In basic mathematics, many different symbols exist and are adopted widely. The following is a compilation of the most commonly-used symbols in **arithmetic** and **common mathematics**, along with other symbols whose usage covers multiple subfields of mathematics.

For readability purpose, these symbols are categorized by their **function** into tables. Other comprehensive lists of symbols — as categorized by subject and type — can be also found in the relevant pages below (or in the navigational panel).

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## Mathematical Constants

In common mathematics, **constants** are often used to denote key natural numbers, integers, real numbers and complex numbers. The following table documents the most common of these — along with their name, usage and function.

Symbol Name | Explanation | Example |
---|---|---|

$0$ (Zero) | Additive identity of common numbers | $5 + 0 = 0 + 5 = 5$ |

$1$ (One) | Multiplicative identity of common numbers | $6 \times 1 = 6$ |

$\sqrt{2}$ (Square root of $2$) | Positive number whose square is $2$. Pythagoras’ constant. Approximately $1.414$. | $\sqrt{2}$ is often considered to be the “simplest” irrational number. |

$e$ (Euler’s number) | Base of natural logarithm. Limit of sequence $\left( 1+\frac{1}{n} \right)^n$. Approximately $2.718$. | $\ln e = 1$ |

$\pi$ (Pi, Archimedes’ constant) | Ratio of a circle’s circumference and diameter. Half-circumference of unit circle. | $\pi$ is irrational and approximately $3.1416$. |

$\varphi$ (Phi, golden ratio) | Ratio between two positive numbers $a > b$ such that $\frac{a+b}{a} = \frac{a}{b}$. Positive root of polynomial $x^2-x-1$. | $\varphi = \dfrac{1+\sqrt{5}}{2} \approx 1.618$ |

$i$ (Imaginary unit) | Principal square root of $-1$. Foundational component of complex numbers. | $i^2 = (-i)^2 = -1$ |

## Delimiters

Delimiters are symbols used to signal the **separation** between different independent mathematical entities. These include the common delimiters such as parentheses, brackets and braces, and the use of delimiters in the context of intervals.

### Common Delimiters

Symbol Name | Explanation | Example |
---|---|---|

$.$ | Decimal separator | $15.35 + 8.25 = 23.60$ |

$,$ | Object separator | $\{ 5, 0, 2 \}$ |

$:$ | Ratio indicator | $4 : 3 = 1024 : 768$ |

$(), [], \{ \}$ | Order-of-operation indicators | $\left[(2+3) + 4\right] + 5$ |

$( )$ | Tuple-indicator | $(4, 7, 11, 15)$ |

### Intervals

Symbol Name | Explanation | Example |
---|---|---|

$[a, b]$ | Closed interval from $a$ to $b$ | $\pi \in [3, 5]$ |

$(a, b)$ | Open interval from $a$ to $b$ | $(1, 9) =$ $\{x \in \mathbb{R} \mid \\ 1 < x < 9\}$ |

$[a, b)$ | Right-open interval from $a$ to $b$ | $[e, \pi) \subseteq [1, \infty)$ |

$(a, b]$ | Left-open interval from $a$ to $b$ | $0 \notin (0, 100]$ |

## Operators

Operators are placeholder symbols used to denote **mathematical operations**, which take one or multiple mathematical objects to another similar object. In common mathematics, these include the arithmetic operators, and other number-related unary operators.

### Arithmetic Operators

Symbol Name | Explanation | Example |
---|---|---|

$x + y$ | Sum ($x$ plus $y$) | $\dfrac{3}{5} + \dfrac{2}{3} = \dfrac{19}{15}$ |

$x-y$ | Difference ($x$ minus $y$) | $13-1.\overline{3} = 11.\overline{6}$ |

$-x$ | Additive inverse (negative $x$) | $(-1.5) + 1.5 =0$ |

$x \times y$, $x \cdot y$, $xy$ | Product ($x$ times $y$) | $2 \times (3 + 5) = \\ 6 + 10$ |

$x \div y$, $\, x / y$ | Quotient ($x$ over $y$) | $16 \div 2.5 = 6.4$ |

$\dfrac{x}{y}$ | Fraction of $x$ over $y$ | $\dfrac{3}{8}=0.375$ |

$x^y$ | Power ($x$ raised to $y$) | $3^{10} = 9^5$ |

$\pm$ | Plus-and-minus operator | With the quadratic formula, we have that $x = \dfrac{-b \pm \sqrt{\Delta}}{2a}$. |

$\mp$ | Minus-and-plus operator | $5 \pm (-3) = 5 \mp 3$ |

### Number-related Unary Operators

Symbol Name | Explanation | Example |
---|---|---|

$\sqrt{x}$ | Principal square root of $x$ | $\sqrt{30}= \\ \sqrt{2 \cdot 3 \cdot 5}$ |

$\sqrt[n]{x}$ | nth root of $x$ | $\sqrt[3]{125}=5$ |

$|x|$ | Absolute value of $x$ | $|-5| = |5| = 5$ |

$x \%$ | $x$ percent | $5 \% \doteq \dfrac{5}{100}$ |

## Relational Symbols

In mathematics, relational symbols are used to denote **mathematical relations**, which take one or multiple mathematical objects to form full mathematical sentences. In arithmetic and common mathematics, these include the relational symbols related to equality and comparison.

### Equality-based Relational Symbols

Symbol Name | Explanation | Example |
---|---|---|

$x \doteq y$, $x \overset{df}{=} y$, $x := y$ | $x$ is defined as $y$ | $\mathbb{R}_+ \doteq \\ \{ x \in \mathbb{R} \mid x > 0 \}$ |

$x = y$ | $x$ is equal to $y$ | $ \pi = \dfrac{C}{d}$ |

$x \ne y$ | $x$ is not equal to $y$ | $\sqrt{3} \ne 1.7$ |

$x \approx y$ | $x$ is approximately equal to $y$ | $\dfrac{5}{7} \approx 0.714$ |

$f(x) \propto g(x)$ | Function $f$ is directly proportional to function $g$ | $\dfrac{\pi}{2} x^2 \propto 3x^2$ |

### Comparison-based Relational Symbols

Symbol Name | Explanation | Example |
---|---|---|

$x < y$ | $x$ is less than $y$ | $2 < e$ |

$x > y$ | $x$ is greater than $y$ | $\dfrac{13}{4} > 3$ |

$x \le y$ | $x$ is less than or equal to $y$ | $1 \le n^2$ |

$x \ge y$ | $x$ is greater than or equal to $y$ | $n! \ge 2^n$ for $n \ge 4$ |

## Notational Symbols

Notational symbols are often **conventions** and **shorthands** which don’t fall into the categories of constants, delimiters, operators and relational symbols. The following table documents some of these in the context of common mathematics — along with their usage and meaning.

Symbol Name | Explanation | Example |
---|---|---|

$\ldots, \cdots$ | Horizontal ellipsis symbols | $3 + 7 + 11 + \cdots + 43$ |

$\infty$ | Infinity symbol | $\dfrac{1}{1} + \dfrac{1}{2} + \cdots = \infty$ |

$Q. E. D.$, $\square$, $\blacksquare$ | QED / End-of-the-proof symbols | Hence $1 + \cdots + n = \frac{n(n+1)}{2}$, as desired. $\, \blacksquare$ |

※, ⨳ | Contradiction symbols | Squaring both sides of the equation yields that $2 < 1$. ⨳ |

For the master list of symbols, see mathematical symbols. For lists of symbols categorized by **type** and **subject**, refer to the relevant pages below for more.

- Arithmetic and Common Math Symbols
- Geometry and Trigonometry Symbols
- Logic Symbols
- Set Theory Symbols
- Greek, Hebrew, Latin-based Symbols
- Algebra Symbols
- Probability and Statistics Symbols
- Calculus and Analysis Symbols

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## Additional Resources

**Definitive Guide to Learning Higher Mathematics**: A standalone, 10-principle framework for tackling higher mathematical learning, thinking and problem solving efficiently**10 Commandments of Higher Mathematical Learning**: An illustrated web guide on 10 scalable rules for learning higher mathematics