Table of Contents
Last modified on July 12th, 2024
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.
A set of Natural numbers (or counting numbers) includes all positive numbers without fractions or decimals that start from 1 and continue indefinitely.
ℕ = {1, 2, 3, 4, …}
A set of Whole numbers consists of all natural numbers, including 0.
𝕎 = {0, 1, 2, 3, 4, …}
A set of integers includes all positive and negative natural numbers and zero.
ℤ = {…, -4, -3, -2, -1, 0, 1, 2, 3, 4, …}
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.
ℚ = ${\{ \dfrac{p}{q} \ | \ p,q\in \mathbb{Z} \ and \ q\neq 0\}}$
Here, ${\dfrac{2}{3}}$ ∈ ℚ
A set of irrational Numbers consists of real numbers that are not rational.
ℚ’ = {x | x ∉ ℚ}
Here, ${\sqrt{2}}$ ∈ ℚ’
The set of real numbers includes all rational and irrational.
ℝ = {x | x ∈ ℚ ∪ ℚ’}
Here, ${\dfrac{2}{3}}$ and ${\sqrt{2}}$ ∈ ℝ
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.
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
A set of transcendental numbers includes any numbers that are not algebraic.
{x ∈ ℂ | x ∉ 𝔸}
Here, e and π are two transcendental numbers.
A set of imaginary numbers includes the numbers that, when squared, result in a negative number.
𝕀 = {bi | b ∈ ℝ}
Here, 9i and -4.7i ∈ 𝕀
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.
ℂ = {z ∈ ℂ | z = a + bi, and a, b ∈ ℝ}
2 + 3i and -5 + 7i ∈ ℂ
Thus,
We can summarize their relationship in a Venn diagram.
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’) |
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 non-zero integers {…, −3, −2, −1, 1, 2, 3, …} can be denoted by ℤ*
Last modified on July 12th, 2024