Computer algebra system

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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 mathematicians and scientists. The development of the computer algebra systems in the second half of the 20th century is part of the discipline of "computer algebra" or "symbolic computation", which has spurred work in algorithms over mathematical objects such as polynomials.

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.

General-purpose computer algebra systems aim to be useful to a user working in any scientific field that requires manipulation of mathematical expressions. To be useful, a general-purpose computer algebra system must include various features such as:

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 divisors 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. Significant systems include Axiom, GAP, Maxima, Magma, Maple, Mathematica, and SageMath.


A Texas Instruments TI-Nspire calculator that contains a computer algebra system

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. Another early system was FORMAC.

Using 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 ("mathematical laboratory") should not be confused with MATLAB ("matrix laboratory"), 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, and it was possible, for the first time in a calculator,[1] 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 Texas Instruments company in 1995 released the TI-92 calculator with a CAS based on the software Derive; the TI-Nspire series replaced Derive in 2007. The TI-89 series, first released in 1998, also contains a CAS.

Casio released their first CAS calculator with the CFX-9970G and succeeded it with the Algebra FX Series in 1999-2003 and the current ClassPad Series.[citation needed]

The first popular computer algebra systems were muMATH, Reduce, Derive (based on muMATH), and Macsyma; a popular copyleft version of Macsyma called Maxima is actively being maintained. Reduce became free software in 2008.[2] As of today,[when?] the most popular commercial systems are Mathematica[3] and Maple, which are commonly used by research mathematicians, scientists, and engineers. Freely available alternatives include SageMath (which can act as a front-end to several other free and nonfree CAS). Other significant systems include Axiom, GAP, Maxima and Magma.

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.[4]

More recently, computer algebra systems have been implemented using artificial neural networks, though as of 2020 they are not commercially available.[5]

Symbolic manipulations[edit]

The symbolic manipulations supported typically include:

In the above, the word some indicates that the operation cannot always be performed.

Additional capabilities[edit]

Many also include:

Some include:

Some computer algebra systems focus on specialized disciplines; these are typically developed in academia and are free. They can be inefficient for numeric operations as compared to numeric systems.

Types of expressions[edit]

The expressions manipulated by the CAS typically include polynomials in multiple variables; standard functions of expressions (sine, exponential, etc.); various special functions (Γ, ζ, erf, Bessel functions, etc.); arbitrary functions of expressions; optimization; derivatives, integrals, simplifications, sums, and products of expressions; truncated series with expressions as coefficients, matrices of expressions, and so on. Numeric domains supported typically include floating-point representation of real numbers, integers (of unbounded size), complex (floating-point representation), interval representation of reals, rational number (exact representation) and algebraic numbers.

Use in education[edit]

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.[10] 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.[11]

Computer algebra systems have been extensively used in higher education.[12][13] 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.[14][15]

CAS-equipped calculators are not permitted on the ACT, the PLAN, and in some classrooms[16] 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, Chemistry, Physics, and Statistics exams.[17]

Mathematics used in computer algebra systems[edit]

See also[edit]


  1. ^ Nelson, Richard. "Hewlett-Packard Calculator Firsts". Hewlett-Packard. Archived from the original on 2010-07-03.
  2. ^ "REDUCE Computer Algebra System at SourceForge". Retrieved 2015-09-28.
  3. ^ Interview with Gaston Gonnet, co-creator of Maple Archived 2007-12-29 at the Wayback Machine, SIAM History of Numerical Analysis and Computing, March 16, 2005.
  4. ^ Bhattacharya, Jyotirmoy (2022-05-12). "Wolfram|Alpha: a free online computer algebra system". The Hindu. ISSN 0971-751X. Retrieved 2023-04-26.
  5. ^ Ornes, Stephen. "Symbolic Mathematics Finally Yields to Neural Networks". Quanta Magazine. Retrieved 2020-11-04.
  6. ^ "dsolve - Maple Programming Help". Retrieved 2020-05-09.
  7. ^ "DSolve - Wolfram Language Documentation". Retrieved 2020-06-28.
  8. ^ "Basic Algebra and Calculus — Sage Tutorial v9.0". Retrieved 2020-05-09.
  9. ^ "Symbolic algebra and Mathematics with Xcas" (PDF).
  10. ^ "Teaching kids real math with computers". Retrieved 2017-08-12.
  11. ^ "Mathematics - Manitoba Education". Retrieved 2017-08-12.
  12. ^ "Mathematica for Faculty, Staff, and Students : Information Technology - Northwestern University". Retrieved 2017-08-12.
  13. ^ "Mathematica for Students - Columbia University Information Technology". Retrieved 2017-08-12.
  14. ^ "Mathematica for Higher Education: Uses for University & College Courses". Retrieved 2017-08-12.
  15. ^ "MathWorks - Academia - MATLAB & Simulink". Retrieved 2017-08-12.
  16. ^ ACT's CAAP Tests: Use of Calculators on the CAAP Mathematics Test Archived August 31, 2009, at the Wayback Machine
  17. ^ "AP Exams Calculator Policy". AP Students. College Board. Retrieved 2024-05-24.
  18. ^ a b B. Buchberger; G.E. Collins; R. Loos (2013-06-29). Computer Algebra: Symbolic and Algebraic Computation. Springer Science & Business Media. ISBN 978-3-7091-3406-1.
  19. ^ Joachim von zur Gathen; Jürgen Gerhard (2013-04-25). Modern Computer Algebra. Cambridge University Press. ISBN 978-1-107-03903-2.
  20. ^ Keith O. Geddes; Stephen R. Czapor; George Labahn (2007-06-30). Algorithms for Computer Algebra. Springer Science & Business Media. ISBN 978-0-585-33247-5.

External links[edit]