Complex, Algebraic, and Pure Imaginary Numbers

This is an excellent question that gets to the heart of how numbers are classified in mathematics. The relationship is one of nested sets, like Russian dolls. The following diagram shows how these number sets relate to each other:

Now, let's break down each group in detail.

1. Complex Numbers (ℂ)

Definition

A complex number is any number that can be written in the form: a + bi where:

a and b are Real Numbers, and i is the imaginary unit, defined by the property i² = -1.

In this form, a is called the real part and b is called the imaginary part.

Key Takeaway: This is the broadest category. It encompasses every other number type on this list.

Examples

3 + 4i (Has both a real and imaginary part)

5 (Which is 5 + 0i - a Real Number)

-2i (Which is 0 - 2i - a Pure Imaginary Number)

√2 + i (An Algebraic Number)

Applications

Electrical Engineering: The fundamental tool for analyzing AC circuits. Complex numbers represent impedance, voltage, and current, simplifying calculations with phase shifts and oscillations.

Signal Processing: Used in Fourier transforms to break down complex signals (like audio, images, radio waves) into their constituent sine waves. Your digital music and WiFi rely on this.

Quantum Mechanics: The state of a quantum system is described by a "wave function," which is inherently a complex-valued function.

Control Theory: Used to analyze and design stable systems, from autopilots to cruise control.

Fluid Dynamics: Describes potential flow in two dimensions.

Computer Graphics: Can be used for rotations and fractal generation (like the famous Mandelbrot set).

2. Algebraic Numbers (𝔸)

Definition

An algebraic number is any complex number (real or non-real) that is a root of a non-zero polynomial equation with integer coefficients.

In simpler terms: It's a number that can be a solution to an equation like: aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀ = 0 where all the a's are integers (..., -2, -1, 0, 1, 2, ...) and aₙ ≠ 0.

Key Takeaway: This is a vast subset of Complex numbers. It includes all rational numbers and many irrationals, but excludes the "weird" transcendentals like π and e.

Examples

7 (Solution to x - 7 = 0)

2/3 (Solution to 3x - 2 = 0)

√2 (Solution to x² - 2 = 0)

The Golden Ratio, φ (Solution to x² - x - 1 = 0)

i itself (Solution to x² + 1 = 0)

3 + 4i (It's a solution to x² - 6x + 25 = 0)

Non-Examples (Transcendental Numbers)

π (Pi): There is no finite polynomial with integer coefficients that has π as a root.

e (Euler's Number): Same as π.

Applications

Number Theory: The study of algebraic numbers is a central field in itself, exploring the properties of numbers and the solutions to polynomial equations.

Computer Algebra Systems: Software like Mathematica and Maple must be able to manipulate and simplify algebraic numbers symbolically.

Field Theory (Abstract Algebra): Algebraic numbers form "field extensions" of the rational numbers, which is a key concept in higher algebra.

Coding Theory: Some error-correcting codes are constructed using algebraic numbers and properties of algebraic number fields.

3. Pure Imaginary Numbers

Definition

A complex number where the real part is zero. That is, any number of the form: bi where b is a non-zero real number.

If b = 0, you have 0i, which is just 0, a real number. So by convention, we exclude this to talk about purely imaginary numbers.

Key Takeaway: This is a specific type of complex number. It's a subset of Complex Numbers but is distinct from Real Numbers (except for the number 0, where they intersect).

Examples

2i

-5i

iπ (Since π is a real number)

i√3

Non-Examples

3 + 4i (Has a non-zero real part)

7 (Has no imaginary part)

Applications

Pure imaginary numbers don't typically have unique applications separate from complex numbers as a whole. Their utility comes from their role within the complex number system.

They represent a 90-degree rotation or a pure phase shift in the complex plane.

In the AC circuit example, a purely imaginary impedance represents a perfect capacitor or inductor, storing and releasing energy without dissipating it as heat (which is represented by the real part).

Summary Comparison

Feature	Complex Numbers	Algebraic Numbers	Pure Imaginary Numbers
Definition	Any number of the form a + bi	A root of a polynomial with integer coefficients	Any number of the form bi (where b ≠ 0)
Set Notation	ℂ	𝔸	A subset of ℂ
Contains	Everything: Real, Imaginary, Algebraic, Transcendental	All Rationals, some Irrationals, and some Complex numbers (e.g., i)	Only numbers with zero real part
Does NOT Contain	Nothing (it's the largest set)	Transcendental numbers (π, e)	Numbers with a non-zero real part
Primary Application	General tool for 2D problems, oscillations, waves	Number theory, abstract algebra, computer algebra	A component of complex numbers representing 90° rotations
Example	3 + 4i, 5, -2i, π, e	5, 2/3, √2, i, 3+4i	2i, -iπ