Mathematics is both an art and a science- and pure mathematics lies at its heart. Pure mathematics explores the boundary of mathematics and pure reason. It has been described as “that part of mathematical activity that is done without explicit or immediate consideration of the direct application,” although what is “pure” in one era often becomes applied later. Finance and cryptography are current examples of areas to which pure mathematics is applied in significant ways. Pure mathematics is driven by abstract problems and used to design a mathematical model of real-world problems. Much of what’s pursued by pure mathematicians can have their roots in concrete physical problems, but a deeper understanding of these phenomena brings about problems and technicalities. These abstract problems and technicalities are what pure mathematics attempts to solve and these attempts have led to major discoveries for mankind, including the Universal Turing Machine. Computational mathematics involves mathematical research in areas of science, where computing plays a central and essential role, emphasizing algorithms, numerical methods and symbolic methods.