This is a confused and confusing treatment of lower-division college mathematics. According to the Preface it is aimed at those “who are struggling with mathematics, either as first-year undergraduates taking math as a subsidiary to a science course, or students taking AP courses”. From the body of the book it appears to be a refresher course that is not intended for those who have never seen the material before (it treats two years of college math in 150 pages).

The introductory chapters cover numbers in general, unit conversion, and dimensional analysis. These are not bad, except that the unit conversions are done ad hoc and no method is apparent, and there’s no mention of approximate values or significant digits. For example, on p. 13 we find that the age of the Earth is 4000000000 years, this being claimed to be equivalent to 4 x 10^{9} years. Throughout the book numerical values are rounded off without any discussion.

The two chapters on calculus are the longest and weakest of the book. The explanation of the limit concept (p. 53) doesn’t make any sense, even if you already know what limits are, and it occurs out of context, ten pages before it is used. The weak explanation is not helped by the statement in two places (pp. 56, 73) that the limit of sin θ as θ goes to 0 is θ (not 0). The book makes reference to infinitesimals without explaining them (e.g., p. 98).

The chapters on Matrix Algebra and Statistics are not bad, although very skimpy. The book suggests on p. 122 that the way to solve a system of 100 equations in 100 variables is to find the matrix inverse, which nobody would do in real life. The book states incorrectly on p. 138 that the chi-square test is nonparametric, but its coverage is otherwise reliable.

At times the book seems self-defeating. For example, on p. 25 it states, “To a large extent the overt use of logarithms can now be avoided because of the power of the computer,” and immediately launches into a five-page discussion of the properties of logarithms. The book doesn’t mention that there are many applications where the items of interest grow logarithmically, and in fact it has such an example (the pH of a chemical solution) in the exercises for this section.

Bottom line: An extremely concise, but mathematically shaky, refresher course that might be useful to those who understood the material well at one time. Others should seek lengthier or more specialized texts.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.