An Introduction to Modern Mathematical Computing
Thirty years ago mathematical, as opposed to applied numerical, computation was difficult to perform and so relatively little used. Three threads changed that:
• The emergence of the personal computer, identified with the iconic Macintosh but made ubiquitous by the IBM PC.
• The discovery of fiber-optics and the consequent development of the modern Internet culminating with the foundation of the World Wide Web in 1989 made possible by the invention of hypertext earlier in the decade.
• The building of the “Three Ms”: MapleTM, Wolfram MathematicaR, and Matlab. Each of these is a complete mathematical computation workspace with a large and constantly expanding built-in “knowledge base”. The first two are known as “computer algebra” or “symbolic computation” systems, sometimes written CAS.
Of course each of these threads rely on earlier related events and projects, and there are many other open source and commercial software packages. For example, Sage is an open-source CAS, GeoGebra an open-source interactive geometry package, and Octave is an open-source counterpart of Matlab. But this is not the place to discuss the merits and demerits of open source alternatives. For many purposes Mathematica and Maple are interchangeable as adjuncts to mathematical learning. We propose to use the latter. After reading this book, you should find it easy to pick up the requisite skills to use Mathematica  or Matlab.
Many introductions to computer packages aim to teach the syntax (rules and structure) and semantics (meaning) of the system as efficiently as possible [7, 8, 9, 13]. They assume one knows why one wishes to learn such things. By contrast, we intend to persuade that Maple and other like tools are worth knowing assuming only that one wishes to be a mathematician, a mathematics educator, a computer scientist, an engineer, or scientist, or anyone else who wishes/needs to use mathematics better. We also hope to explain how to become an experimental mathematician while learning to be better at proving things. To accomplish this our material is divided into three main chapters followed by a postscript.
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