Pure Mathematics or Pure Mathematics It is the study of mathematical concepts independently of any application outside of mathematics. These concepts may arise in real-world interests, and the obtained results may subsequently turn out to be useful for practical applications, but pure mathematicians are not primarily motivated by these applications. Rather, the appeal is ascribed to the intellectual challenge and beauty of defining the logical consequences of basic principles. Although pure mathematics has existed as an activity since at least ancient Greece, the concept was developed around 1900, After introducing theorems with counterintuitive properties (such as non-Euclidean geometry and Cantor's theorem of infinite sets), and discovering apparent contradictions (such as continuous functions that cannot be distinguished anywhere, Russell's paradox). This introduced the need to renew the concept of mathematical rigor and to rewrite all mathematics accordingly, with the systematic use of axiomatic methods. This led many mathematicians to focus on mathematics for its own good, pure mathematics.