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

mathematical software Mathematical software is software used to 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 subject, but other ki ...

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, structure, space, models, and change. History On ...

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 philosoph ...

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 " 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 example ...

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, "Mat ...

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 Elementary mathematics consists of mathematics topics frequently taught at the primary or secondary school levels. In the Canadian curriculum, there are six basic strands in Elementary Mathematics: Number, Algebra, Data, Spatial Sense, Finan ...

. 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 ...

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 In mathematics, computer science, and logic, rewriting covers a wide range of methods of replacing subterms of a formula with other terms. Such methods may be achieved by rewriting systems (also known as rewrite systems, rewrite engines, or reduc ...

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 li ...

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

s and special functions. 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, 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 natural sa ...

, Maple, Mathematica, 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, numbe ...

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 machines, as opposed to intelligence displayed by animals and humans. Example tasks in which this is done include speech r ...

. A prime example for the first development was the pioneering work conducted by the later Nobel Prize laureate in physics

Martinus Veltman Martinus Justinus Godefriedus "Tini" Veltman (; 27 June 1931 – 4 January 2021) was a Dutch theoretical physicist. He shared the 1999 Nobel Prize in physics with his former PhD student Gerardus 't Hooft for their work on particle theory. Biogr ...

, 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 SIMH is a free and open source, multi-platform multi-system emulator. It is maintained by Bob Supnik, a former DEC engineer and DEC vice president, and has been in development in one form or another since the 1960s. History SIMH was based on ...

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 25,400 ...

. 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 Mathematica and Maple, 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 as well, whereas the

HP Prime The HP Prime Graphing Calculator is a graphing calculator introduced by Hewlett-Packard in 2013 and currently manufactured by HP Inc. It was designed with features resembling those of smartphones, such as a full-color touchscreen display and t ...

adopted the Xcas 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 The TI-92 series of graphing calculators are a line of calculators produced by Texas Instruments. They include: the TI-92 (1995), the TI-92 II (1996), the TI-92 Plus (1998, 1999) and the Voyage 200 (2002). The design of these relatively large ...

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. 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 In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several ''factors'', usually smaller or simpler objects of the same kind ...

, rewriting as partial fractions,

constraint satisfaction In artificial intelligence and operations research, constraint satisfaction is the process of finding a solution through a set of constraints that impose conditions that the variables must satisfy. A solution is therefore a set of values for th ...

, rewriting trigonometric functions as exponentials, transforming logic expressions, etc. * partial 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 re ...

*some indefinite and definite integration (see symbolic integration), 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 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 in ...

operations such as expansion, summation and products *matrix operations including

products Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...

, inverses, etc. * statistical computation *

theorem proving Automated theorem proving (also known as ATP or automated deduction) is a subfield of automated reasoning and mathematical logic dealing with proving mathematical theorems by computer programs. Automated reasoning over mathematical proof was a maj ...

and

verification Verify or verification may refer to: General * Verification and validation, in engineering or quality management systems, is the act of reviewing, inspecting or testing, in order to establish and document that a product, service or system meets ...

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 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 li ...

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 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 findin ...

*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 mathemati ...

such as physics, bioinformatics, computational chemistry 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, an ...

s Some include: * graphic production and editing such as computer-generated imagery and signal processing as image processing *

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

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,

exponential Exponential may refer to any of several mathematical topics related to exponentiation, including: *Exponential function, also: **Matrix exponential, the matrix analogue to the above *Exponential decay, decrease at a rate proportional to value *Expo ...

, etc.); various special functions ( Γ, ζ, erf, Bessel functions, etc.); arbitrary functions of expressions; optimization; derivatives, integrals, simplifications, sums, and products of expressions; truncated

with expressions as coefficients,

matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

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 languag ...

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 (e.g. ). The set of all rat ...

(exact representation) and algebraic numbers.

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, 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 cover ...

, 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 r ...

, and Statistics exams.

Mathematics used in computer algebra systems

Knuth–Bendix completion algorithm The Knuth–Bendix completion algorithm (named after Donald Knuth and Peter Bendix) is a semi-decision algorithm for transforming a set of equations (over terms) into a confluent term rewriting system. When the algorithm succeeds, it effectively ...

Root-finding algorithm In mathematics and computing, a root-finding algorithm is an algorithm for finding zeros, also called "roots", of continuous functions. A zero of a function , from the real numbers to real numbers or from the complex numbers to the complex numbers ...

s * Symbolic integration via e.g. Risch algorithm or Risch–Norman algorithm * Hypergeometric summation via e.g.

Gosper's algorithm In mathematics, Gosper's algorithm, due to Bill Gosper, is a procedure for finding sums of hypergeometric terms that are themselves hypergeometric terms. That is: suppose one has ''a''(1) + ... + ''a''(''n'') = ''S''(''n'')&nb ...

* 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) 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'' is ...

via e.g. Euclidean algorithm * Gaussian elimination *

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 Gröbn ...

via e.g.

Buchberger's algorithm In the theory of multivariate polynomials, Buchberger's algorithm is a method for transforming a given set of polynomials into a Gröbner basis, which is another set of polynomials that have the same common zeros and are more convenient for extract ...

; 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 * Diophantine equations *

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 \ldots" can be viewed as a question "When is there an x such t ...

over real numbers via e.g. Tarski's method/

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

* 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, and ...

s and special functions. (e.g. See derivatives of the incomplete gamma function.) *

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. *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