Glossary of mathematical symbols

A mathematical symbol is a figure or a group of figures that stands for something in math. It can show a mathematical object, an action, a relationship, or help organize other symbols in a formula or mathematical expression. Symbols make it easier to write and understand math.

The simplest symbols are the decimal digits (0 through 9) and letters from the Latin alphabet. In the past, big letters stood for points in geometry, and small letters showed variables and fixed numbers. As math grew, the Greek alphabet and even Hebrew letters were added. Different styles like boldface, script typeface, German fraktur, and blackboard bold were created. These styles help group symbols by what they do—like using boldface for vectors and big letters for matrices.

Letters alone are not enough. Mathematicians also use symbols from punctuation marks and diacritics, changed letter forms, or brand-new signs like the plus sign (+) and the equals sign (=). These symbols let us write complicated ideas in a small, universal way, making math simpler to learn and more useful.

Layout of this article

This article explains many symbols used in mathematics. 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.

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

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. 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 groups of things.

The symbol ∅ stands for the empty set, a set that has nothing in it. The symbol # can show how many things are in a set.

We also use symbols like ∈ to show that something 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

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. It helps people know what numbers we are talking about.

For example, the symbol R stands for all real numbers.

Other symbols include N for natural numbers (like 1, 2, 3…), 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, 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 is useful in many areas of math and computer science.

The symbol □^⊥ has a few uses. It can mean the orthogonal complement of a linear subspace. It can also refer to special linear forms in more advanced math.

Advanced group theory

In group theory, symbols help us talk about 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 useful tools for mathematicians studying groups!

Infinite numbers

The infinity symbol ∞ shows something that never ends. In math, it means a 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 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 math. Their meanings depend on their shapes and what they enclose.

Parentheses (□) are used to group parts of a math problem. They show that those parts should be treated 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 too many parentheses. They also have special meanings, such as showing the whole number 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 in math to show ideas and relationships. 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! equals 5 × 4 × 3 × 2 × 1, which is 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 problems.