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 A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

s and

scientist A scientist is a person who conducts scientific research to advance knowledge in an area of the natural sciences. In classical antiquity, there was no real ancient analog of a modern scientist. Instead, philosophers engaged in the philosophica ...

s. The development of the computer algebra systems in the second half of the 20th century is part of the discipline of "

computer algebra In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expression ...

" or " symbolic computation", which has spurred work in

algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...

s over mathematical objects such as

polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...

s. 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 Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...

group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...

, 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: *a

user interface In the industrial design field of human–computer interaction, a user interface (UI) is the space where interactions between humans and machines occur. The goal of this interaction is to allow effective operation and control of the machine f ...

allowing a user to enter and display mathematical formulas, typically from a keyboard, menu selections, mouse or stylus. *a

programming language A programming language is a system of notation for writing computer programs. Most programming languages are text-based formal languages, but they may also be graphical. They are a kind of computer language. The description of a programming l ...

and an interpreter (the result of a computation commonly has an unpredictable form and an unpredictable size; therefore user intervention is frequently needed), *a simplifier, which is a rewrite system for simplifying mathematics formulas, *a memory manager, including a garbage collector, needed by the huge size of the intermediate data, which may appear during a computation, *an

arbitrary-precision arithmetic In computer science, arbitrary-precision arithmetic, also called bignum arithmetic, multiple-precision arithmetic, or sometimes infinite-precision arithmetic, indicates that calculations are performed on numbers whose digits of precision are l ...

, needed by the huge size of the integers that may occur, *a large library of mathematical

s and

special functions Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defined ...

. 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 An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy o ...

, Maxima,

Magma Magma () is the molten or semi-molten natural material from which all igneous rocks are formed. Magma is found beneath the surface of the Earth, and evidence of magmatism has also been discovered on other terrestrial planets and some natura ...

Maple ''Acer'' () is a genus of trees and shrubs commonly known as maples. The genus is placed in the family Sapindaceae.Stevens, P. F. (2001 onwards). Angiosperm Phylogeny Website. Version 9, June 2008 nd more or less continuously updated since ht ...

Mathematica Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimi ...

, and

SageMath SageMath (previously Sage or SAGE, "System for Algebra and Geometry Experimentation") is a computer algebra system (CAS) with features covering many aspects of mathematics, including algebra, combinatorics, graph theory, numerical analysis, nu ...

History

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 Artificial intelligence (AI) is intelligence—perceiving, synthesizing, and inferring information—demonstrated by machine A machine is a physical system using Power (physics), power to apply Force, forces and control Motion, moveme ...

. 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 A lisp is a speech impairment in which a person misarticulates sibilants (, , , , , , , ). These misarticulations often result in unclear speech. Types * A frontal lisp occurs when the tongue is placed anterior to the target. Interdental lispi ...

as the programming basis,

Carl Engelman Carl Engelman (October 21, 1929 – November 26, 1983) was an American computer scientist. Carl is best known as the creator of MATHLAB. He was employed by Mitre Corporation and Symbolics. He was a visiting professor at the University of Turin ...

created

MATHLAB MATHLAB is a computer algebra system created in 1964 by Carl Engelman at MITRE and written in Lisp. "MATHLAB 68" was introduced in 1967 and became rather popular in university environments running on DECs PDP-6 and PDP-10 under TOPS-10 or TE ...

in 1964 at

MITRE The mitre (Commonwealth English) (; Greek: μίτρα, "headband" or "turban") or miter (American English; see spelling differences), is a type of headgear now known as the traditional, ceremonial headdress of bishops and certain abbots in t ...

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 MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementa ...

("matrix laboratory"), which is a system for numerical computation built 15 years later at the

University of New Mexico The University of New Mexico (UNM; es, Universidad de Nuevo México) is a public research university in Albuquerque, New Mexico. Founded in 1889, it is the state's flagship academic institution and the largest by enrollment, with over 25, ...

. The first popular computer algebra systems were muMATH, Reduce, Derive (based on muMATH), and Macsyma; a popular

copyleft Copyleft is the legal technique of granting certain freedoms over copies of copyrighted works with the requirement that the same rights be preserved in derivative works. In this sense, ''freedoms'' refers to the use of the work for any purpose, ...

version of Macsyma called Maxima is actively being maintained. Reduce became free software in 2008. As of today, the most popular commercial systems are

and

, which are commonly used by research mathematicians, scientists, and engineers. Freely available alternatives include

(which can act as a front-end to several other free and nonfree CAS). 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, 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 HP may refer to: Businesses and organisations * HP Inc., an American technology company ** Hewlett-Packard, the predecessor to HP Inc. * HP Foods ** HP Sauce, formerly made by HP Foods * Handley Page, an aircraft company * Hindustan Petroleum * ...

as well, whereas the HP Prime adopted the

Xcas Xcas is a user interface to Giac, which is an open source computer algebra system (CAS) for Windows, macOS and Linux among many other platforms. Xcas is written in C++. Giac can be used directly inside software written in C++. Xcas has a c ...

system in 2013. The

Texas Instruments Texas Instruments Incorporated (TI) is an American technology company headquartered in Dallas, Texas, that designs and manufactures semiconductors and various integrated circuits, which it sells to electronics designers and manufacturers globa ...

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 The TI-89 and the TI-89 Titanium are graphing calculators developed by Texas Instruments (TI). They are differentiated from most other TI graphing calculators by their computer algebra system, which allows symbolic manipulation of alge ...

, first released in 1998, also contains a CAS.

Casio is a Japanese multinational electronics manufacturing corporation headquartered in Shibuya, Tokyo, Japan. Its products include calculators, mobile phones, digital cameras, electronic musical instruments, and analogue and digital watches. I ...

released their first CAS calculator with the

CFX-9970G Graphic calculators made by Casio include the touchscreen ClassPad 300 as well as the models with traditional buttons which can be divided into two main generations listed below. Casio produced the world's first graphing calculator, the fx-700 ...

and succeeded it with the Algebra FX Series in 1999-2003 and the current ClassPad Series. More recently, computer algebra systems have been implemented using artificial neural networks.

Symbolic manipulations

The symbolic manipulations supported typically include: *simplification to a smaller expression or some standard form, including automatic simplification with assumptions and simplification with constraints * substitution of symbols or numeric values for certain expressions *change of form of expressions: expanding products and powers, partial and full factorization, rewriting as

partial fraction In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as ...

s, constraint satisfaction, rewriting

trigonometric functions In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in ...

as exponentials, transforming logic expressions, etc. * partial and total differentiation *some

indefinite Indefinite may refer to: * the opposite of definite in grammar ** indefinite article ** indefinite pronoun * Indefinite integral, another name for the antiderivative * Indefinite forms in algebra, see definite quadratic forms * an indefinite matr ...

and definite integration (see

symbolic integration In calculus, symbolic integration is the problem of finding a formula for the antiderivative, or ''indefinite integral'', of a given function ''f''(''x''), i.e. to find a differentiable function ''F''(''x'') such that :\frac = f(x). This is al ...

), including multidimensional integrals *symbolic constrained and unconstrained global optimization * solution of linear and some non-linear equations over various domains *solution of some differential and difference equations *taking some

limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...

s *integral transforms * series operations such as expansion, summation and products *matrix operations including products, inverses, etc. * statistical computation * theorem proving and verification which is very useful in the area of experimental mathematics * optimized code generation In the above, the word ''some'' indicates that the operation cannot always be performed.

Additional capabilities

Many also include: *a

, allowing users to implement their own algorithms * arbitrary-precision numeric operations *exact integer arithmetic and number theory functionality * Editing of mathematical expressions in two-dimensional form *plotting graphs and parametric plots of functions in two and three dimensions, and animating them *drawing charts and diagrams *

APIs Apis or APIS may refer to: * Apis (deity), an ancient Egyptian god * Apis (Greek mythology), several different figures in Greek mythology * Apis (city), an ancient seaport town on the northern coast of Africa **Kom el-Hisn, a different Egyptian ci ...

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 matching and

searching Searching or search may refer to: Computing technology * Search algorithm, including keyword search ** :Search algorithms * Search and optimization for problem solving in artificial intelligence * Search engine technology, software for findi ...

*add-ons for use in

applied mathematics Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathemat ...

such as physics,

bioinformatics Bioinformatics () is an interdisciplinary field that develops methods and software tools for understanding biological data, in particular when the data sets are large and complex. As an interdisciplinary field of science, bioinformatics combin ...

computational chemistry Computational chemistry is a branch of chemistry that uses computer simulation to assist in solving chemical problems. It uses methods of theoretical chemistry, incorporated into computer programs, to calculate the structures and properties of mo ...

and packages for physical computation *solvers for

differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, a ...

s Some include: *

graphic Graphics () are visual images or designs on some surface, such as a wall, canvas, screen, paper, or stone, to inform, illustrate, or entertain. In contemporary usage, it includes a pictorial representation of data, as in design and manufacture, ...

production and editing such as

computer-generated imagery Computer-generated imagery (CGI) is the use of computer graphics to create or contribute to images in art, printed media, video games, simulators, and visual effects in films, television programs, shorts, commercials, and videos. The image ...

and

signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...

image processing An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimension ...

* sound synthesis 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

The expressions manipulated by the CAS typically include

s in multiple variables; standard functions of expressions (

sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opp ...

, exponential, etc.); various special functions ( Γ, ζ, erf,

Bessel function Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary ...

s, 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,

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

s (of unbounded size), complex (floating-point representation), interval representation of reals,

rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...

(exact representation) and

algebraic number An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the p ...

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. 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. Computer algebra systems have been extensively used in higher education. 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. CAS-equipped calculators are not permitted on the ACT, the

PLAN A plan is typically any diagram or list of steps with details of timing and resources, used to achieve an objective to do something. It is commonly understood as a temporal set of intended actions through which one expects to achieve a goal ...

, and in some classroomsACT's CAAP Tests: Use of Calculators on the CAAP Mathematics Test
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 Advanced Placement (AP) Calculus (also known as AP Calc, Calc AB / Calc BC or simply AB / BC) is a set of two distinct Advanced Placement calculus courses and exams offered by the American nonprofit organization College Board. AP Calculus AB cov ...

, Chemistry,

Physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rel ...

, and Statistics exams.

Mathematics used in computer algebra systems

* Knuth–Bendix completion algorithm * Root-finding algorithms *

Symbolic integration In calculus, symbolic integration is the problem of finding a formula for the antiderivative, or ''indefinite integral'', of a given function ''f''(''x''), i.e. to find a differentiable function ''F''(''x'') such that :\frac = f(x). This is al ...

via e.g.

Risch algorithm In symbolic computation, the Risch algorithm is a method of indefinite integration used in some computer algebra systems to find antiderivatives. It is named after the American mathematician Robert Henry Risch, a specialist in computer algebra w ...

Risch–Norman algorithm In symbolic computation, the Risch algorithm is a method of indefinite integration used in some computer algebra systems to find antiderivatives. It is named after the American mathematician Robert Henry Risch, a specialist in computer algebra ...

* Hypergeometric summation via e.g. Gosper's algorithm * Limit computation via e.g. Gruntz's algorithm *

Polynomial factorization In mathematics and computer algebra, factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field or in the integers as the product of irreducible factors with coefficients in the same d ...

via e.g., over finite fields,

Berlekamp's algorithm In mathematics, particularly computational algebra, Berlekamp's algorithm is a well-known method for factoring polynomials over finite fields (also known as ''Galois fields''). The algorithm consists mainly of matrix reduction and polynomial GC ...

Cantor–Zassenhaus algorithm In computational algebra, the Cantor–Zassenhaus algorithm is a method for factoring polynomials over finite fields (also called Galois fields). The algorithm consists mainly of exponentiation and polynomial GCD computations. It was invented by ...

. *

Greatest common divisor In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' i ...

via e.g.

Euclidean algorithm In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is an ...

Gaussian elimination In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of operations performed on the corresponding matrix of coefficients. This method can also be used ...

Gröbner basis In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating set of an ideal in a polynomial ring over a field . A Grö ...

via e.g. Buchberger's algorithm; generalization of Euclidean algorithm and Gaussian elimination * Padé approximant * Schwartz–Zippel lemma and testing polynomial identities *

Chinese remainder theorem In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of the ...

Diophantine equation In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to ...

s * Quantifier elimination over real numbers via e.g. Tarski's method/ Cylindrical algebraic decomposition * Landau's algorithm (nested radicals) * Derivatives of

elementary function In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, a ...

s and

. (e.g. See

derivatives of the incomplete gamma function In mathematics, the upper and lower incomplete gamma functions are types of special functions which arise as solutions to various mathematical problems such as certain integrals. Their respective names stem from their integral definitions, which ...

.) * Cylindrical algebraic decomposition

References

External links

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 Columbus () is the state capital and the most populous city in the U.S. state of Ohio. With a 2020 census population of 905,748, it is the 14th-most populous city in the U.S., the second-most populous city in the Midwest, after Chicago, an ...

. *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

'' {{DEFAULTSORT:Computer Algebra System Computer algebra systems, Algebra education

History

Symbolic manipulations

Additional capabilities

Types of expressions

Use in education

Mathematics used in computer algebra systems

See also

References

External links