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User:Phlsph7/Algebra - Definition

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Definition

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Algebra is the branch of mathematics that studies algebraic operations[a] and algebraic structures.[2] An algebraic structure is a non-empty set of mathematical objects, such as the real numbers, together algebraic operations defined on that set, such addition and multiplication.[3] Algebra explores the laws and general characteristics of algebraic structures and what types of algebraic structures there are. Within certain algebraic structures, it studies the use of variables in equations and how to manipulate these equations.[4]

Algebra is often understood as a generalization of arithmetic.[5] Arithmetic studies arithmetic operations, like addition, subtraction, multiplication, and division, in a specific domain of numbers, like the real numbers.[6] Elementary algebra constitutes the first level of abstraction. Like arithmetic, it restricts itself to specific types of numbers and operations. It generalizes these operations by allowing indefinite quantities in the form of variables in addition to numbers.[7] A higher level of abstraction is achieved in abstract algebra, which is not limited to a specific domain and studies different classes of algebraic structures, like groups and rings. These algebraic structures are not restricted to typical arithmetic operations and cover other binary operations besides them.[8] Universal algebra is still more abstract in that it is not limited to binary operations and not interested in specific classes of algebraic structures but investigates the characteristics of algebraic structures in general.[9]

The term "algebra" is sometimes used in a more narrow sense to refer only to elementary algebra or only to abstract algebra.[10] When used as a countable noun, an algebra is a specific type of algebraic structure that involves a vector space equipped with a certain type of binary operation.[11] Depending on the context, "algebra" can also refer to other algebraic structures, like a Lie algebra or an associative algebra.[12]

Notes

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  1. ^ When understood in the widest sense, an algebraic operation is mapping from a Cartesian power of a set into that set, expressed formally as . Addition of real numbers is an example of an algebraic operations: it takes two numbers as input and produces one number as output. It has the form .[1]

References

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  • Golan, Jonathan S. (1995). "Algebras Over A Field". Foundations of Linear Algebra. Springer Netherlands. ISBN 978-94-015-8502-6.
  • Walz, Guido (1 December 2016). "Algebra". Lexikon der Mathematik: Band 1: A bis Eif (in German). Springer-Verlag. ISBN 978-3-662-53498-4.
  • Weisstein, Eric W. (2003). CRC Concise Encyclopedia of Mathematics (2nd ed.). Chapman & Hall/CRC. ISBN 1-58488-347-2.
  • Renze, John; Weisstein, Eric W. "Algebra". Wolfram MathWorld. Wolfram. Retrieved 12 January 2024.
  • Gilbert, William J.; Nicholson, W. Keith (30 January 2004). Modern Algebra with Applications. John Wiley & Sons. ISBN 978-0-471-46989-6.
  • Grillet, Pierre Antoine (2007). "Universal Algebra". Abstract Algebra. Springer. ISBN 978-0-387-71568-1.
  • EoM Staff (2020a). "Arithmetic". Encyclopedia of Mathematics. Springer. Retrieved 23 October 2023.
  • MW Staff (2023). "Definition of Arithmetic". www.merriam-webster.com. Retrieved 19 October 2023.
  • HC Staff (2022). "Arithmetic". www.ahdictionary.com. HarperCollins. Retrieved 19 October 2023.
  • Romanowski, Perry (2008). "Arithmetic". In Lerner, Brenda Wilmoth; Lerner, K. Lee (eds.). The Gale Encyclopedia of Science (4th ed.). Thompson Gale. ISBN 978-1-4144-2877-2.
  • Burgin, Mark (2022). Trilogy Of Numbers And Arithmetic - Book 1: History Of Numbers And Arithmetic: An Information Perspective. World Scientific. ISBN 978-981-12-3685-3.
  • Wagner, Sigrid; Kieran, Carolyn (7 December 2018). Research Issues in the Learning and Teaching of Algebra: the Research Agenda for Mathematics Education, Volume 4. Routledge. ISBN 978-1-135-43421-2.
  • Maddocks, J. R. (2008). "Algebra". In Lerner, Brenda Wilmoth; Lerner, K. Lee (eds.). The Gale Encyclopedia of Science (4th ed.). Thompson Gale. ISBN 978-1-4144-2877-2.
  • Fiche, Georges; Hebuterne, Gerard (1 March 2013). Mathematics for Engineers. John Wiley & Sons. ISBN 978-1-118-62333-6.
  • Pratt, Vaughan (2022). "Algebra". The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. Retrieved 11 January 2024.
  • EoM Staff (2017). "Algebra". Encyclopedia of Mathematics. Springer. Retrieved 11 January 2023.
  • EoM Staff (2020). "Algebra(2)". Encyclopedia of Mathematics. Springer. Retrieved 11 January 2023.
  • EoM Staff (2023). "Algebraic operation". Encyclopedia of Mathematics. Springer. Retrieved 11 January 2023.
  1. ^ EoM Staff 2023, Lead Section
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    • Pratt 2022, Lead Section, § 1. Elementary Algebra, § 2. Abstract Algebra, § 3. Universal Algebra
    • EoM Staff 2020, § The subject matter of algebra, its principal branches and its connection with other branches of mathematics.
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  12. ^ EoM Staff 2017
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