The Student's Introduction to MATHEMATICA: A Handbook for Precalculus, Calculus, and Linear Algebra

This is a cookbook about how to use Mathematica to do the computational tasks that traditionally have been done by hand in high-school algebra, single- and several-variable calculus, and linear algebra. The book is task-oriented and goes through each topic covered in a traditional course in these subjects.

The goals are very modest: there's no use of the computer for experimentation or discovery, the book does not teach you any mathematics, and in fact you only learn a small fragment of what's available in Mathematica. You do learn how to skip all the hand calculation in these courses: the computer draws your graphs for you, factors polynomials for you, finds eigenvalues for you, and so on.

Opinions among teachers will probably be divided on whether skipping this hand work is a good thing; overall I think it is good. The book does take the time to explain what you are seeing in these graphs and factorizations. The exercises are not just rote application of the methods, but ask you to think about what you are seeing; for example, one exercise is to graph a rational function whose denominator has real roots and then explain why the graph has no vertical asymptotes.

There's quite a lot of material on graphing, and it is well done, although sometimes I thought the authors succumbed to polishing the picture to make it look pretty even after we understood what it was showing.

There are a few awkward spots in the exposition. The book circles around the concept of machine-precision calculations without really explaining it, and makes an incorrect claim on p. 11 that doing a calculation with machine-precision numbers gives the same result as doing an exact calculation and then converting to machine-precision. That's actually true for the types of problems covered in this book, but is certainly not true in general, and this sort of explanation obscures the reason for having exact calculations. There's an amazingly long discussion (pp. 162–171) attempting to justify Mathematica's convention that the cube root of –8 is not –2 but is approximately 1 + 1.73205i. The real reason is that we take the branch of z^1/3 in the complex plane that is real for positive z, but you can't really explain that to students at this level. The book attempts the explanation anyway and makes a mess of it.

The book claims that solutions to every exercise can be freely downloaded at the book's web site, but when I tried this there was nothing there except the first author's email address.

Overall it's a good book. I would have liked something more challenging and less cook-booky, but it's a book that most students can benefit from.

Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at MathNerds.com, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.