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{{Short description|Mathematical software}}
{{Use dmy dates|date=April 2019|cs1-dates=y}}
{{Redirect|Symbolic algebra|the algebra of logic|Symbolical algebra}}
A '''computer algebra system''' ('''CAS''') or '''symbolic algebra system''' ('''SAS''') is any [[mathematical software]] with the ability to manipulate [[mathematical expressions]] in a way similar to the traditional manual computations of [[mathematician]]s and [[scientist]]s. The development of the computer algebra systems in the second half of the 20th century is part of the discipline of "[[computer algebra]]" or "
Computer algebra systems may be divided into two classes: specialized and general-purpose. The specialized ones are devoted to a specific part of mathematics, such as [[number theory]], [[group theory]], or teaching of [[elementary mathematics]].
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The library must not only provide for the needs of the users, but also the needs of the simplifier. For example, the computation of [[polynomial greatest common divisor]]s is systematically used for the simplification of expressions involving fractions.
This large amount of required computer capabilities explains the small number of general-purpose computer algebra systems.
== History ==
[[File:
In the 1950s, while computers were mainly used for numerical computations, there were some research projects into using them for symbolic manipulation. Computer algebra systems began to appear in the 1960s and evolved out of two quite different sources—the requirements of theoretical physicists and research into [[artificial intelligence]].
A prime example for the first development was the pioneering work conducted by the later Nobel Prize laureate in physics [[Martinus Veltman]], who designed a program for symbolic mathematics, especially high-energy physics, called [[Schoonschip]] (Dutch for "clean ship") in 1963.
Using [[Lisp (programming_language)|Lisp]] as the programming basis, [[Carl Engelman]] created [[MATHLAB]] in 1964 at [[MITRE]] within an artificial-intelligence research environment. Later MATHLAB was made available to users on PDP-6 and PDP-10 systems running TOPS-10 or TENEX in universities. Today it can still be used on [[SIMH]] emulations of the PDP-10. MATHLAB ("'''math'''ematical '''lab'''oratory") should not be confused with [[MATLAB]] ("'''mat'''rix '''lab'''oratory"), which is a system for numerical computation built 15 years later at the [[University of New Mexico]].
In 1987, [[Hewlett-Packard]] introduced the first hand-held calculator CAS with the [[HP-28 series]]
The first popular computer algebra systems were [[muMATH]], [[Reduce computer algebra system|Reduce]], [[Derive (computer algebra system)|Derive]] (based on muMATH), and [[Macsyma]]; a popular [[copyleft]] version of Macsyma called [[Maxima (software)|Maxima]] is actively being maintained. [[Reduce (computer algebra system)|Reduce]] became free software in 2008.<ref>{{Cite web|title = REDUCE Computer Algebra System at SourceForge|url = http://reduce-algebra.sourceforge.net|website = reduce-algebra.sourceforge.net|access-date = 2015-09-28}}</ref> As of today,{{when|date=October 2016}} the most popular commercial systems are [[Mathematica]]<ref>[http://history.siam.org/oralhistories/gonnet.htm Interview with Gaston Gonnet, co-creator of Maple] {{webarchive|url=https://web.archive.org/web/20071229044836/http://history.siam.org/oralhistories/gonnet.htm |date=2007-12-29 }}, SIAM History of Numerical Analysis and Computing, March 16, 2005.</ref> and [[Maple (software)|Maple]], which are commonly used by research mathematicians, scientists, and engineers. Freely available alternatives include [[SageMath]] (which can act as a [[Front and back ends|front-end]] to several other free and nonfree CAS).▼
| last = Coons | first = Albert
| date = October 1999
| department = Technology Tips
| doi = 10.5951/mt.92.7.0620
| issue = 7
| journal = The Mathematics Teacher
| jstor = 27971125
| pages = 620–622
| title = Getting started with symbolic mathematics systems: a productivity tool
| volume = 92}}</ref>
▲The first popular computer algebra systems were [[muMATH]], [[Reduce computer algebra system|Reduce]], [[Derive (computer algebra system)|Derive]] (based on muMATH), and [[Macsyma]]; a
▲In 1987, [[Hewlett-Packard]] introduced the first hand-held calculator CAS with the [[HP-28 series]], and it was possible, for the first time in a calculator,<ref name="Nelson">{{cite web | title=Hewlett-Packard Calculator Firsts | first=Richard | last=Nelson | publisher=Hewlett-Packard | url=http://h20331.www2.hp.com/Hpsub/cache/392617-0-0-225-121.html | archive-url=https://web.archive.org/web/20100703031935/http://h20331.www2.hp.com/Hpsub/cache/392617-0-0-225-121.html | archive-date=2010-07-03}}</ref> to arrange algebraic expressions, differentiation, limited symbolic integration, Taylor series construction and a ''solver'' for algebraic equations. In 1999, the independently developed CAS [[Erable]] for the [[HP 48 series]] became an officially integrated part of the firmware of the emerging [[HP 49/50 series]], and a year later into the [[HP 40 series]] as well, whereas the [[HP Prime]] adopted the [[Xcas]] system in 2013.
The movement to web-based applications in the early 2000s saw the release of [[WolframAlpha]], an online search engine and CAS which includes the capabilities of [[Mathematica]].<ref>{{Cite news |last=Bhattacharya |first=Jyotirmoy |date=2022-05-12 |title=Wolfram{{!}}Alpha: a free online computer algebra system |language=en-IN |work=The Hindu |url=https://www.thehindu.com/sci-tech/technology/wolframalpha-a-free-online-computer-algebra-system/article65401003.ece |access-date=2023-04-26 |issn=0971-751X}}</ref>
More recently, computer algebra systems have been implemented using [[artificial neural networks]], though as of 2020 they are not commercially available.<ref>{{Cite web |last=Ornes |first=Stephen |title=Symbolic Mathematics Finally Yields to Neural Networks |url=https://www.quantamagazine.org/symbolic-mathematics-finally-yields-to-neural-networks-20200520/ |access-date=2020-11-04 |website=Quanta Magazine |date=20 May 2020 |language=en}}</ref>▼
▲More recently, computer algebra systems have been implemented using [[artificial neural networks]].<ref>{{Cite web|last=Ornes|first=Stephen|title=Symbolic Mathematics Finally Yields to Neural Networks|url=https://www.quantamagazine.org/symbolic-mathematics-finally-yields-to-neural-networks-20200520/|access-date=2020-11-04|website=Quanta Magazine|language=en}}</ref>
==Symbolic manipulations==
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*[[Application programming interface|APIs]] for linking it on an external program such as a database, or using in a programming language to use the computer algebra system
*[[string manipulation]] such as [[string matching|matching]] and [[string searching|searching]]
*add-ons for use in [[applied mathematics]] such as physics, [[bioinformatics]], [[computational chemistry]] and packages for [[computational physics|physical computation]]<ref>{{
*solvers for [[differential equation]]s<ref>{{Cite web|title=dsolve - Maple Programming Help|url=https://www.maplesoft.com/support/help/Maple/view.aspx?path=dsolve|website=www.maplesoft.com|access-date=2020-05-09}}</ref><ref>{{Cite web|title=DSolve - Wolfram Language Documentation|url=https://reference.wolfram.com/language/ref/DSolve.html|website=www.wolfram.com|access-date=2020-06-28}}</ref><ref>{{Cite web|title=Basic Algebra and Calculus — Sage Tutorial v9.0|url=http://doc.sagemath.org/html/en/tutorial/tour_algebra.html|website=doc.sagemath.org|access-date=2020-05-09}}</ref><ref>{{Cite web|title=Symbolic algebra and Mathematics with Xcas|url=http://www-fourier.ujf-grenoble.fr/~parisse/giac/cascmd_en.pdf}}</ref>
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==Use in education==
There have been many advocates for increasing the use of computer algebra systems in primary and secondary-school classrooms. The primary reason for such advocacy is that computer algebra systems represent real-world math more than do paper-and-pencil or hand calculator based mathematics.<ref>{{cite web|url=http://www.ted.com/talks/conrad_wolfram_teaching_kids_real_math_with_computers?language=en|title=Teaching kids real math with computers|website=Ted.com|date=15 November 2010 |access-date=12 August 2017}}</ref>
This push for increasing computer usage in mathematics classrooms has been supported by some boards of education. It has even been mandated in the curriculum of some regions.<ref>{{cite web|url=http://www.edu.gov.mb.ca/k12/cur/math/outcomes/|title=Mathematics - Manitoba Education|website=Edu.gov.mb.ca|access-date=12 August 2017}}</ref>
Computer algebra systems have been extensively used in higher education.<ref>{{cite web|url=http://www.it.northwestern.edu/software/mathematica-fac/|title=Mathematica for Faculty, Staff, and Students : Information Technology - Northwestern University|website=It.northwestern.edu|access-date=12 August 2017}}</ref><ref>{{cite web|url=https://cuit.columbia.edu/mathematica-students|title=Mathematica for Students - Columbia University Information Technology|website=cuit.columbia.edu|access-date=12 August 2017}}</ref> Many universities offer either specific courses on developing their use, or they implicitly expect students to use them for their course work. The companies that develop computer algebra systems have pushed to increase their prevalence among university and college programs.<ref>{{cite web|url=https://www.wolfram.com/solutions/education/higher-education/uses-for-education.html|title=Mathematica for Higher Education: Uses for University & College Courses|website=Wolfram.com|access-date=12 August 2017}}</ref><ref>{{cite web|url=http://www.mathworks.com/academia/|title=MathWorks - Academia - MATLAB & Simulink|website=Mathworks.com|access-date=12 August 2017}}</ref>
CAS-equipped calculators are not permitted on the [[ACT (test)|ACT]], the [[PLAN (test)|PLAN]], and in some classrooms<ref>[http://www.act.org/caap/sample/calc.html ACT's CAAP Tests: Use of Calculators on the CAAP Mathematics Test<!-- Bot generated title -->] {{webarchive |url=https://web.archive.org/web/20090831032437/http://www.act.org/caap/sample/calc.html |date=August 31, 2009 }}</ref> though it may be permitted on all of [[College Board]]'s calculator-permitted tests, including the [[SAT]], some [[SAT Subject Tests]] and the [[AP Calculus]], [[AP Chemistry|Chemistry]], [[AP Physics|Physics]], and [[AP Statistics|Statistics]] exams.<ref>{{cite web |title=AP Exams Calculator Policy |url=https://apstudents.collegeboard.org/exam-policies-guidelines/calculator-policies |website=AP Students |publisher=College Board |access-date=24 May 2024}}</ref>
== Mathematics used in computer algebra systems ==
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* [[Chinese remainder theorem]]
* [[Diophantine equation]]s
* [[Quantifier elimination]] over real numbers via e.g. Tarski's method/[[Cylindrical algebraic decomposition]]▼
* [[Landau's algorithm]] (nested radicals)
* Derivatives of [[elementary function]]s and [[special functions]]. (e.g. See [[derivatives of the incomplete gamma function]].)
* [[Cylindrical algebraic decomposition]]
▲* [[Quantifier elimination]] over real numbers via
==See also==
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==External links==
*[http://www.ericdigests.org/2003-1/age.htm Curriculum and Assessment in an Age of Computer Algebra Systems] {{Webarchive|url=https://web.archive.org/web/20091201030924/http://www.ericdigests.org/2003-1/age.htm |date=2009-12-01 }} - From the [[Education Resources Information Center]] Clearinghouse for Science, Mathematics, and Environmental Education, [[Columbus, Ohio]].▼
▲*[http://www.ericdigests.org/2003-1/age.htm Curriculum and Assessment in an Age of Computer Algebra Systems] - From the [[Education Resources Information Center]] Clearinghouse for Science, Mathematics, and Environmental Education, [[Columbus, Ohio]].
*Richard J. Fateman. "Essays in algebraic simplification." Technical report MIT-LCS-TR-095, 1972. ''(Of historical interest in showing the direction of research in computer algebra. At the MIT LCS website: [https://web.archive.org/web/20060917023934/http://www.lcs.mit.edu/publications/specpub.php?id=663])''
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