
Pure mathematics is one of the two main branches of mathematics, the other being applied mathematics. The latter is about relating mathematical concepts and results to the real world; pure mathematics is about obtaining these results in the first place, in a context which is divorced from the distractions of the real world. Although mathematics is truly a unity, and the two branches rely on one another, the real world often gives the wrong impression about what is really going on in general, and so pure mathematics tries to ignore the assumptions which the real world puts into our minds and which seem intuitively obvious, in case they turn out not to be generally true, (The obvious case in point is that of Euclidean geometry, where the conclusions that were reached that seemed to be based on idealization of the properties of lines and points in the real world were not challenged until the 19th century, and today it is even thought to be only an approximation on a small scale to the truth in the universe.)
Pure mathematics is, by its very nature, an extremely abstract study, with its own language of symbolic logic being more fully developed than most technical vocabularies. It is not, however, only a game, as many have thought; it has again and again found results and invented areas of study which, though apparently divorced from any connection with the world, have had applications found for them. It is a field studied for its own sake, with no thought given to the application of the results, and, indeed, pure mathematicians often look down on their applied counterparts, feeling that they alone do the truly original work. This feeling is not really justified; many of the most fertile branches of pure mathematics would have never come into being without the existence of the applied variety; noone, for example, would have studied nonlinear differential equations without the spur of chaos theory and chaotic behaviour in the physical world. The main branches have grown up around such applications, and from the urge to understand the earliest systems abstracted from Nature. SMcL
See also algebra; analysis; geometry; numbers; set theory.Further reading J.H. Panlos, Innumeracy: Mathematical Illiteracy and its Consequences. 
