Concrete Mathematics

by Ronald Graham, Donald Knuth, and Oren Patashnik

Please Be Discrete

July 2001

What is "concrete" math, as opposed to other types of math? The authors explain that the title comes from the blending of CONtinuous and disCRETE math, two branches of math that many seem to like to keep asunder, though each occurs in the foundation of the other. The topics in the book, such as sums, generating functions, and number theory, are actually standard discrete math topics; however, the treatment in this text shows the inherent continuous (read: calculus) undergirding of the topics. Without calculus, generating functions would not have come to mind and their tremendous power could not be put to use in figuring out series.

The smart-aleck marginal notes notwithstanding, this is a serious math book for those who are willing to dot every i and cross every t. Unlike most math texts (esp. graduate math texts), nothing is omitted along the way. Notation is explained (=very= important), common pitfalls are pointed out (as opposed to the usual way students come across them -- by getting back bleeding exams), and what is important and what is =not= as important are indicated.

Still, I cannot leave the marginal notes unremarked; some are serious warnings to the reader. For example, in the introduction, one note remarks "I would advise the casual student to stay away from this course." Notes that advise one to skim, and there are a few, should be taken seriously. All the marginal notes come from the TAs who had to help with the text, and thus have a more nitty-gritty understanding of the difficulties students are likely to face. Still, there are plenty of puns and bad jokes to amuse the text-reader for hours: "The empty set is pointless," "But not Imbesselian," and "John .316" made me chuckle, but you have to find them for yourself.

To someone who has been through the rigors of math grad school, this book is a delight to read; to those who have not, they must keep in mind that this is a serious text and must be prepared to do some real work. Very bright high school students have gotten through this text with little difficulty. I want to note ahead of time - some of the questions in the book are serious research topics. They don't necessarily tell you that when they give you the problem; if you've worked on the problem for a week, you should turn to the answers in the back to check that there really is a solution.

That said, I would highly recommend this book to math-lovers who want some rigorous math outside of the usual fare. The formulas in here can actually come in handy "in real life", especially if one has to use math a lot.

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Mary Pat Campbell, last updated July 2001