# Symbols in mathematics

This section covers:

International standards and resources

The American Mathematical Society provides detailed information for authors of mathematical information.

Australian conventions and resources

We have not identified any specific Australian resources.

## Writing mathematical symbols

Use italics for any letters used as symbols (eg constants, variables) to distinguish them from ordinary text:

x     y     M     Q

Preferably, use 1-letter symbols (with or without subscripts and superscripts) for variables to avoid confusion with a product of 2 symbols (which are shown closed up when there is no multiplication operator):

2x     ab     5xyz

Use plain text rather than italics for mathematical functions and variables where abbreviations of whole words are used:

log     exp     sin     avg     min

APG = net increase (births – deaths) + NOM
where APG = annual population growth, NOM = net overseas migration

Set vectors either in bold (roman) or in italics with an arrow above. Matrices and tensors are set in bold; however, note that there are more specific standards in particular fields regarding roman or italic type, and upper or lower case, so consult an expert:

Vector: f or $$\vec{f}$$

Tensor: S

Italicise Greek symbols:

A = πr2

## Writing fractions

Depending on the type of equation, a slash (acting as a division sign) can be used to separate the parts of the fraction. Although the slash normally does not have space inserted around it, a thin space or hairspace may be needed to make the fraction look visually balanced, depending on the particular characters on either side and the font used; it may be possible to use a dedicated equation editor to automatically adjust the spacing:

a + b/[with no space added]

a + b /[with a thin space added before the slash]

$$a + b/c$$ [using an equation editor]

(This also shows that equations entered using an equation editor often have a different ‘look’ from those entered manually. Establishing an overall policy on how mathematical material is to be written can be a good idea.)

Fractions can be built up using a horizontal line to indicate the division sign (‘display’ format):

$a+\frac{b}{c}$

## Mathematical operators

Use a nonbreaking space (or ¼ em space, from the ‘special characters’ of the symbol set) before and after all mathematical operators, such as +, –, =, >, <, ≤ and ±, apart from a slash (which has no spaces either side).

However, this only applies when the symbol is being used as an operator; expressions of numbers that include such symbols, such as >20 or –3, have the symbol closed up to the number (see Principles for using unit symbols).

How to insert a nonbreaking space:
Use Ctrl+Shift+Space (Windows), or Option+Space (Mac).

Caution! The multiplication symbol (×) is a specific symbol (unicode 00D7), not a lower-case x.

## Brackets, delimiters and fences

Brackets, which come in many types, are used to group things. In mathematics, they may be called delimiters, enclosures or fences. They are never italicised. They are nested in the order {[()]}.

$x + \left\{ 1+ \left[ 3+ 2 \times \left( 4 + 5 \right) \right] \right\}$

Take care with fences when writing mathematics inline.

The equation

$z=\frac{x-2}{y}$

is the same as $$z=\left(x-2\right)/y$$. It is not the same as $$z=x-2/y$$. In the second case, $$x$$ is not being divided by $$y$$.

Do not use spaces between numbers or variables and brackets:

$$2(a-b)$$
not
$$2~(a-b)$$

Except for a very few specialist cases, an opened fence must be closed by another of the same type, and they must be nested correctly.

Use full-height fences for displayed equations. This requires using an equation editor:

$$\left(\frac{A+B}{A} \right)$$   not   $$( \frac{A+B}{A} )$$

$$\left[ \frac{A+B}{A}+\ln\left( C+D\right) \right]$$   not   $$[ \frac{A+B}{A}+\ln ( C+D ) ]$$

## Subscripts and superscripts in mathematical expressions

Do not use spaces between terms and their subscript or superscripts, or after the subscript or superscript:

y3z     πr2     [entered from the keyboard]
$$y^3z$$     $$\pi{}r^2$$    [using an equation editor]

Equation editors determine the spacing for you, and in general manual adjustment is tricky and undesirable.

## Words and abbreviations in mathematical expressions

When formatting equations manually, use a nonbreaking space (or ¼ em space, from the ‘special characters’ of the symbol set) on either side of a word or abbreviation:

a sin y     log b

But close up such expressions if quantities before or after the functions are enclosed in brackets:

(ac)sin32y     exp(a +b)

Tip. A good equation editor will make these adjustments automatically.

## Calculus

Many symbols used in mathematics, including calculus, take complex accessory symbols, which are placed above or below the main symbol in display text, but can be placed adjacent to the symbols in running text:

Displayed Inline
$\lim_{a \rightarrow \infty}f(x) = b$ $$\lim_{a \rightarrow \infty}f(x) = b$$
$\int_0^1 f(x) dx = 3$ $$\int_0^1 f(x) dx = 3$$
$\sum_{i=0}^n a_i = 1$ $$\sum_{i=0}^n a_i = 1$$

In the symbols $$dx$$, $$dy$$ and so on that are common in calculus, the $$d$$ is italicised as well as the variable.

## Matrices

Matrices are arrays of numbers in rows and columns. In running text, an overall symbol can be used (eg matrix Y), or general symbols or vectors can be used to represent the components of the matrix.

$$M=\left[\begin{matrix}\mathbf{a}&\mathbf{b}&\mathbf{c}\end{matrix}\right]$$     becomes $M=\left[\begin{matrix}a_1&b_1&c_1\\a_2&b_2&c_2\\a_3&b_3&c_3\end{matrix}\right]$

where $\mathbf{a}=\left[\begin{matrix}a_1\\a_2\\a_3\end{matrix}\right],\qquad \mathbf{b}=\left[\begin{matrix}b_1\\b_2\\b_3\end{matrix}\right],\qquad \mathbf{c}=\left[\begin{matrix}c_1\\c_2\\c_3\end{matrix}\right]$

[the matrix consists of vectors $$\mathbf{a}$$, $$\mathbf{b}$$ and $$\mathbf{c}$$, each with components as noted]

$$M=\left[m_{ij}\right]$$     becomes $M=\left[ \begin{matrix}m_{11}&m_{12}&m_{13}\\m_{21}&m_{22}&m_{23}\end{matrix} \right]$

[the matrix consists of elements $$m_{ij}$$, where $$i$$ gives the row number and $$j$$ the column]

Did you know? In mathematics, the plural of matrix is matrices (not matrixes).

A matrix that is only 1 line (1 × n) or 1 column (n × 1) is known as a vector; (1 × n) matrices are also sometimes called row vectors or covectors.

## Ellipses in mathematical expressions

Three dots are used to indicate omitted values or terms, but the ellipsis character (…; unicode 2026) will usually not have enough space between the dots. Instead, use regular full stops separated by spaces.

In many cases, centred dots are preferable. As with lowered dots, to space them correctly you may need to use 3 single centred dots (·; unicode 00b7) rather than the midline ellipsis character (⋯; unicode 22ef). Use centred dots if the surrounding characters are vertically centred and lowered dots if the surrounding characters are lowered:

x1 + x2 + · · · + xn   not   x1 + x2 + . . . + xn
x1x2, . . . , xn   not   x1x2, · · ·, xn

When using 3 dots in the middle of an expression, the characters on either side should generally be the operator or separator that is repeated:

x1 + x2 + · · · + xn   not   x1 + x2 + x3 · · · xn

x1x2, . . . , xn   not   x1x2. . . xn