Multiple related examples may also be given together, as in Adjunction formula § Applications to curves. 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. Representative examples and applications help to illustrate definitions and theorems and to provide context for why they might be interesting. 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. 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. Sometimes this statement will be in the lead, for example: When the topic is a theorem, the article should provide a precise statement of the theorem.
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. Then f is called continuous if, for every open set O in T, the preimage f −1( O) is an open set in S. Let S and T be topological spaces, and let f be a function from S to T. If an article requires extensive notation, consider introducing the notation as a bulleted list or separating it into a section titled "Notation".Īn 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, an article that uses x^ n or x** n to denote exponentiation (instead of x n) should define the notations. If an article requires non-standard or uncommon notation, they should be defined. The latter group will be helped by an aside like ".where Δ( K) is the discriminant of the field K". 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. When in doubt, articles should define the notation they use. Readers have differing levels of experience and knowledge. Motivation or applications, which can illuminate the use of the topic and its connections to other areas of mathematics or other non-mathematical subjects.Include a physical or geometric analogy or diagram if it can help introduce the topic. The informal introduction should clearly state that it is informal, and that it is only stated to introduce the formal approach. The appropriate audience for the overview will vary by article, but it should be as basic as reasonable.
An informal introduction to the topic, without rigor, suitable for a general audience.The origin of the subject's name should be explained if it is not self-evident. Historical motivation, including names and dates, especially if the article does not have a "History" section.The lead section should include, when appropriate: In Euclidean plane geometry, Apollonius's problem is to construct circles that are tangent to three given circles in a plane. 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. The lead sentence should informally define or describe the subject. A reader can be assumed to be ignorant of any topics outside of that scope or more advanced topics. 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. 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. The lead sentence should state that the article is about a topic in mathematics, unless the title already does so. In general, the lead sentence should include the article title, or some variation thereof, in bold along with any alternate names, also in bold. In the latter case the LaTeX source is displayed without the tags and. For having formulae displayed when hovering, they must be written in raw html (without template ), or in LaTeX (inside. 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. The lead should, as much as possible, be accessible to a general reader, so specialized terminology and symbols should be avoided. and summarize the article's most important points.Main page: Wikipedia:Manual of Style/Lead sectionĪrticles should start with a short introductory section, called the "lead".
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