User:The Editor's Apprentice/sandbox/1

Structure

Probably the hardest part of writing a Wikipedia article on a mathematical topic, and generally any Wikipedia article, is addressing the 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 as the article later on.

Article introduction

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.

In general, the lead sentence should include the article title, or some variation thereof, in bold along with alternative names, also be 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. Any topics outside of that scope or more advanced that them a reader can be assumed to be ignorant of. 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.

Article body

Readers have differing levels of experience and knowledge. When in doubt, articles should define the notation they uses. For example, some readers will immediately recognize that Δ(K) is a common notation for the discriminant of a number field as well as what it implies. On the other hand, other readers will have never previously encountered the notation. The latter group will be helped by a statement 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, the notations used should be defined within it. For example, an article that uses x^n or x**n to denote the exponentiation, usually written as xⁿ, the notations should be defined. If an article requires extensive notation, consider introducing the notation as a bulleted list or separating it into a section titled "Notation".

When an article is about a mathematical object, the article should provide an exact definition, such as 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 phrase "formal definition" may help to flag where the actual definition of a concept is in an article, possibly after being prefaced by section(s) of motivation. This may seem repetitive, as in mathematical contexts a formal definition is often a normal definition and a formal proof is just a proof, but it can help a reader navigate the article.

When an article is about 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 of the theorem into its own section of the article. This is especially true if the statement is long, as it is for the Poincaré–Birkhoff–Witt theorem, or has multiple equivalent formulations, as is the case for Nakayama's lemma.

Representative examples and applications are often helpful to readers. These serve both to expand on definitions and theorems and to provide context for why they might be interesting. 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 is the case for 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, such as those found at Lambert W function § Applications. It may also be beneficial to list non-examples—things which come close to satisfying the definition, but do not—in order to refine the reader's intuition. It is important to remember when including examples, however, that the purpose of an encyclopedia is to inform rather than instruct (see WP:NOTTEXTBOOK for details). 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 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.

Mathematics articles should not simply present a formula without explanation under the assumption that a reader will be able to understand it completely. Readers are likely to skip formulae in most cases or be unable to understand it in isolation. Careful thought should be given to each formula included, and words should be used instead if possible.^[why?] In particular, the English words "for all", "exists", and "in" should be preferred to corresponding symbols the ∀, ∃, and ∈. Similarly, definitions should be highlighted 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 details on its motivation.

Concluding matters

Most mathematical ideas are capable of some form of generalization. If appropriate, such material 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 "See also" section, which connects to related subjects, or to pages which could provide more insight into the contents of the current 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 the section § Including literature and references.