Pure mathematics is the study of (theoretical) concepts of mathematics independently of any other applications. These concepts may originate in real-world concerns, and hence the results obtained became useful for practical applications. However, the pure mathematicians are not motivated by such applications.
The concept of pure mathematics was elaborated upon around the year 1900, after the introduction of theories with counter-intuitive properties (such as non-Euclidean geometries and Cantor's theory of infinite sets), and the discovery of apparent paradoxes (such as continuous functions that are nowhere differentiable). This directed mathematicians for the need of rewriting all mathematics accordingly, with a systematic use of axiomatic methods which led many mathematicians to focus on mathematics for its own sake, that is, pure mathematics.
We formulate four sub-areas for pure mathematics consisting of :
Combinatorics, Graph Theory, and Cryptography.
Analysis and Approximation Theory.
Geometry and Topology.