What is the purpose of a concept page?

A concept page is designed to help readers quickly understand and apply a concept encountered in a problem. It is a reference page, not a complete lesson or a textbook chapter.

How do you write a concept page?

All concept pages must be written in LaTeX and follow the outline below to ensure consistency and so that everyone can find their way around quickly.
For illustrative purposes, you can refer to the pages on the cross product, the conjugate of a complex number, or a group, which are among the most well-developed concept pages.

Intuitive Definition

To avoid diving too quickly into mathematical abstraction, it’s best to start with a short introduction (two or three sentences) that presents a simple way to understand the concept. For example, you might discuss why this concept was introduced, where it can be found, and possibly offer a brief tip on how to use it.

Rigorous Definition(s)

After the intuitive explanation, it’s time to precisely define your concept. Here, a clear definition is required—without oversimplification—including all necessary assumptions and notation.
However, some concepts have truly technical rigorous definitions (sets, integers, $\in$,$\dots$), so in such cases it is necessary to provide several definitions, arranged in ascending order of difficulty. Keep in mind that it won’t be seasoned mathematicians who need this definition the most, so don’t skimp on the pedagogy!

A Few Notes

Some concepts have several equivalent definitions at the same level of abstraction, or certain elementary properties (an odd function vanishes at $0$, no need for the assumption $f(0)=0$ for a linear function, etc.), which may be worth noting.

Add a Few Examples and Diagrams

One to three well-chosen examples are usually sufficient. Include a counterexample when it highlights an important limitation of the definition. You can also add a diagram if necessary.

The Role of Exercises

To get a feel for a concept, it’s important to create simple exercises related to it. For educational purposes, it’s best not to require the user to think any harder than is necessary to understand the concept. For a concept such as a Theorem, you can also include a proof of the theorem itself, or simply a proof of a basic property stated in a note.
Unlike problems, exercises should guide the user through the entire proof; the goal is not so much to explore new ideas as it is to reinforce one or more methodological points.

External Links and Sources

For concepts that do not have a universal definition, are at the research level, or for which you have drawn heavily on the insight provided by an external source, it is, of course, essential to cite that source.
Use the editor’s Link tool for this purpose, even if the page remains incomplete!

Status: Stub, Usable, Reviewed

Each concept has three review levels that are accessible only to trusted users and other moderators. Any newly created concept is first considered a draft; then, once it meets the previously established format requirements, a moderator can mark it as “Usable.”
Finally, a concept that includes at least one or two examples and one or two well-chosen exercises can be designated as Reviewed.

Summary

Pedagogy is key—imagine explaining the concept to someone who knows absolutely nothing about it but would like to quickly understand what it’s all about.

$\hookrightarrow$ Intuitive definition: highly recommended
$\hookrightarrow$ Rigorous definition: required, with different levels of abstraction if necessary.
$\hookrightarrow$ Notes and diagrams: optional but preferred
$\hookrightarrow$ Examples: Two or three are recommended.
$\hookrightarrow$ Exercises: At least one, required.