Wikipedia provides one of the more prominent resources on the Web for factual information about contemporary mathematics, with over 20,000 articles on mathematical topics. It is natural that many readers use Wikipedia for the purpose of self-study in mathematics and its applications. Some readers will be simultaneously studying mathematics in a more formal way, while others will rely on Wikipedia alone. There are certain points that need to be kept in mind by anyone using Wikipedia for mathematical self-study, in order to make the best use of what is here, perhaps in conjunction with other resources.
Studying mathematics from a reference source is not ideal. Unless you consult Wikipedia to answer a specific question, it is not reasonable to expect instant results. If you are a student who is studying for school curriculum, you should give first priority to the textbooks. Try to learn from them first, but if you find any concept or any problem hard to understand or solve respectively, then you can jump to Wikipedia for that particular topic. You can get good knowledge about that concept as the content present on Wikipedia is a cumulative contribution of a lot of people. You can also learn about the topics that are related to that particular concept with the help of those hyperlinks, so you should consider Wikipedia as a resource to understand certain things but not the entire subject. When it comes to solving a particular problem it is not always true that you will find the solution on Wikipedia, so you should also have other tools in hand on which you can rely.
Mathematics textbooks are conventionally built up carefully, one chapter at a time, explaining what mathematicians would call the prerequisites before moving to a new topic. For example, you may think you can study Chapter 10 of a book before Chapter 9, but reading a few pages may then show you that you are wrong. Because Wikipedia's pages are not ordered in the same way, it may be less clear what the prerequisites are, and where to find them, if you are struggling with a new concept.
There is no quick way around the need for prerequisite knowledge. When King Ptolemy asked for an easier way of learning mathematics, Euclid is famously said to have replied, "there is no royal road to geometry". Some background reading is expected when learning a new mathematical subject, and different readers will have greatly different needs regarding introductory material. Therefore:
The best advice for retaining definitions of mathematical terms is to draw images or write examples that include the definitions.
Mathematics is something that is done rather than read. A mathematics textbook will contain many exercises, and doing them is an essential part of learning mathematics. Wikipedia does not include exercises; by design, Wikipedia is an encyclopedic reference, not a textbook.
When it comes to more advanced topics, mathematics is developed, and largely hangs together, by means of the large body of quite formal proofs that exist in the mathematical literature. Wikipedia does not attempt to condense all of these proofs into encyclopedic form, for reasons that are discussed at length in another essay. Wikipedia assembles the facts uncovered by mathematical investigation, and the definitions underlying the abstract theories. In common with other mathematical encyclopedias, it omits most proofs.
Although learning mathematics involves memorization of the sort of factual knowledge that Wikipedia provides, memorization is not enough to master the field. To become a mathematician, you must acquire the skills of creating proofs and doing calculations for yourself, to internalize the material; therefore, you must go beyond the outline a Wikipedia article can supply. We hope that Wikipedia articles can provide a good starting point for the process, along with a reference for topics you have already learned.
Remember that any source may contain errors, so do not put too much trust into a single account. Verify proofs and calculations yourself. Because anyone can edit Wikipedia, you can correct any errors you find; this can be a very powerful learning experience.
There are some mathematical concepts for which different authors use different definitions. For example, some authors count zero as a natural number while others do not. These differences can affect the way that mathematical theorems are stated. Therefore, double-check the definitions in each article to see whether they match those to which you are accustomed.
The main way to use Wikipedia is to search for an article on a topic that interests you. Follow the wikilinks to articles that explain any terms you don't understand or want to explore further. In addition:
If you are in doubt, ask at the mathematics reference desk. No one on Wikipedia is going to do your math homework for you, but if you ask the right question they might point you to some information that will enable you to do it for yourself.
For those engaged in self-study, some of Wikipedia's sister projects may help. These have different and definite purposes: