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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==

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.

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 [[algorithm]]s 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]].

 

 

[[Calculus]] is the mathematical subject that studies rates of change of functions.  There are two main branches of calculus: [[Derivative (calculus)|differential]] calculus,  and [[integral]] calculus.  There are subfields of these: single [[variable]] calculus, [[exterior calculus]], and multi [[variable]] calculus.

====Chaos Theory====

====Chaos Theory====

The study of discrete structures, such as graphs, Latin squares, and block designs. Discrete mathematics can be studied from a pure, theoretical standpoint, or by studying its applications, such as those to theoretical computer science and combinatorial optimization.

The study of discrete structures, such as graphs, Latin squares, and block designs. Discrete mathematics can be studied from a pure, theoretical standpoint, or by studying its applications, such as those to theoretical computer science and combinatorial optimization.

====[[Combinatorics]]====

====[[Combinatorics]]====

====[[Set theory]]====

====[[Set theory]]====

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

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 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.

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]]====

== See also ==

== See also ==

*[[Additive property of equality]]

*[[Additive property of equality]]

*[[Amédée Mannheim]]

*[[Amédée Mannheim]]