Computer algebra system

A computer algebra system (CAS) or symbolic algebra system (SAS) is a type of mathematical software that helps people work with math problems. It can change and simplify math expressions much like a human mathematician or scientist would do by hand. These tools are very important because they let us solve complex math problems that would be too hard or take too long to do manually.

The idea of computer algebra systems started in the second half of the 20th century. This led to a new area of study called "computer algebra" or "symbolic computation." It focuses on creating special algorithms for working with math objects like polynomials.

There are two main types of computer algebra systems. Some are made for one specific area of math, like number theory, group theory, or teaching elementary mathematics. Others are general-purpose tools meant to help anyone who needs to work with math expressions, no matter their field. To work well, these systems need many features. They need a good user interface so people can type in and see math formulas. They also need a programming language and an interpreter to understand what the user wants. There must be a simplifier to make formulas easier, a memory manager to handle big amounts of data, and a way to work with very large numbers using arbitrary-precision arithmetic. They also need many algorithms and special functions to solve different kinds of math problems.

Because so many capabilities are needed, there aren't many general-purpose computer algebra systems. Some of the most well-known ones are Axiom, GAP, Maxima, Magma, Maple, Mathematica, SageMath, and SymPy. These tools help scientists, engineers, and students explore math in new and powerful ways.

History

In the 1950s, researchers began exploring ways to use computers for more than just numbers—they wanted computers to handle symbols and math expressions like people do. This led to the creation of computer algebra systems in the 1960s. These systems grew from the needs of physicists and early artificial intelligence research.

Important early systems include Schoonschip, developed by physicist Martinus Veltman in 1963, and MATHLAB created by Carl Engelman in 1964. Later, handheld calculators with these abilities appeared, such as the HP-28 series. Popular systems like Mathematica and Maple became widely used, and today there are free options like SageMath. Over time, these systems have evolved, even moving online with tools like WolframAlpha.

Symbolic manipulations

A computer algebra system can handle many kinds of math problems by working with symbols instead of just numbers. It can simplify expressions, substitute values, change how expressions look (like turning products into sums), and do calculus operations such as finding slopes and areas.

It can also solve equations, work with special math functions, perform matrix calculations, and even help prove mathematical ideas. Not all operations can always be done, but these systems can tackle a wide range of symbolic math tasks.

Main article: symbolic integration

Additional capabilities

Many computer algebra systems come with extra tools. They often have a programming language that lets users create their own math rules. They can handle very large numbers exactly and let you edit math expressions to look nice on screen.

These systems can also draw graphs and parametric plots of functions in two or three dimensions, create charts and diagrams, and connect to other programs through APIs. They support tasks like string manipulation, solving differential equations, and presenting results in standard math notation through pretty-printing. Some even help with bioinformatics, computational chemistry, and physical computation. Additionally, certain systems can produce graphic outputs, including computer-generated imagery and signal processing like image processing and sound synthesis.

Types of expressions

A computer algebra system can work with many kinds of math problems. It can handle polynomials with many variables, common functions like sine and exponential, and special functions such as the Gamma function. It can also deal with optimization, calculus operations like derivatives and integrals, and work with matrices and series.

These systems support different number types, including regular decimals, very large whole numbers, complex numbers, numbers within ranges, exact fractions, and solutions to algebraic equations.

Use in education

Many people believe that computer algebra systems should be used more in schools because they help students understand real-world math better than traditional methods. Some education boards have supported this idea, and in some places, it is now part of the school curriculum.

These systems are also widely used in colleges and universities. Many schools offer special classes to teach how to use them, and students are often expected to use them for their math and science work. However, these tools are not allowed on certain standardized tests like the ACT, PLAN, and SAT, though they may be permitted on some Advanced Placement exams like AP Calculus, Chemistry, Physics, and Statistics.

Mathematics used in computer algebra systems

Computer algebra systems use many important math ideas to work with symbols and solve problems. Some of these include algorithms for finding roots of equations, integrating functions symbolically, and factoring polynomials. They also use methods like the Euclidean algorithm for finding greatest common divisors and special techniques for handling complex math expressions. These tools help computers handle math much like a human mathematician would on paper.

Main article: Computer algebra system