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Mathematics is the [[science]] of [[numbers]], it involves such areas as counting quantity, studying relationships between numbers and relationships in space, and study of structure, rates of change, and time. It includes the fields of arithmetic (simple mathematical calculations such as addition, subtraction, multiplication, division, and extracting square roots) [[Algebra]] (relationships between numbers), [[Geometry]] (relationships in space), [[Trigonometry]] (the study of triangles), [[Calculus]] (rates of change), [[Number Theory]] (integers), and [[Category Theory]] (categories). Most mathematical concepts are designed for practical applications such as financial accounting, geometric construction of building, astronomical calculations, calendar dating, telling time, engineering, physics, chemistry, etc. But some mathematicians have developed mathematical concepts without practical applications in mind. The former is called applied mathematics, the latter is called pure mathematics.

Mathematics is the study and practice of rigorous inference about abstract structures. It includes many practical results concerning quantity and measure, such as calculations involving numbers, financial accounting, geometric construction of building, astronomical calculations, calendar dating, telling time, engineering, physics, chemistry, etc. but also more abstract issues such as establishing the conditions under which certain kinds of equations and formulas have solutions. [[Applied mathematics]] concerns the use of mathematical methods for practical purposes. [[Pure mathematics]] involves reasoning about abstract structures. The main areas are

 

* Algebra. Broadly, speaking algebra concerns 'addition' and 'multiplication', but in the widest possible sense. The objects that are being added or multiplied can be numbers, as in [[Number theory]], but they can also can be more general structures such as [[matrices]], [[functions]], [[polynomials]], or [[vectors]]. Concentrating on addition an multiplication does not exclude subtraction or division, since subtraction is formally considered to be the addition of an [[additive inverse]] and division is considered to be multiplication by a [[multiplicative inverse]]. That is, subtracting 3 from 2 is rigorously defined to adding the number -2 to 3. Minus two is called the additive inverse of +2. Similarly, dividing 3 by 2 is formally defined in terms of multiplying 3 by (1/2), where 1/2 is the multiplicative inverse of 2.

 

* Analysis. Analysis is concerned with limits. This subject includes the theory of limits of sequences and series and all forms of [[calculus]], including the calculus of several variables, [[vector calculus]] and [[tensor calculus]]. Also included is [[numerical analysis]], the study of error propagation in algorithms carried out to finite precision.

 

* Geometry and Topology. Geometry was defined by [[Felix Klein]] as the study of [[invariants]] under [[group]]s of [[transformation]]s. For example, the [[Euclidean transformation]]s are [[translation]], [[rotation]] and [[reflection]]. The quantities that are not altered by these transformations are things like angles and distances, so these are the subjects of interest in [Euclidean geometry]]. Other types of transformations, such as the [[affine transformation]]s, define other types of geometry. Topology is concerned with the connectedness of objects, rather than the distance between them. It is sometimes called 'rubber sheet geometry', as it concerns properties of objects that would be preserved even if a diagram of them were to be stretched or shrunk. Topology began with [[Leonard Euler]]'s consideration of the [[Königsberg Bridges Problem]]. [[Beck's map of the London Underground]] in 1933 used a topological distortion of the locations of the subway stations in order to produce a more useful and artistic map.

 

* Logic and [[set theory]]. All of mathematics can be expressed in terms of [[set]]s. Sets are defined by a collection of [[axiom]]s called the [[Zermelo-Fraenkel axioms]]. One ofthe axioms, the [[Axiom of Choice]] has been the subject of much discussion.

 

[[Category:Mathematics| ]]

[[Category:Mathematics| ]]