MATH 131-01, Mathematics Through Advanced Software, Fall 2010

Abstract

A computer algebra system is a software program that facilitates symbolic mathematics. Though the main purpose of a computer algebra system is manipulation of mathematical expressions in symbolic form, over the past few decades computer algebra systems have evolved to include formidable numerical and data analysis capabilities. By “symbolic manipulation,” I mean simplification of certain expressions to some standard form (with user-designated constraints often being possible). Computer algebra systems permit the user to substitute symbols or numerical values in place of certain expressions, as well as to make changes in the form of those expressions (partial and full factorization, representation as partial fractions, rewriting trigonometric functions in terms of exponential functions, etc.). Many standard tasks of calculus are well-handled by computer algebra systems: the symbolic computation of limits, derivatives, integrals, and infinite sums; symbolic constrained and unconstrained local and global optimization; and the calculation of Taylor expansions of analytic functions. Increasingly, computer algebra systems are being used to power automated theorem-proving and theorem verification, two tasks that are central components of the growing field of experimental mathematics. Mathematica is just one example of a computer algebra system, but there are many others, including Maple, Derive, Reduce, MuPAD, Magma, Axiom, and Maxima. Though proprietary (and, arguably, expensive), Mathematica is the computer algebra system most frequently used at Rhodes College (and, in fact, at many colleges, universities, corporations, and research laboratories). It was originally conceived by Stephen Wolfram, a MacArthur fellow, physicist, and author of A New Kind of Science, as a computer program called SMP (symbolic manipulation program). Mathematica 1.0, the successor to SMP, was released in 1988 and has improved incrementally with new major releases in 1991, 1996, 1999, and 2003. With the release of Mathematica 6.0 in 2007, Mathematica experienced enormous improvements in system stability, software documentation, dynamic interactivity, and data computability. In 2009, Rhodes College upgraded its ten-year-old Mathematica 4.0 license to a new Mathematica 7.0 license. Today at Rhodes College, Mathematica is used for undergraduate and faculty research, projects in upper-level courses in mathematics and the natural sciences, and instruction in Math 115 (applied calculus).