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'''Mathematics''' is the rigorous analysis of abstract structures, including numeric and logical systems.  The earliest known beginning of this topic is about 2400 B.C., the date of the oldest extant mathematical tablets.<ref>Davis & Hersh, ''The Mathematical Experience'' xi (Mariner Books 1981)</ref>

'''Mathematics''' is the rigorous analysis of abstract structures, including numeric and logical systems.  The earliest known beginning of this topic is about 2400 B.C., the date of the oldest extant mathematical tablets.<ref>Davis & Hersh, ''The Mathematical Experience'' xi (Mariner Books 1981)</ref>

Mathematics 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.

Mathematics 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.

==Symbols, Equations, and Theories==

==Symbols, Equations, and Theories==

Mathematics is expressed with [[symbol]]s. Some of the most commonly used are the numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. These are symbols used to express our intuitive notion of [[quantity]]. Other symbol used in elementary mathematics are the equality ( = ), addition ( + ), subtraction ( - ), multiplication ( x ), less than ( < ), greater than ( > ), etc. More advanced branches of mathematics have their own symbols.

Mathematics is expressed with [[symbol]]s. Some of the most commonly used are the numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. These are symbols used to express our intuitive notion of [[quantity]]. Other symbol used in elementary mathematics are the equality ( = ), addition ( + ), subtraction ( - ), multiplication ( x ), less than ( < ), greater than ( > ), etc. More advanced branches of mathematics have their own symbols.

An [[equation]] is a mathematical statement that asserts the equality of two expressions. Some equations, like (x + 2 = 5), express the equality of two quantities. Other equations, called [[differential equation]]s, express the equality of two functions.

An [[equation]] is a mathematical statement that asserts the equality of two expressions. Some equations, like (x + 2 = 5), express the equality of two quantities. Other equations, called [[differential equation]]s, express the equality of two [[function]]s.

 

A mathematical theory is expressed as a set of sentences, called [[axiom]]s. These axioms should be self consistent, that is, they must not contradict with each other. From these axioms, new results can be derived adhering strictly to mathematical logic. These derived results are called [[theorem]s. It is important to note that, according to the [[Godel's Incompleteness Theorems]], it is impossible to state a self consistent set of axioms from which the whole mathematics can be derived.

==Pure and Applied Mathematics==

==Pure and Applied Mathematics==

====[[Algebra]]====

====[[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 [[matrix|matrices]], [[function]]s, [[polynomial]]s, [[vector]]s or many others. Concentrating on addition and multiplication does not exclude subtraction or division, since subtraction is formally considered to be 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.  Abstract algebra is the study of [[algebraic structures]] such as [[Group (mathematics)|group]]s, [[Ring (mathematics)|rings]], and [[Field (mathematics)|fields]].

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 [[matrix|matrices]], [[function]]s, [[polynomial]]s, [[vector]]s or many others. Concentrating on addition and multiplication does not exclude subtraction or division, since subtraction is formally considered to be 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.  Abstract algebra is the study of algebraic structures such as [[Group (mathematics)|group]]s, [[Ring (mathematics)|rings]], and [[Field (mathematics)|fields]].

====[[Analysis]]====

====[[Analysis]]====

Analysis is concerned with limits and other infinite processes. This subject includes the theory of limits of sequences and series and all forms of [[statistics]], and [[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.  Additional topics in analysis include [[real analysis]] and [[complex analysis]].

Analysis is concerned with limits and other infinite processes. This subject includes the theory of limits of sequences and series and all forms of [[statistics]], and [[calculus]], including the calculus of several variables, vector calculus and tensor calculus. Also included is numerical analysis, the study of error propagation in [[algorithm]]s carried out to finite precision.  Additional topics in analysis include [[real analysis]] and [[complex analysis]].

 

 

 

Chaos theory is the study of systems that are highly sensitive to initial conditions. That is, small differences in the initial conditions can produce large variations in the long time behaviour. This is known popularly as the Butterfly effect.

====[[Chaos Theory]]====

====[[Discrete Mathematics]]====

Is the study of systems that are highly sensitive to its initial conditions. That is, small differences in the initial conditions can produce large variations in the long time behaviour. This is known popularly as the Butterfly effect.

 

 

Combinatorics is the study of situations in which elements of a set or sets are combined or permuted in various ways. An example of a combinatorics problem would be "in a group of six men and four women, how many possible ways are there to choose three men and two women?" Derangements are another concept in combinatorics. A derangement is a re-ordering of a set so that no element ends up where it was originally; a derangement problem typically asks how many arrangements are possible for a given set, meeting given conditions.

====[[Game Theory]]====

====[[Game Theory]]====

Is the mathematical study of strategic situations, in which the success of an individual making choices depend on the choices of others.

Game theory is the mathematical study of strategic situations, in which the success of an individual making choices depend on the choices of others.

====[[Geometry]]====  

====[[Geometry]]====  

Geometry was defined by [[Felix Klein]] as the study of [[invariant]]s under [[Group (mathematics)|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 (that is, [[arc]]s between [[node]]s in [[network]]s) of objects that would be preserved even if a diagram of them were to be stretched or shrunk.  

Geometry is the study of shapes and special relationships. It was defined by Felix Klein as the study of [[invariant]]s under [[Group (mathematics)|group]]s of [[transformation]]s. For example, the Euclidean transformations 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 transformations, 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 (that is, [[arc]]s between [[node]]s in [[network]]s) of objects that would be preserved even if a diagram of them were to be stretched or shrunk.

 

Logic is the study of [[reasoning]]. It examines general forms that [[argument]]s can take, and determines which forms are valid, and which forms are fallacies. In mathematics, it is the study of inferences within one formal language.

====[[Logic]] and [[set theory]]====

====[[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 of the axioms, the [[Axiom of Choice]], has been the subject of much discussion

Set theory is the mathematical study of collections of objects. It is one of the most fundamental areas of mathematics, since 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 of the axioms, the [[Axiom of Choice]], has been the subject of much discussion.

====[[Probability and Statistics]]====

====[[Probability and Statistics]]====

====[[Topology]]====

====[[Topology]]====

Topology began with [[Leonard Euler]]'s consideration of the [[Königsberg Bridges Problem]], which also introduced [[Graph Theory]]. [[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.  [[Differential geometry]] is a specialized field of its own.

Topology is the study of special properties that are preserved under continuous deformation of objects. Put more simply, it studies the properties that don't change unless you poke a hole in the object. One of the most popular examples is that of a coffee cup that can be continuously deformed into a doughnut. Then, the cup and the doughnut are said to be “topologically equivalent”. Topology began with [[Leonard Euler]]'s consideration of the Königsberg Bridges Problem, which also introduced Graph Theory. 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.  [[Differential geometry]] is a specialized field of its own.

====[[Trigonometry]]====

====[[Trigonometry]]====

In its more basic sense, trigonometry is the study of the relationships between the sides and the angles of [[triangle]]s. However, trigonometric functions are widely used outside their original realm of describing triangles. For example, senoidal functions are used to describe oscillatory motion and waves.

In its more basic sense, trigonometry is the study of the relationships between the sides and the angles of [[triangle]]s. However, trigonometric functions are widely used outside their original realm of describing triangles. For example, sinusoidal functions (sine and cosine) are used to describe oscillatory motion and waves. In fact, there are extremely deep and fruitful connections among those functions, the exponential function, complex numbers, [[Fourier transform|Fourier]] and Laplace transforms, and signal processing.

 

 

 

== References ==

== References ==

[[Category:Mathematics]]

[[Category:Mathematics]]