Mathematics involves more and more subject variations and depths of history, and understanding requires systems to categorize and organize many subjects into the more general field of mathematics . A number of different classification schemes have emerged, and although they share some similarities, there are differences because of some of the different purposes they serve. In addition, as mathematics continues to be developed, this classification scheme must change as well to account for newly created areas or newly discovered links between different areas. Classification is made more difficult by some subjects, often the most active, who straddle the boundary between different areas.
The traditional division of mathematics is into pure mathematics, mathematics studied for its intrinsic importance, and applied mathematics, mathematics that can be directly applied to real-world problems. This division is not always clear and many subjects have been developed as pure mathematics to find unexpected applications later on. Extensive divisions, such as discrete mathematics and computational mathematics, have emerged recently.
The ideal classification system allows the addition of new areas into the organization of prior knowledge, and adjusts surprising findings and unexpected interactions into the outline. For example, the Langlands program has uncovered unexpected connections between previously unrelated regions, at least the Galois groups, Riemann surfaces and number theory.
Arithmetic is the study of the number and nature of operations among them.
Algebra
The study of structures begins with numbers, first numbers and natural integers and their arithmetic operations, recorded in basic algebra. The deeper properties of these numbers are studied in number theory. The investigation of methods for solving equations leads to the field of abstract algebra, which, inter alia, studies rings and planes, structures that generalize the properties possessed by everyday numbers. The long questions about compass and ruler construction were finally solved by Galois theory. Physically important vector concepts, generalizable to vector spaces, are studied in linear algebra.
- Order theory
- For two different real numbers, one must be larger than the other. Order Theory extends this idea to establish in general. This includes ideas such as lattices and ordered algebra structures. See also glossary of order theory and list of order topics.
- Common algebra system
- Given a set, various ways to join or associate members from the set can be determined. If this obeys certain rules, certain algebraic structures are formed. Universal algebra is a more formal study of these structures and systems.
- Number theory
- Number theory is traditionally related to the properties of integers. More recently, it has become concerned with the broader class of problems that naturally arise from the study of integers. This can be divided into basic number theory (where integers are studied without the help of techniques from other fields of mathematics); analytic number theory (in which calculus and complex analysis are used as a tool); the theory of algebraic numbers (which studies the algebraic numbers - polynomial roots with integer coefficients); geometric number theory; combinatorial number theory; the theory of transcendental numbers; and computational number theory. See also list of number theory topics.
- Field theory and polynomial
- Field theory studies the properties of the field. Field is a mathematical entity that addition, subtraction, multiplication and division are well defined. Polynomials are expressions in which constants and variables are combined using only sum, subtraction, and multiplication.
- Commutative and algebraic rings
- In ring theory, abstract algebra branch, commutative ring is a ring in which multiplication operation obeys commutative law. This means that if a and b are any elements of the ring, then aÃÆ' â € "b = bÃÆ' â €" a. Commutative algebra is the field of study of commutative rings and their ideals, modules and algebras. This is the basis both for algebraic geometry and for the theory of algebraic numbers. The most prominent example of a commutative ring is a polynomial ring.
Analysis
In the world of mathematics, analysis is a branch that focuses on change: the rate of change, the accumulation of change, and many things that change relative to (or not dependent) on each other.
Modern analysis is a rapidly expanding branch of mathematics that touches virtually every other division of discipline, finding direct and indirect applications in diverse topics like number theory, cryptography, and abstract algebra. It is also the language of science itself and is used in the fields of chemistry, biology, and physics, from astrophysics to X-ray crystallography.
Combinatorics
Combinatorics is the study of a collection of limited or separate objects that meet specified criteria. In particular, it relates to the "enumerative combinatorics" of the object and by deciding whether a certain "optimal" object exists (extreme combinatorics). These include the graph theory, which is used to describe interconnected objects (graph in this sense is a network, or a collection of connected dots). See also a list of combinatorical topics, chart theory topics and the glossary of graph theory. Combinatorial flavors are present in many troubleshooting sections.
Geometry and topology
Source of the article : Wikipedia
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