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Although mathematics itself is not usually considered a natural science, the specific structures that are investigated by mathematicians often have their origin in the natural sciences, most commonly in physics. However, mathematicians also define and investigate structures for reasons purely internal to mathematics, because the structures may provide, for instance, a unifying generalization for several subfields, or a helpful tool for common calculations. Finally, many mathematicians study the areas they do for purely aesthetic reasons, viewing mathematics as an art form rather than as a practical or applied science. Einstein referred to the subject as the Queen of the Sciences in his book Ideas and Opinions. Mathematics is considered absolute, without any reference.
Mathematics is often abbreviated as math ( American English) or maths ( British EnglishBritish English (or UK English (en-GB according to RFC 3066) is a collective term for the forms of English spoken in the British Isles. In particular, when used by other English speakers, it often refers to the written Standard English and the pronunciati).
See the article on the history of mathematicsSee Timeline of mathematics for a timeline of events in mathematics. See mathematician for a list of biographies of mathematicians. Also see The Nine Chapters on the Mathematical Art for information about the development of mathematics in China. The word for details.
The word "mathematics" comes from the GreekThe Greek language ( /Elini'k{/) is an Indo-European language which has existed from around the 14th century BC in the Cretan inscriptions called Linear B. Mycenaean Greek of this period is distinguished from later Classical or Ancient Greek of the 8th ce μάθημα (mαthema) which means "science, knowledge, or learning"; μαθηματικός (mathematikσs) means "fond of learning".
The major disciplines within mathematics arose out of the need to do calculations in commerce, to measure land and to predict astronomical events. These three needs can be roughly related to the broad subdivision of mathematics into the study of structure, space and change.
The study of structure starts with numberA number is an abstract entity used to describe quantity. There are different types of numbers. The most familiar numbers are the whole numbers {0, 1, 2,. denoted by W and the natural numbers {1, 2, 3,. used for counting and denoted by N . If the negatives, first the familiar natural numberNatural number can mean either a positive integer ( 1, 2, 3, 4,. or a non-negative integer ( 0, 1, 2, 3, 4,. Natural numbers have two main purposes: they can be used for counting ("there are 3 apples on the table"), or they can be used for ordering ("thiss and integerThe integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3,. and the number zero. The set of all integers is usually denoted in mathematics by Z (or Z in blackboard bold, ), which sts and their arithmeticArithmetic Arithmetic or arithmetics in common usage is a branch of (or the forerunner of) mathematics which records elementary properties of certain operations on numerals, though in usage by professional mathematicians, it often is treated as synonym foal operations, which are recorded in elementary algebraAlgebra Elementary algebra is the most basic form of algebra taught to students who are presumed to have no knowledge of mathematics beyond the basic principles of arithmetic. While in arithmetic only numbers and their arithmetical operations (such as +,. The deeper properties of whole numbers are studied in number theory. The investigation of methods to solve equations leads to the field of abstract algebra, which, among other things, studies rings and fields, structures that generalize the properties possessed by the familiar numbers. The physically important concept of vectors, generalized to vector spaces and studied in linear algebra, belongs to the two branches of structure and space.
The study of space originates with geometry, first the Euclidean geometry and trigonometry of familiar three-dimensional space (also applying to both more and less dimensions), later also generalized to non-Euclidean geometries which play a central role in general relativity. Several long standing questions about ruler and compass constructions were finally settled by Galois theory. The modern fields of differential geometry and algebraic geometry generalize geometry in different directions: differential geometry emphasizes the concepts of functions, fiber bundles, derivatives, smoothness, and direction, while in algebraic geometry geometrical objects are described as solution sets of polynomial equations. Group theory investigates the concept of symmetry abstractly and provides a link between the studies of space and structure. Topology connects the study of space and the study of change by focusing on the concept of continuity.
Understanding and describing change in measurable quantities is the common theme of the natural sciences, and calculus was developed as a most useful tool for that. The central concept used to describe a changing variable is that of a function. Many problems lead quite naturally to relations between a quantity and its rate of change, and the methods to solve these are studied in the field of differential equations. The numbers used to represent continuous quantities are the real numbers, and the detailed study of their properties and the properties of real-valued functions is known as real analysis. For several reasons, it is convenient to generalise to the complex numbers which are studied in complex analysis. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions, laying the groundwork for quantum mechanics among many other things. Many phenomena in nature can be described by dynamical systems and chaos theory deals with the fact that many of these systems exhibit unpredictable yet deterministic behavior.
In order to clarify and investigate the foundations of mathematics, the fields of set theory, mathematical logic, and model theory were developed.
When computers were first conceived, several essential theoretical concepts were shaped by mathematicians, leading to the fields of computability theory, computational complexity theory, information theory, and algorithmic information theory. Many of these questions are now investigated in theoretical computer science.
Discrete mathematics is the common name for those fields of mathematics useful in computer science.An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis and prediction of phenomena and is used in all sciences. Numerical analysis investigates the methods of efficiently solving various mathematical problems numerically on computers and takes rounding errors into account.