This guideline is a part of the English Wikipedia's Manual of Style. It is a generally accepted standard that editors should attempt to follow, though it is best treated with common sense, and occasional exceptions may apply. Any substantive edit to this page should reflect consensus. When in doubt, discuss first on the talk page. |

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This subpage of the Manual of Style contains guidelines for writing and editing clear, encyclopedic, attractive, and interesting articles on mathematics and for the use of mathematical notation in Wikipedia articles on other subjects. For matters of style not treated on this subpage, follow the main Manual of Style and its other subpages to achieve consistency of style throughout Wikipedia.

## Contents

- 1 Suggested structure
- 2 Writing style in mathematics
- 3 Mathematical conventions
- 4 Proofs
- 5 Algorithms
- 6 Including literature and references
- 7 Typesetting of mathematical formulae
- 8 Graphs and diagrams
- 9 See also
- 10 Notes
- 11 Further reading

## Suggested structure[edit]

Probably the hardest part of writing a mathematical article (actually, *any* article) is the difficulty of addressing the level of mathematical knowledge on the part of the reader. For example, when writing about a field, do we assume that the reader already knows group theory? A general approach is to start simple, then move toward more abstract and technical statements as the article proceeds.

### Article introduction[edit]

The article should start with a short introductory section (often referred to as the **lead**). The purpose of this section is to describe, define, and give context to the subject of the article, to establish why it is interesting or useful, and to summarize the most important points. The lead should as far as possible be accessible to a general reader, so specialized terminology and symbols should be avoided as much as possible.

In general, the lead sentence should include the **article title** in bold along with **alternative names**, also in bold. The lead sentence should inform non-mathematicians that the article is about a topic in mathematics, unless the title already does that. The lay reader knows that arithmetic, algebra, geometry, and calculus are topics in mathematics, but does not know that functional analysis or category theory or combinatorics belongs to mathematics. The lead sentence should informally define or describe the subject. For example:

Topology(from the Greek τόπος, "place", and λόγος, "study") is a major area of mathematics concerned with spatial properties that are preserved under continuous deformations of objects, for example, deformations that involve stretching, but not tearing or gluing.

In Euclidean plane geometry,

Apollonius' problemis to construct circles that are tangent to three given circles in a plane.

The lead section should include, where appropriate:

- Historical motivation, including names and dates, especially if the article does not have a separate History section. Explain the origin of the name if it is not self-evident.
- An
*informal introduction*to the topic, without rigor, suitable for a general audience. (The appropriate audience for the overview will vary by article, but it should be as basic as reasonable.) The informal introduction should clearly state that it is informal, and that it is only stated to introduce the formal and correct approach. If a physical or geometric analogy or diagram will help, use one: many of the readers may be non-mathematical scientists. *Motivation*or*applications*, which can illuminate the use of the mathematical idea and its connections to other areas of mathematics.

### Article body[edit]

Readers come to our articles with differing levels of experience and knowledge. When in doubt, your article should define the notation it uses. For example, some readers will recognize immediately that Δ(*K*) is a common notation for the discriminant of a number field and will understand what that means, while others will have never before encountered that idea. The latter group will be helped by a statement like, "...where Δ(*K*) is the discriminant of the field *K*."

Use standard notation if you can. If the article requires non-standard or uncommon notation, it should define those notations. For example, an article that uses *x*^*n* or *x****n* to denote the exponentiation operation *x ^{n}* should define those notations. If the article requires extensive notation, consider organizing the notation as a bulleted list or separating it into a notation section.

When the article is about a mathematical object, the article should provide an exact definition, often in a *Definition(s)* section. For example,

Let

SandTbe topological spaces, and letfbe a function fromStoT. Thenfis calledcontinuousif, for every open setOinT, the preimagef^{−1}(O) is an open set inS.

Using the term "formal definition" may help to flag where the actual definition is to be found, after some sections of motivation. (Cf. rigged Hilbert space.) This may seem rather empty to a mathematician (a formal definition is a mathematician's definition, as a formal proof is just a proof); but it may help the reader navigate the article.

When the article is about a theorem, the article should provide a precise statement. Sometimes this statement will be in the lead, for example,

Lagrange's theorem, in the mathematics of group theory, states that for any finite groupG, the order (number of elements) of every subgroupHofGdivides the order ofG.

Other times, it may be better to separate the statement of the theorem into its own section of the article. This is especially true if the statement is long (Poincaré–Birkhoff–Witt theorem) or has multiple equivalent formulations (Nakayama's lemma).

Representative examples and applications are helpful to readers. These serve both to expand on the definitions and theorems and to provide context for why one might be interested. The organization of the examples depends upon their number and length. Some examples may fit into the main exposition of the article, such as the discussion at Algebraic number theory#Failure of unique factorization. Others may benefit from being given their own section, such as Chain rule#First example. Multiple related examples can be given together, as in Adjunction formula#Applications to curves. Occasionally, it is appropriate to give a large number of computationally-flavored examples, such as Lambert W function#Applications. You might also want to list non-examples—things which come close to satisfying the definition but do not—in order to refine the reader's intuition more precisely. It is important to remember when composing examples, however, that the purpose of an encyclopedia is to *inform* rather than *instruct* (see WP:NOTTEXTBOOK). Examples should therefore strive to maintain an encyclopedic tone, and should be *informative* rather than merely *instructional*.

A picture is a great way of bringing a point home, and often it could even precede the mathematical discussion of a concept. How to create graphs for Wikipedia articles has some hints on how to create graphs and other pictures, and how to include them in articles.

A person editing a mathematics article should not fall into the temptation that "this formula says it all". A non-mathematical reader will skip the formulae in most cases, and often a mathematician reading outside her or his research area will do the same. Careful thought should be given to each formula included, and words should be used instead if possible. In particular, the English words "for all", "exists", and "in" should be preferred to the ∀, ∃, and ∈ symbols. Similarly, highlight definitions with words such as "is defined by" in the text.

If not included in the introductory paragraph, a section about the *history* of the concept is often useful and can provide additional insight and motivation.

### Concluding matters[edit]

Most mathematical ideas are amenable to some form of *generalization*. If appropriate, such material can be put under a *Generalizations* section. As an example, multiplication of the rational numbers can be generalized to other fields, etc.

It is good to have a *See also* section, which connects to related subjects, or to pages which could provide more insight into the contents of the current article.

Lastly, a well-written and complete article should have a references section. This topic will be discussed in detail below.

## Writing style in mathematics[edit]

There are several issues of writing style that are particularly relevant in mathematical writing.

In the interest of clarity, sentences should not begin with a symbol. Here are some examples of what *not* to do:

- Suppose that
*G*is a group.*G*can be decomposed into cosets, as follows. - Let
*H*be the corresponding subgroup of*G*.*H*is then finite.

Instead, one could write this:

- A group
*G*may be decomposed into cosets as follows. - Let
*H*be the corresponding subgroup of*G*. Then*H*is finite.

Mathematics articles are often written in a conversational style, as if a lecture is being presented to the reader, and the article is taking the place of the lecturer's whiteboard. However, an article that "speaks" to the reader runs counter to the ideal encyclopedic tone of most Wikipedia articles. Article authors should avoid referring to "we" or addressing the reader directly. While opinions vary on how far this guideline should be taken in mathematics articles—an encyclopedic tone can make advanced mathematical topics more difficult to learn—authors should try to strike a balance between simply presenting facts and formulae, and relying too much on directing the reader or using such clichés as Note that, It should be noted that, It must be mentioned that, It must be emphasized that, Consider that, and We see that.

Such introductory phrases are often unnecessary and can be omitted without affecting semantics. Rather than repeatedly attempting to draw the reader's attention to crucial pieces of information that have been appended almost as an afterthought, try to reorganize and rephrase the material such that crucial information comes first. There also should be no doubt as to the reader's willingness to continue reading and taking note of whatever information is presented; the reader does not need to be implored to take note of each thing being pointed out.

The articles should be accessible, as much as possible, to readers not already familiar with the subject matter. Notations that are not entirely standard should be properly introduced and explained. Whenever a variable or other symbol is defined in a formula, make sure that it is clear that this is a definition introducing a notation, and not, for example, just another equation. Also identify the nature of the entity being defined. So don't write this:

- Multiplying
*M*by*u*=*v*−*v*_{0}, ...

Instead, write:

- Multiplying
*M*by the vector*u*defined by*u*=*v*−*v*_{0}, ...

In definitions, the symbol "=" is preferred over "≡" or "**:**=".

When defining a term, do not use the phrase "if and only if". For example, instead of

- A function
*f*is*even*if and only if*f*(−*x*) =*f*(*x*) for all*x*

write

- A function
*f*is*even*if*f*(−*x*) =*f*(*x*) for all*x*.

If it is reasonable to do so, rephrase the sentence to avoid the use of the word "if" entirely. For example,

- An
*even function*is a function*f*such that*f*(−*x*) =*f*(*x*) for all*x*.

Avoid, as far as possible, phrases such as:

- It is easily seen that ...
- Clearly ...
- Obviously ...

The reader might not find what you write obvious. This kind of statement does not add new information and thus detracts from the clarity of the article. Instead, it may be helpful to the reader if a hint is provided as to *why* something must hold, such as:

- It follows directly from this definition that ...
- By a straightforward, if lengthy, algebraic calculation, ...

When lecturing using a blackboard, it is common to use abbreviations including *wrt* (with regard to) and *wlog* (without loss of generality), and to use quantifier symbols ∀ and ∃ instead of *for all* and *there exists* in prose. Some authors, including Paul Halmos, use the abbreviation *iff* for *if and only if* in print. On Wikipedia, all such abbreviations should be avoided. In addition to compromising the formal tone expected of an encyclopedia, these abbreviations are a form of jargon that may be unfamiliar to the reader.

The plural of *formula* is either *formulae* or *formulas*. Both are acceptable, but an article should be internally consistent. If an article is consistent, then editors should not change the article from one style to another.

## Mathematical conventions[edit]

A number of conventions have been developed to make Wikipedia's mathematics articles more consistent with each other. These conventions cover choices of terminology, such as the definitions of *compact* and *ring*, as well as notation, such as the correct symbols to use for a subset.

These conventions are suggested in order to bring some uniformity between different articles, to aid a reader who moves from one article to another. However, each article may establish its own conventions. For example, an article on a specialized subject might be more clear if written using the conventions common in that area. Thus the act of changing an article from one set of conventions to another should not be undertaken lightly.

Each article should explain its own terminology as if there are no conventions, in order to minimize the chance of confusion. Not only do different articles use different conventions, but Wikipedia's readers come to articles with widely different conventions in mind. These readers will often not be familiar with our conventions, which may differ greatly from the conventions they see outside Wikipedia. Moreover, when our articles are presented in print or on other websites, there may be no simple way for readers to check what conventions have been employed.

### Terminology conventions[edit]

#### Natural numbers[edit]

"The set of natural numbers" has two common meanings: , which may also be called *non-negative integers*, and , which may also be called *positive integers*. Use the sense appropriate to the field to which the subject of the article belongs if the field has a preferred convention. If the sense is unclear, and if it is important whether or not zero is included, consider using one of the alternative phrases rather than *natural numbers* if the context permits.

#### Algebra[edit]

- A ring is assumed to be associative and unital. A structure satisfying all the ring axioms except the existence of a multiplicative identity is called a rng.
^{[1]}There is an exception for rings of operators, such as * algebras, B* algebras, C* algebras, which we do not assume to be unital. - The ring with one element is called the zero ring.
- A local ring is not assumed noetherian (
*contra*Zariski). - For Clifford algebras use
*v*^{2}= +*Q*(*v*).

#### Algebraic geometry[edit]

- An algebraic variety is assumed to be an
*irreducible*algebraic set. - A scheme is not assumed to be separated. The term "prescheme" is not used.

#### Topology[edit]

- A compact space is not assumed to be Hausdorff (
*contra*Bourbaki, who uses quasi-compact for our notion of compactness). - Separation axioms for topological spaces are as described on the separation axiom page.

#### Miscellaneous[edit]

- Directed sets are preordered sets with finite joins, not partial orders as in, e.g., Kelley (
*General Topology*; ISBN 0-387-90125-6). - A lattice need not be bounded. In a bounded lattice, 0 and 1 are allowed to be equal.
- Elliptic functions are written in
*ω*= half-period style. - A weight
*k*modular form follows the Serre convention that*f*(−1/*τ*) =*τ*^{k}*f*(τ), and*q*=*e*^{2πiτ}.

### Notational conventions[edit]

- The abstract cyclic group of order
*n*, when written additively, has notation Z_{n}, or in contexts where there may be confusion with*p*-adic integers,**Z**/*n***Z**; when written multiplicatively, e.g. as roots of unity, C_{n}is used (this does not affect the notation of isometry groups called C_{n}). - The standard notation for the abstract dihedral group of order 2
*n*is D_{n}in geometry and D_{2n}in finite group theory. There is no good way to reconcile these two conventions, so articles using them should make clear which they are using. - Bernoulli numbers are denoted by B
_{n}, and are zero for*n*odd and greater than 1. - In category theory, write Hom-sets, or morphisms from
*A*to*B*, as Hom(*A*,*B*) rather than Mor(*A*,*B*) (and with the implied convention that the category is not a small category unless that is said). - The semidirect product of groups
*K*and*Q*should be written*K*×_{φ}*Q*or*Q*×_{φ}*K*where*K*is the normal subgroup and*φ*:*Q*→ Aut(*K*) is the homomorphism defining the product. The semidirect product may also be written*K*⋊*Q*or*Q*⋉*K*(with the bar on the side of the non-normal subgroup) with or without the*φ*.- The context should clearly state that this is a semidirect product and should state which group is normal.
- The bar notation is discouraged because it is not supported by all browsers.
- If the bar notation is used it should be entered as
`{}`

(⋉) or`{}`

(⋊) for maximum portability.

- Subset is denoted by , proper subset by . The symbol may be used if the meaning is clear from context, or if it is not important whether it is interpreted as subset or as proper subset (for example, might be given as the hypothesis of a theorem whose conclusion is obviously true in the case that ). All other uses of the symbol should be explicitly explained in the text.
- For a matrix transpose, use superscript non-italic capital letter T:
*X*^{T}, or , and not*X*^{T}, , or . - In a lattice, infima are written as
*a*∧*b*or as a product*ab*, suprema as*a*∨*b*or as a sum*a*+*b*. In a pure lattice theoretical context the first notation is used, usually without any precedence rules. In a pure engineering or "ideals in a ring" context the second notation is used and multiplication has higher precedence than addition. In any other context the confusion of readers of all backgrounds should be minimized. In an abstract bounded lattice, the smallest and greatest elements are denoted by 0 and 1. - The scalar or dot product of vectors should be denoted with a centre-dot
**a**⋅**b**, as an inner product ⟨**a**,**b**⟩ or (**a**,**b**), or as a matrix product**a**^{T}**b**, never with juxtaposition**ab**.

## Proofs[edit]

This is an encyclopedia, not a collection of mathematical texts; but we often want to include proofs to explain a theorem or definition. A downside of including proofs is that they may interrupt the flow of the article, whose goal is usually expository. Use your judgment; as a rule of thumb, include proofs when they expose or illuminate the concept or idea; don't include them when they serve only to establish the correctness of a result.

Since many readers will want to skip proofs, it is a good idea to set them apart in some way, for instance by giving them a separate section. Additional discussion and guidelines can be found at Wikipedia:WikiProject Mathematics/Proofs.

## Algorithms[edit]

An article about an algorithm may include pseudocode or in some cases source code in some programming language. Wikipedia does not have a standard programming language or languages, and not all readers will understand any particular language even if the language is well-known and easy to read, so consider whether the algorithm could be expressed in some other way. If source code is used always choose a programming language that expresses the algorithm as clearly as possible.

Articles should not include multiple implementations of the same algorithm in different programming languages unless there is encyclopedic interest in each implementation.

Source code should always use syntax highlighting. For example this markup:^{[2]}

<source lang="Haskell"> primes = sieve [2..] sieve (p : xs) = p : sieve [x | x <- xs, x `mod` p > 0] </source>

generates the following:

` primes = sieve [2..] sieve (p : xs) = p : sieve [x | x <- xs, x `mod` p > 0] `

## Including literature and references[edit]

It is quite important for an article to have a well-chosen list of references and pointers to the literature. Some reasons for this are the following:

- Wikipedia articles cannot be a substitute for a textbook (that is what Wikibooks is for). Also, often one might want to find out more details (like the proof of a theorem stated in the article).
- Some notions are defined differently depending on context or author. Articles should contain some references that support the given usage.
- Important theorems should cite historical papers as an additional information (not necessarily for looking them up).
- Today many research papers or even books are freely available online and thus virtually just one click away from Wikipedia. Newcomers would greatly profit from having an immediate connection to further discussions of a topic.
- Providing further reading enables other editors to verify and to extend the given information, as well as to discuss the quality of a particular source.

The Wikipedia:Cite sources article has more information on this and also several examples for how the cited literature should look.

## Typesetting of mathematical formulae[edit]

One may set formulae using LaTeX (the `<math>`

tag, described in the next subsection) or, in certain cases, using other means of formatting that render in HTML; both are acceptable and widely used, except for *section headings,* which should use HTML only, as LaTeX markup does not appear in the table of contents. Some of the issues presented by using LaTeX or HTML are discussed below.

Large scale formatting changes to an article or group of articles are likely to be controversial. One should not change formatting boldly from LaTeX to HTML, nor from non-LaTeX to LaTeX without a clear improvement. Proposed changes should generally be discussed on the talk page of the article before implementation. If there will be no positive response, or if planned changes affect more than one article, consider notifying an appropriate Wikiproject, such as WP:WikiProject Mathematics for mathematical articles.

For inline formulae, such as *a*^{2} − *b*^{2}, the community of mathematical editors of English Wikipedia currently has no consensus about preferred formatting; see WP:Rendering math for details.

Though, for a formula *on its own line* the preferred formatting is the LaTeX markup, with a possible exception for simple strings of Latin letters, digits, common punctuation marks, and arithmetical operators. Even for simple formulae the LaTeX markup might be preferred if required for the uniformity through an article.

### Using LaTeX markup[edit]

Wikipedia allows editors to typeset mathematical formulae in (a subset of) LaTeX markup (see also TeX); the formulae are, for a default reader, translated into PNG images. They may also be rendered as MathML or HTML (using MathJax), depending on user preferences. For more details on this, see Help:Displaying a formula.

The LaTeX formulae can be displayed inline (like this: ), as well as on their own line:

When displaying formulae on their own line, one should indent the line with one or more colons (:). The above was typeset using

`:<math>\int_0^\pi \sin x\,dx.</math>`

If you find an article which indents lines with spaces in order to achieve some formula layout effect, you should convert the formula to LaTeX markup.

Having LaTeX-based formulae inline has the following drawbacks:

- The font size is larger than that of the surrounding text on some browsers, making text containing inline formulae hard to read.
- Misalignment can result. For example, instead of
*e*^{x}, with "*e*" at the same level as the surrounding text and the*x*in superscript, one may see the*e*lowered to put the vertical center of the whole "*e*^{x}" at the same level as the center of the surrounding text. - The download speed of a page is negatively affected if it contains many formulae.
- Copy-pasting of the inline mathematics images that are generated by LaTeX markup will not work if the application into which you are pasting only accepts text.

If an inline formula needs to be typeset in LaTeX, often better formatting can be achieved with the `\textstyle`

LaTeX command. By default, LaTeX code is rendered as if it were a displayed equation (not inline), and this can frequently be too big. For example, the formula `<math>`

, which displays as , is too large to be used inline. `\sum_{n=1}^\infty 1/n^2 = \pi^2/6`

</math>`\textstyle`

generates a smaller summation sign and moves the limits on the sum to the right side of the summation sign. The code for this is `<math>`

, and it renders as the much more aesthetic . However, the default font for `\textstyle\sum_{n=1}^\infty 1/n^2 = \pi^2/6`

</math>`\textstyle`

is larger than the surrounding text on many browsers.

HTML-generating formatting, as described below, is adequate for most simple inline formulae and better for text-only browsers.

#### Deprecated formatting[edit]

Older versions of the MediaWiki software supported displaying simple LaTeX formulae as HTML rather than as an image. Although this is no longer an option, some formulae have formatting in them intended to force them to display as an image, such as an invisible quarter space (`\,`

) added at the end of the formula, or `\displaystyle`

at the beginning. Such formatting can be removed if a formula is edited and need not be added to new formulae.

#### Alt text[edit]

Images generated from LaTeX markup have alt text, which is displayed to visually impaired readers and other readers who cannot see the images. The default alt text is the LaTeX markup that produced the image. You can override this by explicitly specifying an `alt`

attribute for the `math`

element. For example, `<math alt="Square root of pi">\sqrt{\pi}</math>`

generates an image whose alt text is "Square root of pi". Small and easily explained formulas used in less technical articles can benefit from explicitly specified alt text. More complicated formulas, or formulas used in more technical articles, are often better off with the default alt text.

### Using HTML[edit]

The following sections cover the way of presenting simple inline formulae in HTML, instead of using LaTeX.

Templates supporting HTML formatting are listed in Category:Mathematical formatting templates. Not all however are recommended for use, in particular use of the {{frac}} template to format fractions is discouraged in mathematics articles.

#### Font formatting[edit]

By default, regular text is rendered in a sans serif font.

- The relationship is defined as
`''x'' = −(''y''<sup>2</sup> + 2)`

.

will result in:

- The relationship is defined as
*x*= −(*y*^{2}+ 2).

As TeX uses a serif font to display a formula (both as PNG and HTML), you may use the `{{math}}`

template to display your HTML formula in serif as well. Doing so will also ensure that the text within a formula will not line-wrap, and that the font size will closely match the surrounding text in any skin. Note that certain special characters (equal signs, absolute value bars) require special attention.

- The relationship is defined as
`{} −(''y''<sup>2</sup> + 2)}}`

.

will result in:

- The relationship is defined as
*x*= −(*y*^{2}+ 2).

##### Variables[edit]

To start with, we generally use italic text for variables, but never for numbers or symbols. You can use `''x''`

in the edit box to refer to the variable *x*. Some prefer using the HTML "variable" tag, `<var>`

, since it provides semantic meaning to the text contained within. Others use the {{mvar}} template to show single variables is a serif typeface, to help distinguish certain characters such as I and l. Which method you choose is entirely up to you, but in order to keep with convention, we recommend the wiki markup method of enclosing the variable name between repeated apostrophe marks. Thus we write:

`''x'' = −(''y''<sup>2</sup> + 2)`

,

which results in:

*x*= −(*y*^{2}+ 2) .

While italicizing variables, things like parentheses, digits, equal and plus signs should be kept outside of the double-apostrophed sections. In particular, do not use double apostrophes as if they are `<math>`

tags; they merely denote italics. Descriptive subscripts should not be in italics, because they are not variables. For example, *m*_{foo} is the mass of a foo. SI units are never italicized: *x* = 5 cm.

##### Functions[edit]

Names for standard functions, such as sin and cos, are not in italic font, but we use italic names such as *f* for functions in other cases; for example when we define the function as in *f*(*x*) = sin(*x*) cos(*x*).

##### Sets[edit]

Sets are usually written in upper case italics; for example:

*A*= {*x*:*x*> 0}

would be written:

`''A'' = {''x'' : ''x'' > 0}`

.

##### Greek letters[edit]

Italicize lower-case Greek letters when they are variables (in line with the general advice to italicize variables): the example expression *λ* + *y* = *πr*^{2} would then be typeset as:

`''λ'' + ''y'' = ''πr''<sup>2</sup>`

(It is also possible to enter Greek letters directly.)

For consistency with the (La)TeX style, do not italicize *capital* Greek letters.

##### Common sets of numbers[edit]

Commonly used sets of numbers are typeset in boldface, as in the set of real numbers **R**; see blackboard bold for the types in use. Again, typically we use wiki markup: three apostrophes (`'''`

) rather than the HTML `<b>`

tag for making text bold.

##### Superscripts and subscripts[edit]

Subscripts and superscripts should be wrapped in `<sub>`

and `<sup>`

tags, respectively, with no other formatting info. Font sizes and such should be entrusted to be handled with stylesheets. For example, to write *c*_{3+5}, use

`''c''<sub>3+5</sub>`

.

Do not use special characters like `²`

(`²`

) for squares. This does not combine well with other powers, as the following comparison shows:

- 1 +
*x*+*x*² +*x*^{3}+*x*^{4}(with`²`

) versus - 1 +
*x*+*x*^{2}+*x*^{3}+*x*^{4}(with`<sup>2</sup>`

).

Moreover, the TeX engine used on Wikipedia may format simple superscripts using `<sup>...</sup>`

depending on user preferences. Thus, instead of the image , many users see *x*^{2}. Formulae formatted without using TeX should use the same syntax to maintain the same appearance.

#### Special symbols[edit]

There are list of mathematical symbols, list of mathematical symbols by subject and a list at Wikipedia:Mathematical symbols that may be useful when editing mathematics articles. Almost all mathematical operator symbols have their specific code points in Unicode outside both ASCII and General Punctuation (with notable exception of "+", "=", "|", as well as ",", ":", and three sorts of brackets). As a rule of thumb, specific mathematical symbols shall be used, not similar-looking ASCII or punctuation symbols, even if corresponding glyphs are indistinguishable. The list of mathematical symbols by subject includes markup for LaTeX and HTML, and Unicode code points.

There are two caveats to keep in mind, however.

- Not all of the symbols in these lists are displayed correctly on all browsers (see Help:Special characters). Although the symbols that correspond to named entities are very likely to be displayed correctly, a significant number of viewers will have problems seeing all the characters listed at Unicode Mathematical Operators. One way to guarantee that an uncommon symbol is rendered correctly for all readers is to force the symbol to display as an image, using the <math> environment.
- Not all readers will be familiar with mathematical notation. Thus, to maximize the size of the audience who can read an article, it is better to be conservative in using symbols. For example, writing "
*a*divides*b*" rather than "*a*|*b*" in an elementary article may make it more accessible.

#### Less-than sign[edit]

Although the MediaWiki markup engine is fairly smart about differentiating between unescaped "<" characters that are used to denote the start of an embedded HTML or HTML-like tag and those that are just being used as literal less-than symbols, it is ideal to use `<`

when writing the less-than sign, just like in HTML and XML. For example, to write *x* < 3, use

`''x'' < 3`

,

not

`''x'' < 3`

.

#### Multiplication sign[edit]

Standard algebraic notation is best for formulae, so two variables *q* and *d* being multiplied are best written as *qd* when presented in a formula. That is, when *citing* a formula, don't use `×`

.

However, when *explaining* the formula for a general audience (not just mathematicians), or giving examples of its application, it is prudent to use the multiplication sign: "×", coded as `×`

in HTML. Do not use the letter "x" to indicate multiplication. For example:

- When dividing 26 by 4, 6 is the quotient and 2 is the remainder, because 26 = 6 × 4 + 2.
- −42 = 9 × (−5) + 3

An alternative to the `×`

markup is the dot operator `⋅`

(also encoded `<math>\cdot</math>`

and reachable in the "Math and logic" drop-down list below the edit box), which produces a properly spaced centered dot: "*a* ⋅ *b*".

Do not use the ASCII asterisk (*) as a multiplication sign outside of source code. It is not used for this purpose in professionally published mathematics, and most fonts render it in an inappropriate vertical position (above the midline of the text rather than centered on it). For the dot operator, do not use punctuation symbols, such as a simple interpunct `·`

(the choice offered in the "Wiki markup" drop-down list below the edit box), as in many fonts it does not kern properly. The use of U+2022 • BULLET as an operator symbol is also discouraged except in abstract contexts (e.g. to denote an unspecified operator).

#### Minus sign[edit]

The correct encoding of the minus sign "−" is different from all varieties of hyphen "-‐‑",^{[3]} as well as from en-dash "–". To really get a minus sign, use the "minus" character "−" (reachable via selecting "Math and logic" in the drop-down list below the edit box or using `{{subst:minus}}`

) or use the "`−`

" entity.

#### Square brackets[edit]

Square brackets have two problems; they can occasionally cause problems with wiki markup, and editors sometimes 'fix' the brackets in asymmetrical intervals to make them symmetrical. The nowiki tag can be used as a general solution to problems like this, as in `<nowiki>]</nowiki>`

to have the ] treated as literal text.

The use of intervals for the range or domain of a function is very common. A solution which makes the reason for the different brackets around an interval more plain is to use one of the templates {{open-closed}}, {{closed-open}}, {{open-open}}, {{closed-closed}}. For instance:

`{}`

,

produces

- (−π, π].

These templates use the {{math}} template to avoid line breaks and use the TeX font.

#### Function symbol[edit]

There is a special Unicode symbol, U+0192 ƒ LATIN SMALL LETTER F WITH HOOK (HTML `ƒ`

**·** `ƒ`

), sometimes used as the Florin currency symbol.^{[4]} As of December 2010, this character is not interpreted correctly by screen readers such as JAWS and NonVisual Desktop Access^{[5]}. An italicized letter *f* should be used instead.

### Explanation of symbols in formulae[edit]

A list such as:

Example 1: The foocity is given by

where

bis the barness vector,ais the bazness coefficient,ris the quuxance vector.

should be written as prose:^{[why?]}

Example 2: The foocity is given by

where

bis the barness vector,ais the bazness coefficient, andris the quuxance vector.

An exception would be if some of the definitions are very long (for example, as in Heat equation), but, even in this case, each definition should end with a comma or semicolon, and the last one should end with a period if it terminates a sentence.

### Punctuation after formulae[edit]

Just as in mathematics publications, a sentence which ends with a formula must have a period at the end of the formula.^{[6]} This equally applies to displayed formulae (that is, formulae that take up a line by themselves). Similarly, if the conventional punctuation rules would require a question mark, comma, semicolon, or other punctuation at that place, the formula must have that punctuation at the end.

If the formula is written in LaTeX, that is, surrounded by the `<math>`

and `</math>`

tags, then the punctuation should also be inside the tags, because otherwise the puctuation could wrap to a new line if the formula is at the edge of the browser window. Alternatively—since the previous result can be unaesthetic, especially for inlined formulae presented as an image whose baseline does not line up with that of the running text—the formula can be enclosed using the {{nowrap}} template, as in `This shows that { = 0.5</math>.}}`

.

### Font usage[edit]

#### Multi-letter names[edit]

Functions that have multi-letter names should always be in an upright font. The most well-known functions—trigonometric functions, logarithms, etc.—can be written without parentheses for as long as the result does not become ambiguous. For example:

- (parentheses may be omitted here, as the argument consists of a single term only; typeset from
`<math>2\sin x</math>`

) - (parentheses are required to clarify the intended argument)

but not

- (
*incorrect*—typeset from`<math>2sin x</math>`

).

When operator (function) names do not have a pre-defined abbreviation, we may use `\operatorname`

:

- (typeset from
`<math>2\operatorname{csch}x</math>`

). - (typeset from
`<math>a\operatorname{tr}(A)</math>`

).

`\operatorname`

includes correct spacing that would not be present with other means such as `\rm`

:

- (
*incorrect*—typeset from`<math>2{\rm sin} x</math>`

).

Special care is needed with subscripted labels to distinguish the purpose of the subscript (as this is a common error): variables and constants in subscripts should be italic, while textual labels should be in normal text font (Roman, upright). For example:

- (correct—typeset from
`<math> x_\text{this one} = y_\text{that one}</math>`

),

and

- (correct—typeset from
`<math>\sum_{i=1}^n { y_i^2 }</math>`

),

but not

- (
*incorrect*—typeset from`<math>r = x_{predicted} - x_{observed}</math>`

).

For several years this manual recommended `\mbox`

as a workaround for lack of `\text`

, but this is now considered undesirable. See *An opinion: Why you should never use \mbox within Wikipedia.*

#### Roman versus italic[edit]

For single-letter variables, constants, and operators such as the differential, imaginary unit, and Euler's number, Wikipedia articles usually use an italic font. One writes

- (typeset from
`<math>\int_0^\pi \sin x \, dx ,</math>`

—note the thin space (`\,`

) before`dx`

), - (typeset from
`<math>\frac{dz}{dx} = \frac{dz}{dy} \cdot \frac{dy}{dx} ,</math>`

), - (typeset from
`<math>x+iy,</math>`

), and - (typeset from
`<math>e^{i\theta} .</math>`

).

Some authors prefer to use an upright (Roman) font, as in d, i, and e, and other authors use Roman boldface, as in **i**. Changes from one style to another should be done only to make an article consistent with itself. Formatting changes should not be made solely to make articles consistent with each other, nor to make articles conform to a particular style guide or standards body. It is inappropriate for an editor to go through articles doing mass changes from one style to another. When there is dispute over the correct style to use, follow the same principles as MOS:STYLERET.

Generally, one way to determine which usage is appropriate on Wikipedia is to look at prevalence in reliable sources in addition to relevant style guides, per WP:WEIGHT. For example, the ISO 80000-2 recommends that the mathematical constant *e* should be typeset in an upright Roman font: e. But this guide is rarely followed in reliable mathematical sources, and it is contradicted by other style guides, like Donald Knuth's *TeXbook*. This makes the more common practice to use an italic face for the constant *e*.

#### Blackboard bold[edit]

Certain objects, such as the real numbers **R**, are traditionally printed in boldface. On a blackboard or a whiteboard, boldface type is replaced by blackboard bold. Traditional mathematical typography never used printed blackboard bold because it is harder to read than ordinary boldface. Nowadays, however, some printed books and articles use blackboard bold. A particular concern for the use of blackboard bold on Wikipedia is that these symbols must be rendered as images because the Unicode symbols for blackboard bold characters are not supported by all systems.

An article may use either boldface type or blackboard bold for objects traditionally printed in boldface. As with all such choices, the article should be consistent with itself, and editors should not change articles from one choice of typeface to another except for consistency. Again, when there is dispute, follow MOS:STYLERET.

### Fractions[edit]

In mathematics articles, fractions should always be written either with a horizontal fraction bar (as in ), or with a forward slash and with the baseline of the numbers aligned with the baseline of the surrounding text (as in 1/2). The use of {{frac}} (such as ^{1}⁄_{2}) is discouraged in mathematics articles. The use of Unicode symbols (such as ½) is discouraged entirely, for accessibility reasons among others. Metric units are given in decimal fractions (e.g., 5.2 cm); non-metric units can be either type of fraction, but the fraction style should be consistent throughout the article.

## Graphs and diagrams[edit]

There is no general agreement on what fonts to use in graphs and diagrams. In geometrical diagrams points are normally labelled using upper case letters, sides with lower case and angles with lower case Greek letters.

Recent geometry books tend to use an italic serif font in diagrams as in for a point. This allows easy use in LaTeX markup. However, older books tend to use upright letters as in and many diagrams in Wikipedia use sans-serif upright A instead. Graphs in books tend to use LaTeX conventions, but yet again there are wide variations.

For ease of reference diagrams and graphs should use the same conventions as the text that refers to them. If there is a better illustration with a different convention, though, the better illustration should normally be used.

## See also[edit]

### Help for those writing a formula[edit]

### General information[edit]

- Wikipedia:WikiProject Mathematics
- Wikipedia:Scientific citation guidelines—advice on providing references for mathematical and scientific articles

## Notes[edit]

**^**Currently, ring (mathematics) and related articles attempt to cover both unital rings and non-unital rings: they do not consistently implement this interpretation. This attempt to cover multiple meanings violates WP:DICT#Major differences (homographs).**^**This example, from here [1], is in Haskell, not a well-known language so generally not a good choice when showing an algorithm.**^**Note that, aside of <math>, many templates and parser functions accept the hyphen-minus "-" as a valid representation of the minus sign. Except situations where "-" has to represent the minus sign in a source code (including wiki code), it should not be seen in a rendered page, though.**^**Latin Extended-B, [2]**^**Wikipedia talk:WikiProject Mathematics/Archive 68#ƒ or f?**^**This style, adopted by Wikipedia, is shared by Higham (1998), Halmos (1970), the Chicago Manual of Style, and many mathematics journals.

## Further reading[edit]

A style guide specifically written for mathematics:

- Higham, Nicholas J. (1998),
*Handbook of Writing for the Mathematical Sciences*(second ed.), SIAM (Society for Industrial and Applied Mathematics), ISBN 0-89871-420-6.

More style guidance:

- Halmos, P.R. (1970), "How to Write Mathematics",
*Enseignements Mathématiques*,**16**: 123–152, doi:10.5169/seals-43857. Reprinted in ISBN 0821800558

Some finer points of typography are discussed in:

- Knuth, Donald E. (1984),
*The TeXbook*, Reading, Massachusetts: Addison-Wesley, ISBN 0-201-13448-9.

General style manuals often include advice on mathematics, including

- University of Chicago Press Staff, ed. (2010),
*The Chicago Manual of Style*(16th ed.), University of Chicago Press, ISBN 9780226104201