A computer algebra system (CAS) or symbolic algebra system (SAS) is any

mathematical software Mathematical software is software used to mathematical model, model, analyze or calculate numeric, symbolic or geometric data. Evolution of mathematical software Numerical analysis and symbolic computation had been in most important place of the ...

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 Scientific method, researches to advance knowledge in an Branches of science, area of the natural sciences. In classical antiquity, there was no real ancient analog of a modern scientist. Instead, philosophers engag ...

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 expression (mathematics), ...

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

algorithm In mathematics and computer science, an algorithm () is a finite sequence of Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algo ...

s over

mathematical object A mathematical object is an abstract concept arising in mathematics. Typically, a mathematical object can be a value that can be assigned to a Glossary of mathematical symbols, symbol, and therefore can be involved in formulas. Commonly encounter ...

s such as

polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...

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 is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

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 fro ...

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. Programming languages are described in terms of their Syntax (programming languages), syntax (form) and semantics (computer science), semantics (meaning), usually def ...

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 po ...

, 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 by ...

. 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 In algebra, the greatest common divisor (frequently abbreviated as GCD) of two polynomials is a polynomial, of the highest possible degree, that is a factor of both the two original polynomials. This concept is analogous to the greatest common d ...

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. 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 or ...

, GAP, Maxima,

Magma Magma () is the molten or semi-molten natural material from which all igneous rocks are formed. Magma (sometimes colloquially but incorrectly referred to as ''lava'') is found beneath the surface of the Earth, and evidence of magmatism has also ...

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

Mathematica Wolfram (previously known as Mathematica and Wolfram Mathematica) is a software system with built-in libraries for several areas of technical computing that allows machine learning, statistics, symbolic computation, data manipulation, network ...

, 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, group theory, differentia ...

History

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 Artificial intelligence (AI) is the capability of computer, computational systems to perform tasks typically associated with human intelligence, such as learning, reasoning, problem-solving, perception, and decision-making. It is a field of re ...

. 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. Other early systems include FORMAC. Using

Lisp Lisp (historically LISP, an abbreviation of "list processing") is a family of programming languages with a long history and a distinctive, fully parenthesized Polish notation#Explanation, prefix notation. Originally specified in the late 1950s, ...

as the programming basis, Carl Engelman created MATHLAB in 1964 at

MITRE The mitre (Commonwealth English) or miter (American English; American and British English spelling differences#-re, -er, see spelling differences; both pronounced ; ) is a type of headgear now known as the traditional, ceremonial headdress of ...

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, implementat ...

("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; ) is a public research university in Albuquerque, New Mexico, United States. Founded in 1889 by the New Mexico Territorial Legislature, it is the state's second oldest university, a flagship university in th ...

. In 1987,

Hewlett-Packard The Hewlett-Packard Company, commonly shortened to Hewlett-Packard ( ) or HP, was an American multinational information technology company. It was founded by Bill Hewlett and David Packard in 1939 in a one-car garage in Palo Alto, California ...

introduced the first hand-held calculator CAS with the HP-28 series. Other early handheld calculators with symbolic algebra capabilities included the

Texas Instruments Texas Instruments Incorporated (TI) is an American multinational semiconductor company headquartered in Dallas, Texas. It is one of the top 10 semiconductor companies worldwide based on sales volume. The company's focus is on developing analog ...

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 algebra ...

and TI-92 calculator, and the

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. It ...

CFX-9970G. The first popular computer algebra systems were muMATH, Reduce, Derive (based on muMATH), and

Macsyma Macsyma (; "Project MAC's SYmbolic MAnipulator") is one of the oldest general-purpose computer algebra systems still in wide use. It was originally developed from 1968 to 1982 at MIT's Project MAC. In 1982, Macsyma was licensed to Symbolics and ...

; a

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 is called Maxima. Reduce became free software in 2008. Commercial systems include

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). Other significant systems include

, GAP, Maxima and

. 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

. More recently, computer algebra systems have been implemented using

artificial neural networks In machine learning, a neural network (also artificial neural network or neural net, abbreviated ANN or NN) is a computational model inspired by the structure and functions of biological neural networks. A neural network consists of connected ...

, though as of 2020 they are not commercially available.

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 In mathematics, factorization (or factorisation, see American and British English spelling differences#-ise, -ize (-isation, -ization), English spelling differences) or factoring consists of writing a number or another mathematical object as a p ...

, rewriting as partial fractions, 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 all ...

as exponentials, transforming logic expressions, etc. *

partial Partial may refer to: Mathematics *Partial derivative, derivative with respect to one of several variables of a function, with the other variables held constant ** ∂, a symbol that can denote a partial derivative, sometimes pronounced "partial d ...

and

total differentiation In mathematics, the total derivative of a function at a point is the best linear approximation near this point of the function with respect to its arguments. Unlike partial derivatives, the total derivative approximates the function with res ...

*some indefinite 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 formula for a differentiable function ''F''(''x'') such that :\frac = f(x ...

), 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 equation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...

s *taking some limits *integral transforms *

series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used i ...

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 Experimental mathematics is an approach to mathematics in which computation is used to investigate mathematical objects and identify properties and patterns. It has been defined as "that branch of mathematics that concerns itself ultimately with th ...

* 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 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 *add-ons for use in

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

such as physics,

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

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

and packages for physical computation *solvers for differential equations 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 the data, as in design and manufa ...

production and editing such as

computer-generated imagery Computer-generated imagery (CGI) is a specific-technology or application of computer graphics for creating or improving images in Digital art, art, Publishing, printed media, Training simulation, simulators, videos and video games. These images ...

and

signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, Scalar potential, potential fields, Seismic tomograph ...

image processing An image or picture is a visual representation. An image can be two-dimensional, such as a drawing, painting, or photograph, or three-dimensional, such as a carving or sculpture. Images may be displayed through other media, including a pr ...

sound synthesis A synthesizer (also synthesiser or synth) is an electronic musical instrument that generates audio signals. Synthesizers typically create sounds by generating waveforms through methods including subtractive synthesis, additive synthesis an ...

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 opposite th ...

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

Bessel function Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary complex ...

s, etc.); arbitrary functions of expressions; optimization; derivatives, integrals, simplifications, sums, and products of expressions; truncated

with expressions as coefficients,

matrices Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the ...

of expressions, and so on. Numeric domains supported typically include floating-point representation of real numbers,

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

s (of unbounded size),

complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...

(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 (for example, The set of all ...

(exact representation) and

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

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 Goal, objective to do something. It is commonly understood as a modal logic, temporal set (mathematics), set of intended actions through wh ...

, and in some classrooms though it may be permitted on all of

College Board The College Board, styled as CollegeBoard, is an American not-for-profit organization that was formed in December 1899 as the College Entrance Examination Board (CEEB) to expand access to higher education. While the College Board is not an asso ...

's calculator-permitted tests, including the

SAT The SAT ( ) is a standardized test widely used for college admissions in the United States. Since its debut in 1926, its name and Test score, scoring have changed several times. For much of its history, it was called the Scholastic Aptitude Test ...

, some SAT Subject Tests and the

AP Calculus Advanced Placement (AP) Calculus (also known as AP Calc, Calc AB / BC, AB / BC Calc 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 Calculu ...

Chemistry Chemistry is the scientific study of the properties and behavior of matter. It is a physical science within the natural sciences that studies the chemical elements that make up matter and chemical compound, compounds made of atoms, molecules a ...

Physics Physics is the scientific study of matter, its Elementary particle, 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 whi ...

, and

Statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...

exams.

Mathematics used in computer algebra systems

* Knuth–Bendix completion algorithm *

Root-finding algorithm In numerical analysis, a root-finding algorithm is an algorithm for finding zeros, also called "roots", of continuous functions. A zero of a function is a number such that . As, generally, the zeros of a function cannot be computed exactly nor ...

s *

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 formula for a differentiable function ''F''(''x'') such that :\frac = f(x ...

via e.g. Risch algorithm or Risch–Norman algorithm *

Hypergeometric summation In mathematics, the Gaussian or ordinary hypergeometric function 2''F''1(''a'',''b'';''c'';''z'') is a Special functions, special function represented by the hypergeometric series, that includes many other special functions as special case, spe ...

via e.g.

Gosper's algorithm In mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. ...

* Limit computation via e.g. Gruntz's algorithm * Polynomial factorization via e.g., over finite fields, Berlekamp's algorithm or Cantor–Zassenhaus algorithm. *

Greatest common divisor In mathematics, the greatest common divisor (GCD), also known as greatest common factor (GCF), of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers , , the greatest co ...

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 a ...

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 row-wise operations performed on the corresponding matrix of coefficients. This method can a ...

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 K _1,\ldots,x_n/math> ove ...

via e.g. Buchberger's algorithm; generalization of Euclidean algorithm and Gaussian elimination *

Padé approximant In mathematics, a Padé approximant is the "best" approximation of a function near a specific point by a rational function of given order. Under this technique, the approximant's power series agrees with the power series of the function it is ap ...

* 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 thes ...

Diophantine equation ''Diophantine'' means pertaining to the ancient Greek mathematician Diophantus. A number of concepts bear this name: *Diophantine approximation In number theory, the study of Diophantine approximation deals with the approximation of real n ...

s * 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.) *

Cylindrical algebraic decomposition In mathematics, cylindrical algebraic decomposition (CAD) is a notion, along with an algorithm to compute it, that is fundamental for computer algebra and real algebraic geometry. Given a set ''S'' of polynomials in R''n'', a cylindrical algebraic ...

Quantifier elimination Quantifier elimination is a concept of simplification used in mathematical logic, model theory, and theoretical computer science. Informally, a quantified statement "\exists x such that ..." can be viewed as a question "When is there an x such ...

over real numbers via cylindrical algebraic decomposition

References

External links

Curriculum and Assessment in an Age of Computer Algebra Systems
- From the

Education Resources Information Center The Education Resources Information Center (ERIC) is an online digital library of education research and information. ERIC is sponsored by the Institute of Education Sciences of the United States Department of Education. Description The missio ...

Clearinghouse for Science, Mathematics, and Environmental Education,

Columbus, Ohio Columbus (, ) is the List of capitals in the United States, capital and List of cities in Ohio, most populous city of the U.S. state of Ohio. With a 2020 United States census, 2020 census population of 905,748, it is the List of United States ...

. *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