Customer Reviews

Fibonacci and Lucas Numbers with Applications

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As a longtime subscriber to The Fibonaci Quarterly and a regular contributor to the problem column, I thought I was well-versed in the many applications of the Fibonacci and Lucas numbers. The contents of this book quickly led to an attitude adjustment on my part. It is astonishing and almost mystifying at times to see these numbers appearing in situations where "intuition" would tell us that they have no reason to be involved.
Subjects such as Fibonometry, where Fibonacci and Lucas numbers are related using trigonometric functions, demonstrates that the mathematics world is one where there are some topics that just seem to be attractors for many others. You start at one point and follow a logical path and suddenly, unexpectedly you arrive at the Fibonacci and/or Lucas numbers. The number of identities betwixt and between these two sets of numbers is so large, that one wonders if some of them may forever remain undiscovered.
There is a great deal of material in this book that could be selected for use in mathematics courses at almost all levels, starting at the position of precalculus. The inductive proofs of some of the identities are excellent demonstrations of how induction is used in mathematics. An extensive problem set is given at the end of each chapter, and solutions to the odd-numbered exercises are included in an appendix.
This is one of those books that all mathematicians should own. The problems and demonstrations in this book can be used as fodder for students hungry for interesting problems with solutions that teach them something. It demonstrates how mathematics is interrelated and how even simple definitions can lead to very complex and universal results.

Published in Journal of Recreational Mathematics, reprinted with permission.

Overall, this is a good book, covering many aspects of the fibonacci numbers, however, at about a 100USD, the potential buyer should be warned that there is a lot a typographical errors in the book.

While it is frequent to find a few notation's mistakes in every mathematics book, this one is way over average, most of them fortunately are quite obvious and easy to correct, but, in some cases, it simply makes the demonstration unreadable and the reader is better off working out the proof by himself; luckily, most of the proofs are quite elementary.
For the reader patient enough to go through this book simultaneously reading and correcting it, it definitely is a worthy repository of results involving Fibonacci and Lucas numbers.

There's nothing wrong with this book, exactly, but it suffers from being too technical to be a popular book on Fibonacci and too breezy and gimmicky to be a mathematics text. It flits and floats and flies from sub-topic to sub-topic, sometimes visiting the same sub-topic more than once, without discernable pattern. While I have no doubt that that Koshy knows his stuff and his discussion in each sub-chapter is more or less correct, I wish I had a better sense of what's left out.

Fibonacci has always been a dilettante's heaven, and dilettantes need books, like Koshy's could have been, that give them a sense of what's known and what's not. If you're staring into the fire some winter night and suddenly go, Hark!, you need an easy way to decide if your Hark! is worth pursuing, and if so what's already been done. You don't need mathematical rigor, at least not beyond some bare minimum. You need to know if anyone ever bothered to work out a general formula, for general k, for the coefficients of a(0)F(i)^k + a(1)F(i+1)^k + a(2)F(i+2)^k + . . . + a(m)F(i+m)^k = 0. (someone did: see Lewis, More Power to Fibonacci, Mathematics Gazette July 2003.)

A one-volume (affordable) Fibonacci encyclopedia reliably answering the breadth question would be nearly priceless. Koshy's book is not it. You have to buy this book; there's nothing else out there, that I know about, anyway, matching its breadth or depth. And, I suppose Koshy deserves his royalties simply for stepping into the market niche where I was waiting. But there's something missing.

PS. People who know their stuff have warned us about the many typos in this book. I have found a few myself. So, test those Fibonacci identities before you use them, the same way you would test thin ice.