Number Sets

Number sets classify numbers into various categories, each with unique properties. The range of each number set shows the difference between the highest and lowest values within the sets.

Here are the major number sets commonly used in set theory, along with their symbols, properties, and examples.

Natural Numbers (ℕ) 

A set of Natural numbers (or counting numbers) includes all positive numbers without fractions or decimals that start from 1 and continue indefinitely. 

Examples

ℕ = {1, 2, 3, 4, …}

Whole Numbers (𝕎)

 A set of Whole numbers consists of all natural numbers, including 0. 

Example 

𝕎 = {0, 1, 2, 3, 4, …}

Integers (ℤ)

A set of integers includes all positive and negative natural numbers and zero. 

Example

ℤ = {…, -4, -3, -2, -1, 0, 1, 2, 3, 4, …}

Rational Numbers (ℚ)

A set of rational Numbers contains all numbers that are the quotient or fraction of two integers, where the numerator is an integer and the denominator is a non-zero integer.

Example

ℚ = ${\{ \dfrac{p}{q} \  |  \  p,q\in \mathbb{Z} \  and \  q\neq 0\}}$

Here, ${\dfrac{2}{3}}$ ∈ ℚ

Irrational Numbers (P or ℚ’)

A set of irrational Numbers consists of real numbers that are not rational. 

Example

ℚ’ = {x | x ∉ ℚ}

Here, ${\sqrt{2}}$ ∈ ℚ’

Real Numbers (ℝ)

The set of real numbers includes all rational and irrational. 

Example

ℝ = {x | x ∈ ℚ ∪ ℚ’}

Here, ${\dfrac{2}{3}}$ and ${\sqrt{2}}$ ∈ ℝ

Algebraic Numbers (𝔸)

A set of algebraic numbers consists of any numbers that are the solutions to a polynomial equation with rational coefficients, including all rational and some irrational numbers. 

Example

𝔸 = {α ∈ ℂ | ∃ p(x) ∈ Q[x], p(x) ≠ 0 such that p(α) = 0}

Here, ${\sqrt{2}}$ is an algebraic number since it is a solution of the polynomial equation x² – 2 = 0

Transcendental Numbers

A set of transcendental numbers includes any numbers that are not algebraic.

Example

{x ∈ ℂ | x ∉ 𝔸}

Here, e and π are two transcendental numbers.

Imaginary Numbers (𝕀)

A set of imaginary numbers includes the numbers that, when squared, result in a negative number.

Example

𝕀 = {bi | b ∈ ℝ}

Here, 9i and -4.7i ∈ 𝕀

Complex Numbers (ℂ)

A set of complex numbers combines real and imaginary numbers, represented as a + bi, where ‘a’ and ‘b’ are real numbers, and ‘i’ is the imaginary unit.

Example

ℂ = {z ∈ ℂ | z = a + bi, and a, b ∈ ℝ}

2 + 3i and -5 + 7i ∈ ℂ

Venn Diagram

Thus, 

• Natural numbers are a subset of integers
• Integers are a subset of rational numbers
• Rational numbers are a subset of real numbers
• Real and imaginary numbers together form complex numbers

We can summarize their relationship in a Venn diagram.

How to Use

Here are some algebraic equations and the number set required to solve them:

Equation	Solution	Number Set
2x – 12 = 0	x = 6	Natural number (ℕ)
5x + 15 = 0	x = -3	Integer (ℤ)
3x – 1 = 6	x = ${\dfrac{7}{3}}$	Rational number (ℚ)
x2 – 3 = 0	x = ${\pm \sqrt{3}}$	Irrational number (ℚ’)

Others

We can use an existing set symbol and add ‘+’ in the superscript to indicate positive numbers and ‘*’ in the superscript to signify non-zero values. 

For example,

• A set of positive integers {1, 2, 3, …} can be denoted by the symbol ℤ⁺

A set of non-zero integers {…, −3, −2, −1, 1, 2, 3, …} can be denoted by ℤ^*