Quote by Henri Poincare: “The mathematician does not study pure mathemati...”
One of the joys of being married to a pure mathematician—other than finding coffee-stained notebooks full of integrals lying around the flat—is hearing her try to explain her job to other people. No, sometimes, and no. She rarely uses a computer, traffics more with inequalities than equations, and—like most researchers in her subfield—considers any number larger than 5 to be monstrously big. Pure math research is a weird job, and hard to explain. A: Picture mathematics as a big yin-yang symbol. But instead of light vs.
Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, but pure mathematicians are not primarily motivated by such applications. Instead, the appeal is attributed to the intellectual challenge and aesthetic beauty of working out the logical consequences of basic principles. While pure mathematics has existed as an activity since at least Ancient Greece , the concept was elaborated upon around the year ,  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 , and Russell's paradox. This introduced the need of renewing the concept of mathematical rigor and rewriting all mathematics accordingly, with a systematic use of axiomatic methods.
It seems that nowadays pure mathematics takes a back seat to applied mathematics. Also many mathematicians declare their research applied, even when it is purely mathematical. This note intends to argue with the opinion that pure mathematics is too theoretical and useless. While serious mathematicians understand the importance of pure mathematics and will probably learn nothing new from reading this article, there are still many people who need to be convinced in the importance and usefulness of pure mathematics. Applied mathematics develops mathematical methods that are used in science, business and industry, and applies existing mathematical methods in these fields.