umbers cos^{-1} = cos ^{-1}(x+jy) + -j ln(z(+-) Square root(z-1)) sin^{-1} = sin ^{-1}(x+jy) + -j ln(iz(+-) Square root(z-1))

Explore the origins and definition of complex numbers, starting with the limitations of real numbers in solving certain equations. Learn about the introduction of the imaginary unit i and the contributions of mathematicians like Gerolamo Cardano and Carl Friedrich Gauss.