Broadly speaking, pure mathematics is mathematics that studies completely abstract concepts . This paper is a recognizable category of mathematical activity from the nineteenth century onwards,  at variance with the trend towards meeting the needs of navigation , astronomy , physics , economics , engineering , and so on.
Another view Is That pure mathematics is not Necessarily applied mathematics : it is feasible to study abstract entities with respect to Their Nature and intrinsic not Be Concerned with How They manifest in the real world.  Even though the pure and applied views are distinct philosophical positions, in practice there is much overlap in the activity of pure and applied mathematicians.
To develop accurate models for describing the real world, many mathematical tools and techniques that are often considered to be “pure” mathematics. On the other hand, many pure mathematicians draw on natural and social phenomena as inspiration for their abstract research.
Ancient Greek mathematicians were among the earliest to make distinction between pure and applied mathematics. Plato helped to create the gap between “arithmetic”, now called number theory , and “logistic”, now called arithmetic . Plato regarded logistic (arithmetic) as appropriate for businessmen and men of war who “should learn the art of numbers or [they] will not know how to array [their] troops and arithmetic (number theory) as appropriate for philosophers” because [ they have] to arise out of the sea of change and lay hold of true being. ”  Euclid of AlexandriaWhen he asked the study of geometry, “Since he must make gain of what he learns.”  The Greek mathematician Apollonius of Perga was asked about the usefulness of some of his theorems in Book IV of Conics to which he proudly asserted, 
They are worthy of acceptance for the sake of the demonstrations themselves, in the same way as we accept many other things in mathematics for this reason.
And since many of its results are not applicable to the science or engineering of his day, Apollonius further argued in the preface to the fifth book of Conection that the subject is one of those that “… seem worthy of study for their own sake . ” 
The term itself is enshrined in the full title of the Sadleirian Chair , founded (as a professorship) in the mid-nineteenth century. The idea of a separate discipline of pure mathematics may have emerged at that time. The generation of Gauss made no sweeping distinction of the kind, between pure and applied . In the following years, specialization and professionalisation (particularly in the Weierstrass approach to mathematical analysis ).
At the beginning of the twentieth century mathematicians took the axiomatic method , by David Hilbert ‘s example. The logical formulation of pure mathematics suggéré by Bertrand Russell in terms of a quantified structure of proposals Seemed more and more plausible, as wide parts of mathematics est devenu axiomatised THUS and subject to the single criteria of Rigorous proof .
In fact in an axiomatic setting rigorous adds nothing to the idea of proof . Pure mathematics, according to a view that can be attributed to the Bourbaki group , is what is proved. Pure mathematician has become recognized vocation, achievable through training.
The Case Was Made That pure mathematics is Useful in engineering education : 
- There is a training in habits of thought, points of view, and an intellectual understanding of ordinary engineering problems, which only the study of higher mathematics can give.
Generality and abstraction
One central concept in pure mathematics is the idea of generality; pure mathematics often exhibits a trend towards increased generality. Uses and advantages of generality include the following:
- Generalizing theorems or mathematical structures can lead to deeper understanding of the original theorems or structures
- Generality can simplify the presentation of material, resulting in shorter proofs or arguments that are easier to follow.
- One can use generality to avoid duplication of effort, proving a general result instead of having separate cases independently, or using results from other areas of mathematics.
- Generality can facilitate connections between different branches of mathematics. Category theory is one area of mathematics dedicated to exploring this commonality of structure in some areas of math.
Generality’s impact on intuition is both dependent on the subject and a matter of personal preference or learning style. An intuition is often used as a tool for intuition.
As a prime example of generality, the Erlangen program involved an expansion of geometry to accommodate non-Euclidean geometries and the field of topology , and other forms of geometry, by viewing geometry in the study of a space together with a group of transformations. . The study of numbers , called algebra at the beginning of undergraduate level, extends to abstract algebra at a more advanced level; and the study of functions , called Expired calculus at the college freshman level Becomes mathematical analysis and functional analysisat a more advanced level. Each of These branches of more abstract mathematics-have Many sub-specialties, and there are in fact many connections entre pure mathematics and applied mathematics disciplines. A steep rise in abstraction was seen mid 20th century.
In practice, however, these divergences from physics , particularly from 1950 to 1983. Later this was criticized, for example by Vladimir Arnold , too much Hilbert , not enough Poincaré . The point does not yet seem to be stable, in that string theory one way sweaters, while discrete mathematics pulls back towards proof as central.
Mathematicians have always had differing opinions regarding the distinction between pure and applied mathematics. One of the Most Famous (goal Perhaps misunderstood) modern examples of this debate can be found in GH Hardy ‘s A Mathematician’s Apology .
It is widely believed that Hardy considered applied mathematics to be ugly and dull. Although it is true that hardy preferred pure mathematics, which he often compared to painting and poetry , hardy saw the distinction between pure and accurate mathematics to be that that applied mathematics sought to express physical truth in a mathematical framework, then pure mathematics were independent of the physical world. Hardy made a distinction in mathematics between what he called “real” mathematics, “which has permanent aesthetic value”, and “the dull and elementary parts of mathematics” that have practical use.
Hardy regarded Some physicists, Such As Einstein , and Dirac , to be Among the “real” mathematicians, aim at the Time That He Was writing the Apology he aussi regarded general relativity and quantum mechanics to be “useless”, qui allowed _him_ to hold the opinion that only “dull” mathematics was useful. Moreover, Hardy has come to the forefront of the application of matrix theory and theory to physics had come unexpectedly-the time may come as some kind of beautiful, “real” mathematics may be useful as well.
Another insightful view is offered by Magid:
I’ve always thought that a good model could be drawn from the ring theory. In this subject, one has the subareas of commutative ring theory and one reconciling ring theory. An uninformed observer might think that these represent a dichotomy, but in fact the latter subsumes the train: a noncommutative ring is a not necessarily commutative ring. If we use similar conventions, then we could refer to applied mathematics and nonapplied mathematics, where by the latter we mean not necessarily applied mathematics … [emphasis added] 
Analysis is concerned with the properties of functions. It deals with concepts Such As continuity , limits , differentiation and integration , providing good THUS has Rigorous foundation for the calculus of infinitesimals Introduced by Newton and Leibniz in the 17th century. Real analysis studies functions of real numbers, while complex analysis extends the aforementioned concepts to functions of complex numbers. Functional analysis is a branch of analysis that studies infinite-dimensional vector spaces and views functions in these spaces.
Abstract algebra is not confused with the manipulation of formula that is covered in secondary education. It studies sets with binary operations defined on them. Sets and their binary operations may be classified according to their properties: for instance, if an operation is associative on a set that contains an identity element and inverses for each member of the set, and the set is considered to be a group . Other structures include rings , fields , vector spaces and lattices .
Geometry is the study of shapes and space, in particular, groups of transformations that act on spaces. For example, projective geometry is about the group of projective transformations that act on the real projective plane, and the inverse of geometry is concerned with the group of inverse transformations acting on the extended complex plane.
Number theory is the theory of the positive integers . It is based on ideas such as divisibility and congruence . Its fundamental theorem states that each positive integer has a unique prime factorization. In some ways it is the most accessible discipline in pure mathematics for the general public: for instance the Goldbach conjecture is easily stated (but is yet to be proven or disproved). In other ways it is the least accessible discipline; for example, Wiles ‘proof that Fermat’ s equation has no nontrivial solutions requiring understanding of automorphic forms , which, however, is intrinsic to nature in the field of physics or the general public discourse.
Topology is a modern extension of Geometry. Rather than focusing on the dimensions of objects and their measurement, they are not very important, but they are not very important (but not, for example, tearing or shearing). Topology’s subfields interact with other branches of pure mathematics: these topical uses of ideas from analysis , such as metric spaces, and algebraic topology links to ideas from combinatorics in addition to those of analysis.
- Applied mathematics
- Jump up^ Piaggio, H. “Three Sadleirian Professors: AR Forsyth, EW Hobson and GH Hardy” . MacTutor History of Mathematics archive . St. Andrews University . Retrieved 12 July 2015 .
- ^ Jump up to:a b Andy Magid, Letter from the Editor, in Notices of the AMS , November 2005, American Mathematical Society, p.1173. 
- Jump up^ Boyer, Carl B. (1991). “The age of Plato and Aristotle”. A History of Mathematics (Second ed.). John Wiley & Sons, Inc. p. 86. ISBN 0-471-54397-7 .
Plato is important in the history of mathematics largely for his role as an inspirer and director of others, and perhaps to him is the sharp distinction in ancient Greece between arithmetic (in the sense of the theory of numbers) and logistic (the technique of computation ). Plato considered logistic as appropriate for the businessman and for the man of war, who “must learn the art of numbers or he will not know how to array his troops.” The philosopher, on the other hand, must be an arithmetician “because he has got to get out of the sea of change and lay hold of true being.”
- Jump up^ Boyer, Carl B. (1991). “Euclid of Alexandria”. A History of Mathematics(Second ed.). John Wiley & Sons, Inc. p. 101. ISBN 0-471-54397-7 .
Evidently Euclid did not stress the practical aspects of his subject, for he is a tale told of the subject of his study, “Euclid asked his slave to give the student threepence,” since he must make gain of what he learns. “
- ^ Jump up to:a b Boyer, Carl B. (1991). “Apollonius of Perga”. A History of Mathematics (Second ed.). John Wiley & Sons, Inc. p. 152. ISBN 0-471-54397-7 .
It is in connection with the theorems in this book that Apollonius makes a statement implying that in his day, there are narrow-minded opponents of pure mathematics who pejoratively inquired about the usefulness of such results. The author proudly asserted: “They are worthy of acceptance for the sake of the demonstrations themselves, in the same way as we accept many other things in mathematics for this reason.” (Heath 1961, p.lxxiv).
The preface to Book V, on the subject of maximum and minimum straight lines to a conic, again argues that the subject is one of those that seem “worthy of study for their own sake.” While one must admire the author for his lofty intellectual attitude, it may be reliably pointed out that the day is beautiful theory, with no prospect of applicability to the science or engineering of his time, celestial mechanics.
- Jump up^ AS Hathaway(1901)”Pure mathematics for engineering students”,Bulletin of the American Mathematical Society7 (6): 266-71.