Glossary of mathematical symbols

A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula or a mathematical expression. These symbols are essential for expressing all kinds of mathematics clearly and efficiently.

The most basic symbols include the decimal digits (0 through 9) and letters from the Latin alphabet. Historically, uppercase letters stood for points in geometry, while lowercase letters represented variables and constants. As mathematics grew, the Greek alphabet and even Hebrew letters joined in. To meet the growing needs of math, different styles like boldface, script typeface, German fraktur, and blackboard bold were developed. Each style helps group symbols by their purpose—like using boldface for vectors and uppercase letters for matrices.

Since letters alone aren't enough, mathematicians created symbols from punctuation marks and diacritics, deformed letter forms, or designed entirely new signs like the plus sign (+) and the equals sign (=). These symbols help us write complex ideas in a compact, universal way, making math both easier to learn and more powerful to use.

Layout of this article

This article explains many symbols used in mathematics. Because symbols don't have a natural order, the sections are arranged from basic to more advanced topics. Beginners will find common symbols in the early sections, while more specific symbols appear later.

Most symbols have different meanings depending on the area of mathematics or their position in a formula. This article groups symbols by their most common uses. You can find a symbol by its Unicode character, which makes searching and copying easier. The article also shows how to write symbols using LaTeX, the standard way to display math online.

Arithmetic operators

The plus sign (+) means addition, like in "3 + 2". It can also show that a number is positive, like +2, to make it clear it’s not negative.

The minus sign (−) means subtraction, like in "3 − 2". It can also show the opposite of a number, like −2, which is the same as "the negative of 2".

The multiplication sign (×) means times, like in "3 × 2". In geometry, it can show a cross product.

The dot (·) also means multiplication, like "3 ⋅ 2". In geometry, it can show a dot product.

The plus–minus sign (±) shows two possible values, like 10 ± 2 means the value could be between 8 and 12.

The division sign (÷) means division, like dividing 3 by 2, though it’s not commonly used in math today.

The slash (/) is often used instead, like "3 / 2". It can also show division in other areas, like probability.

The square-root symbol (√) means finding the square root, like √2. With a number above it, like √3, it shows the 7th root of 3.

Equality, equivalence and similarity

The equals sign (=) shows that two things are the same. For example, if we say "let x = 5", it means x is equal to 5. The not-equal sign (≠) tells us when two things are different.

Other symbols show closeness or similarity. The approximately equal sign (≈) means two numbers are nearly the same, like π ≈ 3.14159. The tilde (~) can mean things are about the same size or have the same order of magnitude. It is also used in probability to show a distribution, like X ∼ N(0, 1), meaning X follows a standard normal distribution. The triple bar (≡) can show an identity, a congruence in number theory, or logical equivalence. The symbol (≅) might show that two shapes are congruent or that two mathematical structures are isomorphic.

Comparison

The greater-than sign > shows that one number is bigger than another. For example, it tells us that 5 is greater than 3.

We also have symbols like ≤ (less than or equal to) and ≪ (much less than). These help us compare numbers and groups in different ways. For example, ≤ means that one number or group is either smaller than or exactly the same as another.

Set theory

Mathematical symbols help us work with sets, which are collections of objects.

The symbol ∅ represents the empty set, a set with no elements. The symbol # can show how many elements are in a set or represent special products of prime numbers.

We also use symbols like ∈ to show that an object is part of a set ("is in"), and ∉ to show it is not. Symbols such as ⊂ and ⊆ help us describe how sets relate to each other — for example, whether one set is a subset of another.

Basic logic

Several logical symbols are used in mathematics to show relationships between ideas. For example, the symbol ¬ means "not". If you have a statement E, ¬E is true when E is false. The symbol ∨ means "or". If you have two statements E and F, E ∨ F is true when either E, F, or both are true. The symbol ∧ means "and". E ∧ F is true only when both E and F are true.

Other symbols include ∀ which means "for all", and ∃ which means "there exists". For example, ∀x E means that the statement E is true for every possible value of x. ∃x E means that there is at least one value of x for which E is true. These symbols help mathematicians express complex ideas clearly and concisely.

Main article: List of logic symbols

Blackboard bold

The blackboard bold typeface is used to show important number systems in math. This helps people know what numbers we are talking about, no matter where they see the symbol. For example, the symbol R stands for all real numbers. This is useful because many areas of math use real numbers, even if they don’t study them directly.

Other symbols include N for natural numbers (like 1, 2, 3… or sometimes 0, 1, 2…), Z for all integers (…, -2, -1, 0, 1, 2, …), Q for rational numbers (fractions), and C for complex numbers. There are also special symbols for p‑adic integers, p‑adic numbers, quaternions, octonions, sedenions, and trigintaduonions.

Calculus

Mathematical symbols help us work with numbers and ideas in a clear and organized way. In calculus, we use special symbols to talk about how things change. For example, the symbol f′ (read as "f prime") shows the derivative of a function f. This tells us how quickly f is changing at any point.

We also have symbols like ∂f/∂x, which is called the partial derivative. This helps us understand how a function changes when only one part of it is altered, while the other parts stay the same. These symbols are important tools for solving many kinds of problems in math and science.

Linear and multilinear algebra

The symbol ∑, called capital-sigma notation, is used to show the sum of several numbers or terms. For example, it can add up the squares of numbers from 1 to n, like 1² + 2² + 3² + ... + n².

The symbol transpose, shown as T or superscript T, is used with matrices. If A is a matrix, A^T means we flip the matrix so that its rows become columns and its columns become rows. This helps in many areas of math and computer science.

The symbol □^⊥ has a few uses. It can mean the orthogonal complement of a linear subspace in a space with angles, like finding all vectors that are at right angles to every vector in that subspace. It can also refer to special linear forms in more advanced math.

Advanced group theory

In group theory, symbols help us describe special relationships between groups. The symbol ⊲ means that one group is a normal subgroup of another. This tells us the smaller group fits perfectly inside the larger one.

Other symbols like ⋉ and ⋊ describe how two groups can combine in different ways. For example, they can form a semidirect product, where the groups mix but keep some of their own properties. The symbol ≀ shows the wreath product, a way to build a new group from two existing groups. These symbols are important tools for mathematicians studying groups!

Infinite numbers

The infinity symbol ∞ represents something that goes on forever without ending. In math, it can show that a sum, product, or other calculation has no upper or lower limit.

Other special symbols like 𝔠, ℵ, ℶ, and ω are used to talk about very large numbers that go on forever in different ways. For example, ℵ₀ is the smallest infinite number, representing how many natural numbers there are. These symbols help mathematicians work with ideas that are bigger than any regular number.

Main article: infinity symbol
Main articles: summation, infinite product, integral, projectively extended real line, fraktur, cardinality of the continuum, real numbers, aleph number, beth number, limit ordinal, ordered set, natural numbers, computer science, computational complexity, matrix multiplication, asymptotic growth, Big O notation § Related asymptotic notations, number theory, prime omega function

Brackets

Many types of brackets are used in mathematics. Their meanings depend on the shapes and what they enclose, as well as their arrangement.

Parentheses (□) are used in expressions to group parts together, showing that they should be considered as one unit. They help set the order of operations. For example, in f(x), the parentheses show that x is the input to the function f. Parentheses can also show ordered pairs, like (π, 0), or sequences and matrices.

Square brackets [□] can be used like parentheses to avoid nested parentheses. They also have special meanings, such as showing the integral part of a number or describing sets. Braces { } are used in set-builder notation to list elements or define sets based on rules.

Miscellaneous

Mathematical symbols are special signs used to show ideas and relationships in math. They help us write and solve problems more easily. For example, the symbol "!" after a number, like in "5!", means the factorial of that number. This tells us to multiply the number by every smaller whole number down to 1 (so 5! = 5 × 4 × 3 × 2 × 1 = 120).

Other symbols show if numbers divide each other evenly. The symbol "∣" between two numbers, like "3 ∣ 6", means that the first number is a divisor of the second. Symbols can also show when lines are parallel or perpendicular in geometry, helping us describe shapes and their angles. These signs make math clearer and more powerful for solving all kinds of problems.