Wikipedia:Manual of Style/Mathematics

This article contains some suggestions which are hoped to contribute towards writing clear, pleasant looking, and hopefully interesting Mathematics articles. This guide is meant not to substitute, but rather to complement, the Wikipedia manual of style which has much useful information for a Wikipedia editor.

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 general 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 which 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 have also an informal introduction to the topic, without rigor, suitable for a high school student or a first-year undergraduate, as appropriate. This should 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 have a section for motivation or applications, which can illuminate the use of the mathematical idea and its connections to other areas of mathematics.

Main part

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 = x**n = 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.

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.

A picture is a great way of bringing a point home, and often it could even precede the mathematical discussion of a concept. Wikipedia has a picture tutorial about how to include pictures in articles, and several graphics tutorials to help you make your own pictures.

A person editing a mathematics article should not fall into temptation that "this formula says it all". A non-mathematical reader will skip the formulas in most cases, and often a mathematician reading outside 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.

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.

This is an encyclopedia, not a collection of math texts; but we often want to include proofs, as a way of really exposing the meaning of some theorem, definition, etc. 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.

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.

Typesetting of mathematical formulas

Wikipedia allows editors to typeset mathematical formulas in (a subset of) LaTeX markup (see also TeX); the formulas are normally translated into PNG images. For the mechanics of this, see meta:Help:Formula.

The LaTeX formulas can be displayed in-line (like this: $\mathbf {x} \in \mathbb {R} ^{2}$ ), as well as on their own line:

\int _{0}^{\infty }e^{-x^{2}}\,dx

Having formulas as in-line PNG images, as above, is generally discouraged, for the following reasons.

The font size is somewhat larger than normal, making text containing in-line formulas hard to read.
The download speed of a page is negatively affected if it contains many images.
HTML (as described below) is adequate for most simple in-line formulas and better for text-only browsers.

When displaying formulas on their own line, one should indent the line with one or more colons (:); the above was typeset as

:<math>\int_0^\infty e^{-x^2}\,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.

If you plan on editing LaTeX formulas, it is helpful if you leave your preference settings (link in the upper right corner of this page, underneath your user name) in the "rendering math" section at the default "HTML if very simple or else PNG"; that way, you'll see the page like most users will see it.

The following sections cover the way of presenting simple in-line formulas in HTML.

Choice of text fonts

To start with, we generally use italic text for variables (but never for numbers or symbols). Most editors prefer to use emphasized text (with '', i.e., apostrophe-apostrophe) rather than italic text (with the  tag, resembling HTML), since the former is easier to type and read in the edit box. Some prefer using the HTML "variable" tag, <var>, since it provides semantic meaning to the text contained within. Which method you choose is entirely up to you, but in order to keep with convention, we recommend the double-apostrophe emphasis method. Thus we write:

''x'' = (''y''2 + 2)

which results in:

x = (y² + 2)

Note that the parentheses, equal and plus signs are not emphasized; try to keep them outside of the double-quoted sections. Also, don't forget that descriptive subscripts should not be in italics, because they are not variables. For example, m_foo is the mass of a foo.

Fixed names for functions, such as sin and cos, are not emphasized, but we emphasize f when we define the function by f(x) = sin(x) cos(x).

Sets are usually written in upper case, and emphasized; for example:

A = {x : x > 0}

would be written:

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

Greek letters should not be emphasized; for example, as in λ + ''y'' = π''r''2, for the expression "λ + y = πr²".

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 three apostrophes (''') rather than the  tag for bolding.

Templates for subscripts and superscripts make it possible to write c₃₊₅ by using ''c''{{sub|3+5}} instead of ''c''3+5, and similarly for superscripts.

Using special symbols

You may want to have a look at the table of mathematical symbols. One should keep in mind though, that not all of symbols in that list are displayed correctly on all browsers. Also, as mentioned earlier, it is generally better to be rather conservative with the use of symbols in order to reach a larger audience, for example by writing "x in Y" rather instead of "x ∈ Y".

Very simple formulas

If you enter a very simple formula using the math notations like <math>L^p</math> this will (in the default used by most users) not be displayed using a PNG image but using HTML, like this $L^{p}$ . This is different from typesetting it as ''L''''p''. Compare:

LaTeX:

L^{p}

HTML: L^p

Both forms are acceptable. Do not change one form to the other in other people's writing. They are likely to get annoyed since this seems to be a highly emotional issue. Changing to make a single page consistent is acceptable. However, at all costs avoid in-line PNG images. Even if you use <math>L^p</math> throughout the page, use ''L''∞ to get L^∞ rather than the horrid $L^{\infty }$ .

If you want to force an image output for a simple formula, put a \, (one quarter space in LaTeX) at the end of the formula, or \!\, (one negative quarter space and one quarter space).

Punctuation

It is good style to put a period at the end of a formula, if that formula is at the end of a sentence. If the formula is written in LaTeX, that is, surrounded by the <math> and </math> tags, then the period must be also inside the tags, because otherwise the period can get badly misaligned.

Including literature and references

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.