> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/900-index/980-hierarchy/classification/areas-of-math.md). # Areas of mathematics Mathematics encompasses a growing variety and depth of subjects over history, and comprehension requires a system to categorize and organize the many *subjects* into *more general areas* of mathematics. A number of different classification schemes have arisen, and though they share some similarities, there are differences due in part to the different purposes they serve. In addition, as mathematics continues to be developed, these classification schemes must change as well to account for newly created areas or newly discovered links between different areas. Classification is made more difficult by some subjects, often the most active, which straddle the boundary between different areas. A traditional division of mathematics is into **pure mathematics**, mathematics studied for its intrinsic interest, and **applied mathematics**, mathematics which can be directly applied to real world problems. This division is not always clear and many subjects have been developed as pure mathematics to find unexpected applications later on. Broad divisions, such as **discrete mathematics** and **computational mathematics**, have emerged more recently. An ideal system of classification permits adding new areas into the organization of previous knowledge, and fitting surprising discoveries and unexpected interactions into the outline. For example, the *Langlands program* has found unexpected connections between areas previously thought unconnected, at least Galois groups, Riemann surfaces and number theory. --- # Agent Instructions This documentation is published with GitBook. GitBook is the documentation platform designed so that both humans and AI agents can read, navigate, and reason over technical content effectively. Learn more at gitbook.com. ## Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. Perform an HTTP GET request on the current page URL with the `ask` query parameter: ``` GET https://mandober.gitbook.io/math-debrief/900-index/980-hierarchy/classification/areas-of-math.md?ask= ``` The question should be specific, self-contained, and written in natural language. The response will contain a direct answer to the question and relevant excerpts and sources from the documentation. Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.