Computer algebra system

A computer algebra system (CAS) or symbolic algebra system (SAS) is a type of mathematical software. It helps people work with math problems. It can change and simplify math expressions, just like a person would do by hand. These tools are 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 started looking for ways to make computers do more than just work with numbers. They wanted computers to work with symbols and math expressions, just like people do. This led to the creation of computer algebra systems in the 1960s.

These systems were created to help 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. Today, there are free options like SageMath. These systems have changed over time and are now available online with tools like WolframAlpha.

Symbolic manipulations

A computer algebra system can solve many kinds of math problems by using symbols, not just numbers. It can make math expressions simpler, replace values, and change how expressions look. It can also find slopes and areas.

It can solve equations, work with special math functions, and do matrix calculations. These systems can handle many symbolic math tasks, though not every operation can always be done.

Main article: symbolic integration

Additional capabilities

Many computer algebra systems have extra tools. They often include a programming language that lets users make their own math rules. They can work with very large numbers exactly and help you make math expressions look nice on screen.

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

Types of expressions

A computer algebra system can solve many types of math problems. It works with polynomials, functions like sine and exponential, and special functions. It can also handle calculus, such as derivatives and integrals, and manage matrices and series.

These systems work with many kinds of numbers, including decimals, large whole numbers, complex numbers, numbers in ranges, exact fractions, and answers to algebra problems.

Use in education

Many people think that computer algebra systems should be used more in schools. These tools help students understand math better. Some schools now include them in their lessons.

These systems are also used in colleges and universities. Many schools teach special classes on how to use them. Students often use them for math and science work. But, these tools are not allowed on some big tests like the ACT, PLAN, and SAT. They may be allowed 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. They use algorithms to find answers to equations, add up functions in symbols, and break down math expressions. They also use the Euclidean algorithm to find the biggest common factor and special ways to work with tricky math. These tools help computers solve math problems like a person would with a pencil and paper.

Main article: Computer algebra system