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 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.

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.

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.

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.

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.

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.

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.

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.

"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.

• 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).

• 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.

• 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.

• 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.

• 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).
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