This subpage of the Manual of Style contains guidelines for writing and editing clear, encyclopedic, attractive, and interesting articles on mathematics. 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.

Suggested structure of a mathematics article

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

The article should start with an introductory paragraph (or two), which describes the subject in general terms. Name the field(s) of mathematics this concept belongs to and describe the mathematical context in which the term appears. Write the article title in bold. Include the historical motivation, provide some names and dates, etc. Here is an example.

In topology and related branches of mathematics, a continuous function is, loosely speaking, a function from one topological space to another that preserves open sets. Originally, the idea of continuity was a generalization of the informal idea of smoothness, or lack of discontinuity. The first statement of the idea of continuity was by Euler in 1784, relating to plane curves. Other mathematicians, including Bolzano and Cauchy, then refined and extended the idea of continuity. Continuous functions are the raison d'être of topology itself.

It is a good idea to also have an informal introduction to the topic, without rigor, suitable for a high school student or a first-year undergraduate, as appropriate. For example,

In the case of real numbers, a continuous function corresponds to a graph that can be drawn without lifting pen from paper; that is, without any gaps or jumps.

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.

It is quite helpful to have a section for motivation or applications, which can illuminate the use of the mathematical idea and its connections to other areas of mathematics.

Article body

If you want to introduce some notation, it should be in its own section. You should remember that not everyone understands that, for example, x^n or x**n mean xⁿ; so it is good to use standard notation if you can. If you need to use non-standard notations, or if you introduce new notations, define them in your article.

There should be an exact definition, in mathematical terms; often in a Definition(s) section, 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.

Using the term formal definition 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 to flag where the actual definition is to be found, after some sections of motivation. (Cf. rigged Hilbert space.)

Some representative examples would be nice to have, in a separate section, which could serve to both expand on the definition, and also provide some context as to why one might want to use the defined entity. 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:NOT#TEXTBOOK). 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 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

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

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:

• If G is a group, then G may be decomposed into cosets as follows.
• Let H be the corresponding subgroup of G. Then H must be 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…"
  • "We see that…"

and so on. 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.

• Avoid, as far as possible, using 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. Consider, even when some reassurance may be useful, using:

• "There follows directly..."
• "By straightforward, if lengthy, algebra..."

or the like.

• 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 articles should be consistent with themselves. If an article is consistent, then editors should not change the article from one style to another.

Mathematical conventions

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.