Understanding Number Types

These categories are defined by different properties, so a number can belong to multiple categories at once.

Complex Numbers (ℂ)

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 \( i^2 = -1 \).

Key Idea: The set of complex numbers includes all real numbers and all imaginary numbers. Think of it as the "big box" that contains all the others.

Examples: \( 3 + 4i \) (has both parts), \( 5 \) (which is \( 5 + 0i \), so it's real and complex), \( -2i \) (which is \( 0 - 2i \), so it's imaginary and complex).

Pure Imaginary Numbers

A pure imaginary number is a special type of complex number where the real part is zero. It has the form bi, where b is a non-zero real number.

Key Idea: It lies purely on the imaginary axis of the complex plane.

Examples: \( 2i \), \( -5i \), \( i\sqrt{3} \).

Note: \( 3 + 4i \) is complex but not pure imaginary because its real part is not zero.

Algebraic Numbers

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

Key Idea: It's a "well-behaved" number that can be defined as the solution to a finite equation with integers.

Examples: \( \sqrt{2} \) (root of \( x^2 - 2 = 0 \)), \( \frac{3}{5} \) (root of \( 5x - 3 = 0 \)), \( i \) (root of \( x^2 + 1 = 0 \)).

Transcendental Numbers

A transcendental number is any number (real or complex) that is NOT an algebraic number. There is no non-zero polynomial with integer coefficients that has this number as a root.

Key Idea: These numbers "transcend" the power of algebraic equations. They are often associated with infinite, non-repeating processes.

Examples: \( \pi \) (pi), \( e \) (the base of the natural logarithm).

Note: It is extremely difficult to prove a number is transcendental.

Summary and Relationships

The categories "Complex," "Algebraic," and "Transcendental" overlap. A number can be complex and algebraic (e.g., \( i \)), or complex and transcendental (e.g., \( \pi i \)).

Real Numbers are a subset of Complex Numbers. Pure Imaginary Numbers are a subset of Complex Numbers.

Algebraic and Transcendental are mutually exclusive. A number cannot be both.

Quick Guide

Number	Complex?	Pure Imaginary?	Algebraic?	Transcendental?
\( 5 \)	Yes	No	Yes	No
\( 3 - 2i \)	Yes	No	Yes	No
\( 4i \)	Yes	Yes	Yes	No
\( \sqrt{2} \)	Yes (it's real)	No	Yes	No
\( \pi \)	Yes (it's real)	No	No	Yes
\( i \)	Yes	Yes	Yes	No
\( \pi i \)	Yes	Yes	No	Yes