> 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/systems-of-classification.md).

# Math classification

* [Pure and Applied Mathematics](#pure-and-applied-mathematics)
* [Discrete and Continuous Mathematics](#discrete-and-continuous-mathematics)
* [Computational Mathematics](#computational-mathematics)

## Pure and Applied Mathematics

Considering the *applicability*, mathematics is classified into two traditional categories:

1. **Pure mathematics** is math studied for its intrinsic interest.
2. **Applied mathematics** which is math directly applicable in the real world.

This division is not always clear and many subjects have been developed as pure mathematics to find unexpected applications later on. For example, the study of prime numbers in number theory has found an application in cryptography.

## Discrete and Continuous Mathematics

Considering the *granularity of mathematical objects*, math is broadly classified into two categories:

1. Continuous Mathematics (CM)
2. Discrete Mathematics (DM)

**Continuous mathematics** is based on the continuum - the continuous number line of real numbers. It is characterized by the fact that between any two real numbers there is an infinite set of real numbers. In continuous mathematics a function may be plotted as a smooth curve. Continuous mathematics deals with analog, continuous, quantities.

**Discrete Mathematics** studies discrete, separate and enumerable mathematical objects and structures. Since computers operate on discrete bits of data, discrete mathematics is intertwined with *Computer Science*.

## Computational Mathematics

**Computational mathematics** have emerged more recently with advancemant of computers. Solving a problem computationally requires modeling the world, devising an algorithm, and determining its efficiency and correctness.

Discrete mathematics overlaps with *Computer Science* in its study of:

* algorithms
* algorithmic complexity
* proofs
* formal language
* automata
* computability
* correctness of programming languages
* automated and assisted theorem proving