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Mathematics

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Mathematics is a formal field of study focused on discovering and organizing methods, theories, and theorems. These mathematical truths are developed either to serve the needs of empirical sciences or to advance the discipline itself. Unlike science, which relies on experimentation, the fundamental truths of mathematics are independent of any physical test. Mathematicians use pure reason to establish these truths through a process called a proof. A proof is a sequence of deductive steps applied to already established results. These results, known as theorems, are built upon axioms, which are basic statements assumed to be true.

Arithmetic symbols
Arithmetic symbols

The mechanics of mathematics involve the description and manipulation of abstract objects. These objects can be abstractions taken from nature or purely abstract entities defined by specific properties. To build a theory, a mathematician starts with axioms or definitions. They then apply deductive rules to these starting points to reach new conclusions. This logical chain ensures that if the axioms are true, the resulting theorems must also be true. This rigorous structure allows mathematics to function as a reliable foundation for many other fields.

Historically, mathematics was divided into two primary branches: arithmetic and geometry. Arithmetic focused on the study and manipulation of numbers, while geometry studied shapes and spaces. This division lasted until the 16th and 17th centuries. During this time, algebra and calculus emerged as distinct, powerful fields. The development of new mathematical notation was essential for the rise of modern algebra. The interaction between these mathematical innovations and scientific discoveries caused both fields to grow together.

Bakhshali numerals 2
Bakhshali numerals 2

Number theory is a major area that evolved from the study of natural numbers. It eventually expanded to include integers and rational numbers. While once called arithmetic, modern number theory is a highly abstract discipline. Early progress was made by figures like Euclid and Diophantus of Alexandria. Later, Pierre de Fermat and Leonhard Euler advanced the field into its modern form. It reached full fruition through the work of Adrien-Marie Legendre and Carl Friedrich Gauss.

Carl Friedrich Gauss 1840 by Jensen
Carl Friedrich Gauss 1840 by Jensen

Algebra is often described as the art of manipulating equations and formulas. Its precursors include Diophantus and the 9th-century mathematician al-Khwarizmi. The term "algebra" comes from the Arabic word "al-jabr," meaning the reunion of broken parts. Al-Khwarizmi introduced systematic methods for transforming equations. Later, François Viète introduced variables to represent unknown or unspecified numbers. This allowed mathematicians to describe operations using general formulas rather than specific numbers.

Quadratic formula
Quadratic formula

Geometry has its roots in practical needs like surveying and architecture. The ancient Greeks transformed it by introducing the concept of formal proof. Around 300 BC, Euclid systematized these ideas in his influential book, Elements. This created Euclidean geometry, which studies shapes in planes and three-dimensional spaces. In the 17th century, René Descartes introduced Cartesian coordinates. This allowed for analytic geometry, where points are represented by numbers. This breakthrough enabled algebra to be used to solve complex geometric problems.

Woman teaching geometry
Woman teaching geometry

At the end of the 19th century, mathematics faced a foundational crisis. This period led to the systematic use of the axiomatic method. This change caused an explosion in the number of mathematical subfields. The 2020 Mathematics Subject Classification now lists sixty-three first-level areas. These include modern fields like mathematical logic and foundations. Some areas, such as game theory, are developed alongside their practical applications. Others are developed independently but find uses in the world much later.

FieldsMedalFront
FieldsMedalFront

Today, mathematics is essential to almost every modern profession. It is a vital tool in engineering, medicine, finance, and computer science. It also supports the social sciences through various modeling techniques. Even complex puzzles like Fermat's Last Theorem show the depth of the field. This conjecture was stated in 1637 but required tools from algebraic geometry and category theory to prove in 1994. Whether through simple counting or complex manifold theory, mathematics provides the language for understanding structure.

Julia set (highres 01)
Julia set (highres 01)
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Lock-green.svg
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Rubik's cube.svg
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Quadratic formula.svg
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Sigma summation notation.svg
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Venn A intersect B.svg
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Cauchy sequence illustration.svg
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Supply-demand-equilibrium.svg
File:Arithmetic symbols.svg
Arithmetic symbols.svg
File:Gottfried Wilhelm Leibniz, Bernhard Christoph Francke.jpg
Gottfried Wilhelm Leibniz, Bernhard...
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IllustrationCentralTheorem.png

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