Complex Numbers: Comprehensive Notes & Interactive Quiz

What are Complex Numbers?

Complex numbers are an extension of the real number system that include the imaginary unit i, defined as i = √(-1), or equivalently, i² = -1.

A complex number is expressed in the form:

z = a + bi

where:

a is the real part, denoted as Re(z)
b is the imaginary part, denoted as Im(z)
i is the imaginary unit

The Complex Plane

Complex numbers can be represented geometrically on the complex plane (also called the Argand plane). The horizontal axis represents the real part, and the vertical axis represents the imaginary part.

Example: Plotting Complex Numbers

For the complex number z = 3 + 4i:

The real part (Re(z)) is 3
The imaginary part (Im(z)) is 4

This is represented as a point at coordinates (3, 4) on the complex plane.

History of Complex Numbers

The concept of complex numbers emerged in the 16th century when mathematicians were trying to find solutions to polynomial equations. Initially, mathematical operations involving √(-1) were considered "imaginary" or impossible. The Italian mathematician Gerolamo Cardano first used the concept, while Rafael Bombelli developed rules for working with them.

The term "complex number" was introduced by Carl Friedrich Gauss in the 19th century, and their geometric interpretation as points in a plane was developed by Caspar Wessel, Jean-Robert Argand, and Gauss.

Operations with Complex Numbers

1. Equality

Two complex numbers are equal if and only if both their real and imaginary parts are equal:

a + bi = c + di if and only if a = c and b = d

2. Addition and Subtraction

To add or subtract complex numbers, add or subtract the real and imaginary parts separately:

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

Example: Addition and Subtraction

Given z₁ = 2 + 3i and z₂ = 4 - 5i:

Addition: z₁ + z₂ = (2 + 4) + (3 - 5)i = 6 - 2i

Subtraction: z₁ - z₂ = (2 - 4) + (3 - (-5))i = -2 + 8i

3. Multiplication

To multiply complex numbers, use the distributive property and remember that i² = -1:

(a + bi)(c + di) = ac + adi + bci + bdi²
= ac + adi + bci - bd
= (ac - bd) + (ad + bc)i

Example: Multiplication

Given z₁ = 2 + 3i and z₂ = 4 - 5i:

z₁ × z₂ = (2 × 4 - 3 × (-5)) + (2 × (-5) + 3 × 4)i

= (8 + 15) + (-10 + 12)i

= 23 + 2i

4. Complex Conjugate

The complex conjugate of a complex number z = a + bi is denoted by z̄ (or z*) and is defined as:

z̄ = a - bi

Key properties of complex conjugates:

z + z̄ = 2a (twice the real part)
z - z̄ = 2bi (twice the imaginary part)
z × z̄ = a² + b² (the square of the absolute value)

Example: Complex Conjugate

For z = 2 + 3i, the complex conjugate is z̄ = 2 - 3i.

z + z̄ = (2 + 3i) + (2 - 3i) = 4 + 0i = 4

z × z̄ = (2 + 3i)(2 - 3i) = 4 - 6i + 6i - 9i² = 4 + 9 = 13

5. Division

To divide complex numbers, multiply both numerator and denominator by the complex conjugate of the denominator:

\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)} = \frac{(ac + bd) + (bc - ad)i}{c² + d²}

Example: Division

Compute (2 + 3i) ÷ (4 - 5i):

\frac{2 + 3i}{4 - 5i} = \frac{(2 + 3i)(4 + 5i)}{(4 - 5i)(4 + 5i)}

= \frac{(2 × 4 + 3 × 5) + (3 × 4 - 2 × 5)i}{4² + 5²}

= \frac{8 + 15 + (12 - 10)i}{16 + 25}

= \frac{23 + 2i}{41}

= \frac{23}{41} + \frac{2}{41}i

6. Absolute Value (Modulus)

The absolute value or modulus of a complex number z = a + bi, denoted |z|, is defined as:

|z| = \sqrt{a² + b²}

The absolute value represents the distance from the origin to the point z in the complex plane.

Example: Absolute Value

For z = 3 + 4i:

|z| = \sqrt{3² + 4²} = \sqrt{9 + 16} = \sqrt{25} = 5

Representations of Complex Numbers

1. Rectangular Form

The standard form z = a + bi is known as the rectangular or Cartesian form of a complex number.

2. Polar Form

Any complex number can also be represented in polar form:

z = r(cos θ + i sin θ) = r cis θ

where:

r = |z| = \sqrt{a² + b²} is the modulus (absolute value)
θ = arg(z) = arctan(b/a) is the argument (phase angle)

The argument θ is measured in radians and represents the angle from the positive real axis to the line connecting the origin and the point z, counterclockwise.

Example: Converting from Rectangular to Polar Form

Express z = -3 + 4i in polar form:

r = |z| = \sqrt{(-3)² + 4²} = \sqrt{9 + 16} = \sqrt{25} = 5

θ = arctan(4/(-3)) = arctan(-4/3) ≈ -0.9273 radians

Since z is in the second quadrant, we add π: θ = -0.9273 + π ≈ 2.2143 radians ≈ 126.87°

Therefore, z = 5(cos 2.2143 + i sin 2.2143) = 5 cis 2.2143

3. Exponential Form

Using Euler's formula (e^(iθ) = cos θ + i sin θ), we can express a complex number in exponential form:

z = re^(iθ)

where r and θ are the same as in the polar form.

Example: Converting to Exponential Form

Express z = -3 + 4i in exponential form:

From the previous example, r = 5 and θ ≈ 2.2143 radians

Therefore, z = 5e^(2.2143i)

4. Converting Between Forms

From rectangular form z = a + bi to polar form:

r = \sqrt{a² + b²}
θ = arctan(b/a) (with appropriate quadrant adjustments)

From polar form z = r(cos θ + i sin θ) to rectangular form:

a = r cos θ
b = r sin θ

Quadrant adjustments for θ = arctan(b/a):
- If a > 0, b > 0 (Quadrant I): θ = arctan(b/a)
- If a < 0, b > 0 (Quadrant II): θ = arctan(b/a) + π
- If a < 0, b < 0 (Quadrant III): θ = arctan(b/a) + π
- If a > 0, b < 0 (Quadrant IV): θ = arctan(b/a) + 2π

Properties and Theorems of Complex Numbers

1. De Moivre's Theorem

De Moivre's theorem provides a formula for computing powers of complex numbers in polar form:

[r(cos θ + i sin θ)]^n = r^n [cos(nθ) + i sin(nθ)]

Using exponential form, this becomes:

(re^(iθ))^n = r^n e^(inθ)

Example: Using De Moivre's Theorem

Calculate (1 + i)^6:

First, convert to polar form: 1 + i = \sqrt{2} cis(π/4) = \sqrt{2}e^(iπ/4)

Using De Moivre's theorem: (1 + i)^6 = (\sqrt{2})^6 e^(6iπ/4) = 8e^(6πi/4) = 8e^(3πi/2)

Converting back to rectangular form: 8e^(3πi/2) = 8[cos(3π/2) + i sin(3π/2)] = 8(0 - i) = -8i

2. Roots of Complex Numbers

The nth roots of a complex number z = re^(iθ) are given by:

z^(1/n) = r^(1/n) e^(i(θ+2kπ)/n), k = 0, 1, 2, ..., n-1

These n distinct roots are equally spaced on a circle of radius r^(1/n) in the complex plane.

Example: Finding Cube Roots

Find the cube roots of -8:

Express in polar form: -8 = 8e^(iπ)

Using the formula for cube roots with n = 3:

(-8)^(1/3) = 8^(1/3) e^(i(π+2kπ)/3), k = 0, 1, 2

= 2e^(i(π+2kπ)/3), k = 0, 1, 2

For k = 0: 2e^(iπ/3) = 2(cos(π/3) + i sin(π/3)) = 2(1/2 + i√3/2) = 1 + i√3

For k = 1: 2e^(i(π+2π)/3) = 2e^(i3π/3) = 2e^(iπ) = 2(-1) = -2

For k = 2: 2e^(i(π+4π)/3) = 2e^(i5π/3) = 2(cos(5π/3) + i sin(5π/3)) = 2(1/2 - i√3/2) = 1 - i√3

Therefore, the three cube roots of -8 are: 1 + i√3, -2, and 1 - i√3

3. Euler's Identity

Euler's identity is a special case of Euler's formula when θ = π:

e^(iπ) + 1 = 0

This elegant equation connects five fundamental mathematical constants: 0, 1, π, e, and i.

4. Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that every polynomial equation of degree n with complex coefficients has exactly n complex roots, counting multiplicities.

This means that any polynomial can be completely factored in the complex number system, which is not always possible in the real number system.

Example: Polynomial Factorization

The polynomial p(x) = x² + 1 has no real roots, but in the complex number system, it factors as:

p(x) = x² + 1 = (x + i)(x - i)

The roots are x = i and x = -i.

5. Properties of Complex Conjugates

For any complex numbers z and w:

(z̄) = z (conjugate of conjugate is the original number)
(z + w)̄ = z̄ + w̄
(z × w)̄ = z̄ × w̄
(z/w)̄ = z̄/w̄ (for w ≠ 0)
z + z̄ = 2Re(z)
z - z̄ = 2Im(z)i
|z| = |z̄|
zz̄ = |z|²

6. Algebraic Properties

Complex numbers follow the standard algebraic properties:

Commutative property: z + w = w + z and z × w = w × z
Associative property: (z + w) + v = z + (w + v) and (z × w) × v = z × (w × v)
Distributive property: z × (w + v) = z × w + z × v
Additive identity: z + 0 = z
Multiplicative identity: z × 1 = z
Additive inverse: z + (-z) = 0
Multiplicative inverse: z × (1/z) = 1 (for z ≠ 0)

Applications of Complex Numbers

1. Solving Polynomial Equations

Complex numbers allow us to find solutions to all polynomial equations, including those that have no real roots.

Example: Solving a Quadratic Equation

Solve the equation x² + 2x + 5 = 0:

Using the quadratic formula: x = (-b ± √(b² - 4ac))/2a = (-2 ± √(4 - 4 × 1 × 5))/2 = (-2 ± √(4 - 20))/2 = (-2 ± √(-16))/2

= (-2 ± 4i)/2 = -1 ± 2i

Therefore, the solutions are x = -1 + 2i and x = -1 - 2i

2. Electrical Engineering

Complex numbers are extensively used in AC (alternating current) circuit analysis to represent impedance, which combines resistance and reactance.

The impedance Z of a circuit element is represented by a complex number:

Z = R + jX

where R is the resistance, X is the reactance, and j is used instead of i in engineering contexts.

Example: Complex Impedance in a Series RLC Circuit

In a series RLC circuit with resistance R = 10Ω, inductive reactance X_L = 20Ω, and capacitive reactance X_C = 5Ω, the total impedance is:

Z = R + j(X_L - X_C) = 10 + j(20 - 5) = 10 + j15 Ω

The magnitude of the impedance is |Z| = √(10² + 15²) = √(100 + 225) = √325 ≈ 18.03 Ω

The phase angle is θ = arctan(15/10) ≈ 56.31°

3. Control Theory

Complex numbers are used in control theory to analyze system stability through transfer functions and pole-zero plots.

4. Signal Processing

Complex numbers are fundamental in signal processing, particularly in Fourier transforms and filter design.

The Fourier transform converts a time-domain signal into its frequency-domain representation using complex exponentials.

Example: Discrete Fourier Transform

For a discrete signal x[n], the Discrete Fourier Transform (DFT) X[k] is defined as:

X[k] = Σ x[n]e^(-j2πnk/N), n = 0, 1, ..., N-1

The complex exponential e^(-j2πnk/N) represents sinusoids with different frequencies, and the complex result X[k] encodes both amplitude and phase information.

5. Fluid Dynamics

Complex variables techniques are used to solve problems in fluid dynamics, especially for two-dimensional, inviscid, irrotational flows.

Complex potential functions combine velocity potential and stream functions to describe fluid flow.

6. Quantum Mechanics

Complex numbers are essential in quantum mechanics, where the wave function that describes the quantum state of a system is complex-valued.

7. Fractals and Chaos Theory

Complex numbers are used to generate fascinating mathematical objects like the Mandelbrot set and Julia sets.

The Mandelbrot set is defined as the set of complex numbers c for which the sequence z_n+1 = z_n² + c does not escape to infinity when starting with z₀ = 0.

Example: Julia Set

A Julia set for a complex parameter c is the set of points z in the complex plane for which the orbit of z under the iteration z ← z² + c remains bounded.

Different values of c produce drastically different Julia sets, showcasing the rich geometry of complex dynamics.