Wikipedia:Manual of Style/Mathematics

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.

Structure

Probably the hardest part of writing a Wikipedia article on a mathematical topic, and generally any Wikipedia article, is addressing a reader's level of knowledge. For example, when writing about a field in the context of abstract algebra, is it best to assume that a reader is already familiar with group theory? A general approach to writing an article is to start simple and then move towards more abstract and technical subjects later on in the article.

Article introduction

Main page: Wikipedia:Manual of Style/Lead section

Articles should start with a short introductory section, called the "lead". The purpose of the lead is to

• describe and define the subject,
• provide context regarding the subject,
• and summarize the article's most important points.

The lead should, as much as possible, be accessible to a general reader, so specialized terminology and symbols should be avoided. Formulas should appear in the first paragraph only if necessary, since they will not be displayed in the preview that pops up when hovering over a link. For having formulae displayed when hovering, they must be written in raw html (without templates {{var}} or {{math}}), or in LaTeX (inside <math>...</math>). In the latter case the LaTeX source is displayed without the tags <math> and </math>.

In general, the lead sentence should include the article title, or some variation thereof, in bold along with any alternate names, also in bold. The lead sentence should state that the article is about a topic in mathematics, unless the title already does so. It is safe to assume that a reader is familiar with the subjects of arithmetic, algebra, geometry, and that they may have heard of calculus, but are likely unfamiliar with it. For articles that are on these subjects, or on simpler subjects, it can be assumed that the reader is not familiar with the aforementioned subjects. A reader can be assumed to be ignorant of any topics outside of that scope or more advanced topics.

The lead sentence should informally define or describe the subject. For example:

In mathematics, topology (from the Greek τόπος, 'place', and λόγος, 'study') is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling and bending, but not tearing or gluing.

In Euclidean plane geometry, Apollonius's problem is to construct circles that are tangent to three given circles in a plane.

The lead section should include, when appropriate:

• Historical motivation, including names and dates, especially if the article does not have a "History" section. The origin of the subject's name should be explained 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 approach. Include a physical or geometric analogy or diagram if it can help introduce the topic.
• Motivation or applications, which can illuminate the use of the topic and its connections to other areas of mathematics or other non-mathematical subjects.

Article body

Readers have differing levels of experience and knowledge. When in doubt, articles should define the notation they use. For example, some readers will immediately recognize that Δ(K) means the discriminant of a number field, but others will never have encountered the notation. The latter group will be helped by an aside like "...where Δ(K) is the discriminant of the field K".

Use standard notation when possible. If an article requires non-standard or uncommon notation, they should be defined. For example, an article that uses x^n or x**n to denote exponentiation (instead of xⁿ) should define the notations. If an article requires extensive notation, consider introducing the notation as a bulleted list or separating it into a section titled "Notation".

An article about a mathematical object should provide an exact definition of the object, perhaps in a "Definition" section after section(s) of motivation. For example:

Let S and T be topological spaces, and let f be a function from S to T. Then f is called continuous if, for every open set O in T, the preimage f⁻¹(O) is an open set in S.

The phrase "formal definition" may help to flag the actual definition of a concept for readers unfamiliar with academic terminology, in which "definition" means formal definition, and a "proof" is always a formal proof.

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

Lagrange's theorem, in the mathematics of group theory, states that for any finite group G, the order (number of elements) of every subgroup H of G divides the order of G.

Other times, it may be better to separate the statement into its own section, as for long theorems like the Poincaré–Birkhoff–Witt theorem, or to present multiple equivalent formulations, as for Nakayama's lemma.

Representative examples and applications help to illustrate definitions and theorems and to provide context for why they might be interesting. Shorter examples may fit into the main exposition of the article, such as the discussion at Algebraic number theory § Failure of unique factorization, while others may deserve their own section, as in Chain rule § First example. Multiple related examples may also be given together, as in Adjunction formula § Applications to curves. Occasionally, it is appropriate to give a large number of computationally-flavored examples, as in Lambert W function § Applications. It may also be edifying to list non-examples, which almost-but-not-quite satisfy the definition. In keeping with the purpose and tone of an encyclopedia, examples should be informative rather than instructional (see WP:NOTTEXTBOOK for details).

A picture can really bring home a point, and can often precede the mathematical discussion of a concept. How to create graphs for Wikipedia articles contains some details on how to create graphs and other pictures as well as how to include them in articles.

Formulas tend to repel less mathematical readers, and mathematics articles should take pains to explain (or even replace) them by words if possible. In particular, the English words "for all", "exists", and "in" should be preferred to the corresponding symbols ∀, ∃, and ∈. Similarly, definitions should be highlighted with words such as "is defined by" in the text.

If not included in the introduction, a history section can provide additional context and details on the topic's motivation and connections.

Concluding matters

Most mathematical ideas are capable of 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.

It is also generally good to have a "See also" section in an article. The section should link to related subjects, or to pages which could provide more insight into the contents of the article. More details on "See also" sections can be found at Wikipedia:Manual of Style/Layout § "See also" section. Lastly, a well-written and complete article should have a "References" section. This topic is discussed in detail in the section § Including literature and references.

Writing style in mathematics

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. Do not write:

• 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, write something like:

• 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 similar to a whiteboard lecture. However, a narrative pedagogical style runs counter to Wikipedia's recommended encyclopedic tone. While opinions vary on the most edifying style, authors should generally strike a balance between bare lists of facts and formulae, and relying too much on addressing the reader directly and referring to "we". Also avoid contentless 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. There is no use in imploring the reader to take note of each thing being pointed out. Rather than drawing the reader's attention to crucial information buried in the text, try to reorganize and rephrase to put the crucial part first.

See also: MOS:NOTED

Articles should be as accessible as possible to readers not already familiar with the subject matter. Notations not entirely standard should be properly introduced and explained. Whenever a variable or other symbol is defined by a formula, make sure to say this is a definition introducing a notation, not an equation involving a previously known object. Also identify the nature of the entity being defined. Don't write:

• Multiplying M by u = v − v₀, ...

Instead, write:

• Multiplying M by the vector u defined by u = v − v₀, ...

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, useless phrases such as:

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

The reader might not find what you write obvious. Instead, try to hint why something must hold, such as:

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

Articles should avoid common blackboard abbreviations such as wrt (with respect to), wlog (without loss of generality), and iff (if and only if), as well as quantifier symbols ∀ and ∃ instead of for all and there exists. In addition to compromising the encyclopedic tone, these abbreviations are a form of jargon that may confuse the reader.

Avoid any when verbalizing quantifiers since it is ambiguous. Instead of if any x satisfies F(x) = 0, write if every x satisfies F(x) = 0, or if some x satisfies F(x) = 0, depending on what you wish to express.

The plural of formula is either formulae or formulas. Both are acceptable, but an article should be internally consistent. In an already consistent article, editors should refrain from changing one style to another.

Mathematical conventions

See also: Wikipedia:WikiProject Logic/Standards for notation

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

Natural numbers

"The set of natural numbers" has two common meanings: {0, 1, 2, 3, ...}, which may also be called non-negative integers, and {1, 2, 3, ...}, 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

• 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. 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 v² = +Q(v).

Algebraic geometry

• 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

• 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

• 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^kf(τ), and q = e^2πiτ.

Notational conventions

• 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/nZ; 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 2n 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 {{unicode|⋉}} (⋉) or {{unicode|⋊}} (⋊) for maximum portability.
• Subset is denoted by $\subseteq$, proper subset by $\subsetneq$. The symbol $\subset$ 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, $A\subset B$ might be given as the hypothesis of a theorem whose conclusion is obviously true in the case that $A=B$). All other uses of the $\subset$ symbol should be explicitly explained in the text.
• For a matrix transpose, use superscript non-italic capital letter T: X^T, $X^{\mathrm {T} }$ or $X^{\mathsf {T}}$, and not X^T, $X^{T}$, or $X^{\top }$.
• 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^Tb, never with juxtaposition ab.

Proofs

This is an encyclopedia and 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

See also: Wikipedia:WikiProject Computer science/Manual of style (computer science) § Algorithms

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:

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

generates the following:

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

Including citations and literature references

Per the Wikipedia policy, WP:VERIFY, it is essential for article content to have inline citations, and thus 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

See also: Help:Displaying a formula

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 might cause uneven spacing in the table of contents, as well as the appearance of illegible anchor links to sections. 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 is no positive response, or if planned changes affect more than one article, consider notifying an appropriate Wikiproject, such as WikiProject Mathematics for mathematical articles.

For inline formulae, such as a² − b², the community of mathematical editors of English Wikipedia currently has no consensus about preferred formatting; see WP:Rendering math for details.

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 uniformity within an article. For readability, it is also strongly preferred not to mix HTML and LaTeX markup in the same expression.

Using LaTeX markup

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: $\mathbf {x} \in \mathbb {R} ^{2}$), as well as on their own line: $$\int _{0}^{\pi }\sin x\,dx.$$

A frequent method for displaying formulas on their own line has been to indent the line with one or more colons (:). Although this produces the intended visual appearance, it produces invalid html (see Wikipedia:Manual of Style/Accessibility § Indentation). Instead, formulas may be placed on their own line using <math display=block>. For instance, the formula above was typeset using <math display=block>\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 can be slightly larger than that of the surrounding text on some browsers, making text containing inline formulae harder to read.
• The download speed of a page is negatively affected if it contains many formulas.
• Until bug T263572 is fixed, it will not work in image captions when readers click through to see full-size images.

If an inline formula needs to be typeset in LaTeX, keep the height down by using text-style or horizontal fractions: <math>\tfrac12 x</math> produces ${\tfrac {1}{2}}x$ and <math>x / 2</math> produces $x/2$, but <math>\frac{x}{2}</math> is too tall to fit inline.

Often better formatting can be achieved with <math display=inline> tag, which translates to 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>\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}</math> is too large to be used inline, rendering as

$\sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {\pi ^{2}}{6}}.$

Adding display=inline generates a smaller summation symbol and moves the limits to its right side. The rewritten formula <math display=inline>\sum_{n=1}^\infty 1/n^2 = \pi^2/6</math> renders as ${\textstyle \sum _{n=1}^{\infty }1/n^{2}=\pi ^{2}/6}$, which fits much better inline. Adding display=inline renders exponents lower, especially under square roots, often resulting in a smaller square root which fits better in inline text: compare <math>\sqrt{x^2+y^2}</math> to <math display=inline>\sqrt{x^2+y^2}</math> which render as ${\sqrt {x^{2}+y^{2}}}$ and ${\textstyle {\sqrt {x^{2}+y^{2}}}}$, respectively.

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

Line wrapping

Directly using <math>...</math> tags results in line wrapping points that allow wrapping between it and adjacent text (typically punctuation). Including such text within the tags to avoid this wrapping results in its font being discordant with other text. This can be remedied by wrapping the LaTeX markup in a suitable template (optionally excluding the adjacent punctuation), e.g. ({{nobr|<math>...</math>}}), or by replacing the tags with a template, e.g. ({{tmath|1=...}}). Take care of some necessary substitutions documented for {{tmath}}.

Deprecated formatting

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

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 ${\sqrt {\pi }}$ 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

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 templates are recommended for use; in particular, use of the {{frac}} template to format fractions is discouraged in mathematics articles.

Font formatting

By default, regular text is rendered in a sans serif font. The TeX markup of <math>...</math> uses a serif font to display a formula (whether as SVG or HTML). HTML math expressions should use the {{math}} template so they display in a serif font 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. Using {{math}} and {{mvar}} also helps some spell checkers and screen readers treat math markup properly. Note that certain special characters (equal signs, absolute value bars) require special attention.

The relationship is defined as {{math|''x'' {{=}} −(''y''<sup>2</sup> + 2)}}.

will result in:

The relationship is defined as x = −(y² + 2).

Variables

Use italic text for variables, but never for numbers or symbols. To ensure an italic serif typeface, use the {{mvar}} template to enclose a simple mention of a variable by name. This helps distinguish certain characters such as I and l. Within {{math}} templates (which will set a serif font but not italics), use the wikitext markup of double apostrophes to make variables italic. For example:

{{mvar|x}} is a value on the horizontal axis

displays as:

x is a value on the horizontal axis

and

{{math|''x'' {{=}} −(''y''<sup>2</sup> + 2)}}

results in:

x = −(y² + 2)

While italicizing variables, things like parentheses, digits, equal and plus signs should be kept outside of the double-apostrophed sections. Using double apostrophes for math content instead of {{mvar}} or without {{math}} is undesirable for readers because it will render in a sans serif font; this is especially confusing when other articles or sections render the same variables or equations in a serif font. 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

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

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

A = {x : x > 0}

would be written:

{{math|''A'' {{=}} {''x'' : ''x'' > 0}}} .

Greek letters

Italicize lower-case Greek letters when they are variables or constants (in line with the general advice to italicize variables): the example expression λ + y = πr² would then be typeset by:

{{math|''&lambda;'' + ''y'' = ''&pi;r''<sup>2</sup>}}

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

For consistency with the LaTeX style, do not italicize capital Greek letters; e.g. n! = Γ(n + 1) .

According to the Unicode Consortium, the characters U+00B5 µ MICRO SIGN and U+2126 Ω OHM SIGN are intended for compatibility with legacy character sets, and Unicode-capable environments (like Wikipedia) should use the Greek letters instead (U+03BC μ GREEK SMALL LETTER MU and U+03A9 Ω GREEK CAPITAL LETTER OMEGA). This is also required for "micro" by MOS:UNITSYMBOLS.

Common sets of numbers

Commonly used sets of numbers are typeset in boldface, as in the set of real numbers R. Again, typically we use wiki markup: three apostrophes (''') rather than the HTML <b> tag for making text bold. On chalkboards and increasingly in printed publications, mathematicians alternately use blackboard bold for sets such as the real numbers, which may be encoded in LaTeX as <math>\mathbb{R}</math> (preferred shortcut: <math>\R</math>), which renders as $\mathbb {R} .$ However, the special Unicode characters, such as U+211D (plain text ℝ or math font ℝ) and its adjacent characters should be avoided at present, since these characters are not yet universally supported and may have an inconsistent appearance. Either bold or blackboard bold is acceptable in Wikipedia articles, but should be consistent within each article.

Further information: § Blackboard bold

Superscripts and subscripts

Main page: Wikipedia:Manual of Style (superscripts and subscripts)

Superscripts and subscripts should be wrapped in <sup> and <sub> tags, respectively, with no other formatting info. Font sizes and such should be entrusted to be handled with stylesheets. For example, to write c₃₊₅, use

{{math|''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 + x² + x³ + x⁴ (with ²) versus

1 + x + x² + x³ + x⁴ (with <sup>2</sup>).

Moreover, the TeX engine used on Wikipedia may format simple superscripts using <sup>...</sup> depending on user preferences. Thus, for some users $x^{2}$ will be an image, and for others it will be HTML like x². Formulae formatted without using TeX should use the same syntax to maintain the same appearance.

Special symbols

The list of mathematical symbols, list of mathematical symbols by subject and the list at Wikipedia:Mathematical symbols 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). The list of mathematical symbols by subject includes markup for LaTeX and HTML, and Unicode code points.

Keep in mind:

1. 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 Mathematical operators and symbols in Unicode. 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>...</math> with LaTeX markup.
2. 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.

• For Roman numerals, Basic Latin (ASCII) letters should be used instead of the equivalent Unicode characters in the U+21XX range. For example, L and VI, not Ⅼ, and not precomposed characters like Ⅵ. (The only exception is when discussing the Unicode characters themselves.)

  For when to use or not use Roman numerals, see Wikipedia:Manual of Style/Dates and numbers.

• Use U+2032 ′ PRIME or U+2033 ″ DOUBLE PRIME where the prime symbol is appropriate; do not use the ASCII apostrophe (') or double quote (") in these cases. {{prime}} and {{pprime}} can be useful to prevent overlap with italicized characters.
• Use U+002A * ASTERISK when the character should render like a superscript, and is typical when used as a postfix. This character also appears on keyboards, and is thus easier to type and search for. Example: C*-algebra. Use U+2217 ∗ ASTERISK OPERATOR (&lowast;) for subscripts and when the bottom of the character should roughly align with the baseline of neighboring characters, which is typical when used as a prefix or infix operator, or a standalone character. Usage should be consistent across articles covering the same subfield of mathematics; see Asterisk#Mathematics for a canonical list.
• Use U+002F / SOLIDUS instead of U+2215 ∕ DIVISION SLASH
• Use U+003A : COLON instead of U+2236 ∶ RATIO
• Use U+007E ~ TILDE instead of U+223C ∼ TILDE OPERATOR. Use other symbols to mean "approximately" (and ¬ for negation) in mathematical expressions, because tilde has other mathematical meanings.
• Use <math>... \setminus ...</math> or <math>... \smallsetminus ...</math> instead of U+2216 ∖ SET MINUS or U+005C \ REVERSE SOLIDUS for set substraction. (Either Unicode character can be used where <math>...</math> markup cannot be used for technical reasons.)
• Use <math>... \circ ...</math> ($\circ$) instead of U+2218 ∘ RING OPERATOR (which on some systems is too small and can be confused with interpunct) or U+25CB ○ WHITE CIRCLE (which on some systems is too large).
• Use <math>... \ngtr ...</math> ($\ngtr$) instead of U+226F ≯ NOT GREATER-THAN, which on some systems renders as two separate characters.

See also: Wikipedia:Manual of Style/Abbreviations § Unicode abbreviation ligatures

Less-than sign

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

{{math|''x'' < 3}},

not

{{math|''x'' < 3}}.

Multiplication sign

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 or accessed via {{times}}. Do not use the letter "x" to indicate multiplication.

• 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 × is the dot operator ⋅ (also encoded <math>\cdot</math> and reachable in the "Math and logic" drop-down list below the edit box or via template {{sdot}}), which produces a symmetrically 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).

Metric units often embed the notion of multiplication and division. NIST endorses the half-high dot (⋅) or a bare space for this purpose.

Minus sign

The correct encoding of the minus sign "−" is different from all varieties of hyphen "-‐‑", 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

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:

{{open-closed|−π, π}}

produces

(−π, π].

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

Function symbol

Functions are conventionally denoted by the lowercase italic letters ⁠$f$⁠, ⁠$g$⁠, and ⁠$h$⁠, often with a serif font whose f extends below the baseline. Many readers see Wikipedia body copy in a sans-serif font whose italic f has a different appearance without a "tail". Circa 2010, some editors proposed using the Unicode symbol U+0192 ƒ LATIN SMALL LETTER F WITH HOOK (&fnof;) as a replacement function symbol; however, this symbol should not be used, because the mathematical function symbol is nothing more or less than an ordinary italic f, and its appearance should be consistent with the font otherwise used for mathematical notation in the same article. Furthermore, some screen readers, such as JAWS and NonVisual Desktop Access, pronounce the Unicode ƒ symbol as "florin" (it can be used as a Florin currency symbol) or a numerical code, which can be confusing.

An ordinary italicized letter should be used instead, as in ''f'', {{mvar|f}}, or <math>f</math>, which render as f, f, or $f$, respectively.

Radical symbol

The radical symbol √ can be used when written on its own, but when part of a larger expression, can be problematic. {{radic}} (often seen as {{sqrt}}) is the best way to write such expressions in HTML, but the result is unattractive due to the hole between the overline and the radical symbol in many web browsers:

√9, ³√27

This method should be avoided whenever technically possible to do so. Instead, use <math>...</math> tags and \sqrt{}, even if inline. For example:

${\sqrt {9}},{\sqrt[{3}]{27}}$

Because of Mediawiki bug T263572, <math>...</math> markup is incompatible with the Media Viewer (used for full-screen image viewing on mobile devices), so until that is fixed, the {{radic}} method or √ with no overline should be used in image captions.

The use of √ with no overline is acceptable for simple expressions, as long as the operand is unambiguous.

Explanation of symbols in formulae

A list as in

Example 1: The foocity is given by

$\mathbf {F} =\mathbf {b} \times a\mathbf {r} ,$

where

• $\mathbf {b}$ is the barness vector,
• $a$ is the bazness coefficient,
• $\mathbf {r}$ is the quuxance vector.

should be written as prose, to avoid using more vertical space than necessary:

Example 2: The foocity is given by

$\mathbf {F} =\mathbf {b} \times a\mathbf {r} ,$

where $\mathbf {b}$ is the barness vector, $a$ is the bazness coefficient, and $\mathbf {r}$ is the quuxance vector.

An exception would be if some of the definitions are very long (as in Heat equation, for example). In any 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

Just as in mathematics publications, a sentence which ends with a formula must have a period at the end of the formula. 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 punctuation 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 punctuation can be placed after the </math> tag and then the whole formula (including the punctuation) can be enclosed with the {{nowrap}} template, as in This shows that {{nowrap|<math>\tfrac{1}{2} = 0.5</math>.}}.

Font usage

Multi-letter names

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:

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

$2\sin(x+1)$ (parentheses are required to clarify the intended argument)

but not

$2sinx$   (incorrect—typeset from <math>2sin x</math>).

Note: For potential pitfalls of forms not understood consistently across the board, see order of operations and implied multiplication; if there is any risk that a term could become ambiguous for our readership, use parentheses.

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

$2\operatorname {csch} x$   (typeset from <math>2\operatorname{csch}x</math>).

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

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

$2{\rm {sin}}x$   (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:

$x_{\text{this one}}=y_{\text{that one}}$   (correct—typeset from <math> x_\text{this one} = y_\text{that one}</math>),

and

$\sum _{i=1}^{n}{y_{i}^{2}}$   (correct—typeset from <math>\sum_{i=1}^n { y_i^2 }</math>),

but not

$r=x_{predicted}-x_{observed}$   (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

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

$\int _{0}^{\pi }\sin x\,dx,$   (typeset from <math>\int_0^\pi \sin x \, dx ,</math>—note the thin space (\,) before dx),

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

$x+iy,$   (typeset from <math>x+iy,</math>), and

$e^{i\theta }.$   (typeset from <math>e^{i\theta} .</math>).

Some authors prefer to use an upright (Roman) font for operators, as in d, for the differential operator, as opposed to d for a variable. Upright fonts are sometimes used for standard, nearly universal constants, as in i, e, and π; 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

The blackboard bold letter style originated in the 1960s to distinguish bold letters from ordinary letters on a blackboard or using a typewriter; in professionally typeset documents, bold fonts were used for the same purpose. Since then, blackboard bold has gradually gained currency, and is now commonly used in mathematical printing to denote certain common objects in a style distinct from other uses of bold letters.

Today, either blackboard bold or ordinary bold letters can be used interchangeably to represent the standard number systems ($\mathbb {N} ,$ $\mathbb {Z} ,$ $\mathbb {Q} ,$ $\mathbb {R} ,$ $\mathbb {C} ,$ $\mathbb {H} ,$ $\mathbb {F} _{q},$ $\mathbb {Z} _{p},$ $\mathbb {Q} _{p}$) and for certain other mathematical objects, including affine space $\mathbb {A} ^{n}$, projective space $\mathbb {P} ^{n}$, adele rings $\mathbb {A} _{K}$, the additive and multiplicative group schemes ($\mathbb {G} _{a}$ and $\mathbb {G} _{m}$), and hypercohomology (e.g., $\mathbb {H} ^{i}(X,\Omega _{X}^{\bullet })$). Font preferences vary from one mathematical author or publisher to another.

A particular concern for the use of blackboard bold on Wikipedia is that the Unicode symbols for blackboard bold characters are not supported by all systems, and font substitution in browsers often renders these symbols in discordant fonts. The use of Unicode characters for blackboard bold is discouraged in English Wikipedia; instead, either the LaTeX rendering (for example <math>\mathbb{Z}</math> or <math>\Z</math>) or standard bold fonts should be used. As with all such choices, each article should be consistent with itself, and editors should not change articles from one choice of typeface to another, except to maintain internal consistency. When there is dispute, follow MOS:STYLERET.

Due to a rendering bug, LaTeX blackboard bold currently does not work with numerals. Use bold instead (e.g. {{math|'''1'''}} or <math>\bold{1}</math>), which is more common anyway. If absolutely necessary (e.g. when discussing the notation itself), use the Unicode character (e.g. 𝟙).

Due to bug T263572, <math>...</math> markup should not be used in image captions.

Fractions

See also: Wikipedia:Manual of Style/Dates and numbers § Fractions

In mathematics articles, fractions should always be written either with a horizontal fraction bar (as in $\textstyle {\frac {1}{2}}$), 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 precomposed fractions (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.

Only "/" is used for quotient objects in abstract algebra: R / A or $R/A$  – markup: {{math|''R'' / ''A''}} or <math>R / A</math>

For simple fractional subscripts or superscripts, the horizontal style is visually the least confusing: {{math|x<sup>1/2</sup>}} (x^1/2) or <math>x^{1/2}</math> ($x^{1/2}$).

Graphs and diagrams

The angle CAB is α.
The length of CA is b.

See also: Wikipedia:How to create charts for Wikipedia articles

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 $A$ for a point. This allows easy use in LaTeX markup. However, older books tend to use upright letters as in $\mathrm {A}$ 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

Help for those writing a formula

• Help:Displaying a formula
• Wikipedia:Mathematical symbols
• Wikipedia:Rendering math

General information

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

Further reading

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), doi:10.1137/1.9780898719550, 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