Difference between revisions of "Mathematics"

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(This page as formerly written was wrong on several fronts, and didn't really do justice to the mathematical science. Hence, a rewrite and expansion.)
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'''Mathematics''' is the [[science]] of [[number]]s, and the study of relationships that exist between them. It generally also is considered to contain [[geometry]] and [[topology]] as a subset, as they are related to <math>\mathbb{R}_2</math> and <math>\mathbb{R}_n</math> in general.
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'''Mathematics''' is the [[science]] of structure and change. Mathematics is important to the other sciences because it provides rigourous methods for developing models of complex phenomena. Such phenomena include the spread of computer viruses on a network, the growth of tumours, the risk associated with certain contracts traded on the stock market, and the formation of turbulence around an aircraft. Mathematics provides a kind of "quality control" for the development of trustworthy theories and equations.
  
 
==Overview=={{asy image|<math>1\,2\,3\,4\,5\,6\,7\,8\,9\,0</math>|right|The ten [[digit]]s making up <br /> the base ten number system.}}
 
==Overview=={{asy image|<math>1\,2\,3\,4\,5\,6\,7\,8\,9\,0</math>|right|The ten [[digit]]s making up <br /> the base ten number system.}}
Modern mathematics is normally built around [[base numbers|base 10]], with ten digits. (<math>0,1,2,3,4,5,6,7,8,9</math>) Modern mathematics is separated into two categories: discrete mathematics and non-discrete.
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Modern mathematics is built around a system of axioms, which is a name given to "the rules of the game." Mathematicians then use various methods of formal proof to extend the axioms to come up with surprising and elegant results. Such methods include proof by induction, and proof by contradiction, for example.
===Non-Discrete Mathematics===
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Non-discrete mathematics is study of mathematics that is generally applicable to the "real world", such as [[algebra]], [[geometry|Euclidean geometry]], [[statistics]], and other such topics. (Note that the real world is actually only approximately Euclidean if one studies large areas of it, infinitesimal areas actually are non-Euclidean) There is some controversy over what varieties of algebra are non-discrete, but it is generally agreed that elementary and superior algebra are non-discrete, while [[abstract algebra]] and intermediate topics such as [[field theory|field]] and [[graph theory]] and [[Diophantine]] equations are discrete.
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==Mathematical Subject Classification==
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There are numerous categories and subcategories of mathematics, as shown by the [http://www.ams.org American Mathematical Society's Mathematics] [http://www.ams.org/msc/ Subject Classification scheme].
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A common way of classifying mathematics is into Pure Maths, and Applied Maths. Pure Maths is maths which is studied in order to make mathematics more stable and powerful, and a knowledge of Pure Maths is required to understand the foundations of Applied Maths. Pure Maths is often considered to be divided into the areas of Higher Algebra, Analysis, and Topology.
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Applied Maths consists of taking the techniques from Pure Maths and using them to develop models of "the real world." Applied Maths is sometimes considered to be divided into the areas of Dynamical Systems, Approximation Techniques, and Probability & Statistics. There are also various Applied Mathematical disciplines which use a combination of these areas, but focus on a particular type of application. Examples include Mathematical Physics and Mathematical Biology.
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===Discrete Mathematics===
 
[[Combinatorics]], [[number theory]], and some of the algebraic fields mentioned above are examples of discrete mathematics. Topics of discrete mathematics are generally not directly applicable to the "real world", and if they are, it is only in an abstract fashion.
 
 
==History of Mathematics==
 
==History of Mathematics==
Mathematics was noted by the earliest humans. Over time, as humans evolved, the complexity of mathematics also evolved. There was an astounding discovery on how the numbers correlated with each other, as well as in nature, so well, as they created the concept of numbers.  
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Mathematics was noted by the earliest humans. Over time, as humans evolved, the complexity of mathematics also evolved. There was an astounding discovery on how the numbers correlated with each other, as well as in nature, so well, as they created the concept of numbers. Many cultures throughout the world contributed to the development of mathematics in historical times, from China and India to the Middle East and Greece.
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Modern Mathematics began in Europe during the Rennaissance, after various Arabic texts were translated into European languages during the 12th and 13th centuries. Islamic cultures in the Middle East had preserved various ancient Greek and Hindu texts, and had furthermore extended these old results into new areas. The popularity of the printing press combined with the increasing need for navigational accuracy as various European powers began colonising areas in Africa and Asia provided the final incentive for a huge mathematical boom, which has continued to this day.
  
 
<blockquote>"God created the integers. All the rest is the work of man."</blockquote>  
 
<blockquote>"God created the integers. All the rest is the work of man."</blockquote>  

Revision as of 22:48, 8 June 2008

Mathematics is the science of structure and change. Mathematics is important to the other sciences because it provides rigourous methods for developing models of complex phenomena. Such phenomena include the spread of computer viruses on a network, the growth of tumours, the risk associated with certain contracts traded on the stock market, and the formation of turbulence around an aircraft. Mathematics provides a kind of "quality control" for the development of trustworthy theories and equations.

Overview

$1\,2\,3\,4\,5\,6\,7\,8\,9\,0$

Enlarge.png
The ten digits making up
the base ten number system.

Modern mathematics is built around a system of axioms, which is a name given to "the rules of the game." Mathematicians then use various methods of formal proof to extend the axioms to come up with surprising and elegant results. Such methods include proof by induction, and proof by contradiction, for example.


Mathematical Subject Classification

There are numerous categories and subcategories of mathematics, as shown by the American Mathematical Society's Mathematics Subject Classification scheme.

A common way of classifying mathematics is into Pure Maths, and Applied Maths. Pure Maths is maths which is studied in order to make mathematics more stable and powerful, and a knowledge of Pure Maths is required to understand the foundations of Applied Maths. Pure Maths is often considered to be divided into the areas of Higher Algebra, Analysis, and Topology.

Applied Maths consists of taking the techniques from Pure Maths and using them to develop models of "the real world." Applied Maths is sometimes considered to be divided into the areas of Dynamical Systems, Approximation Techniques, and Probability & Statistics. There are also various Applied Mathematical disciplines which use a combination of these areas, but focus on a particular type of application. Examples include Mathematical Physics and Mathematical Biology.


History of Mathematics

Mathematics was noted by the earliest humans. Over time, as humans evolved, the complexity of mathematics also evolved. There was an astounding discovery on how the numbers correlated with each other, as well as in nature, so well, as they created the concept of numbers. Many cultures throughout the world contributed to the development of mathematics in historical times, from China and India to the Middle East and Greece.

Modern Mathematics began in Europe during the Rennaissance, after various Arabic texts were translated into European languages during the 12th and 13th centuries. Islamic cultures in the Middle East had preserved various ancient Greek and Hindu texts, and had furthermore extended these old results into new areas. The popularity of the printing press combined with the increasing need for navigational accuracy as various European powers began colonising areas in Africa and Asia provided the final incentive for a huge mathematical boom, which has continued to this day.

"God created the integers. All the rest is the work of man."

-Leopold Kronecker


See Also

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