Introduction vii

This text also goes further and presents proofs for many of the results

discussed. For instance, the reader can find, under the relevant entries, a

proof to the fundamental theorem of algebra, a proof of Descartes’s law of

signs, a proof that every number has a unique prime factorization, a proof

of Bretschneider’s formula (generalizing Brahmagupta’s famous formula),

and a derivation of Heron’s formula. Such material is rarely presented in

standard mathematical textbooks. In those instances where the method of

proof is beyond the scope of the text, a discussion as to the methods

behind the proof is at least offered. (For instance, an argument is presented

to show how a formula similar to Stirling’s formula can be obtained, and

the discussion of the Cayley-Hamilton theorem shows that every matrix

satisfies at least some polynomial equation.) This encyclopedia aims to be

satisfying to those at all levels of interest. Each entry contains cross-refer-

ences to other items, providing the opportunity to explore further context

and related ideas. The reader is encouraged to browse.

As a researcher, author, and educator in mathematics, I have always

striven to share with my students the sense of joy and enthusiasm I expe-

rience in thinking about and doing mathematics. Collating, organizing,

and describing the concepts a high-school student or beginning college-

level student is likely to encounter in the typical mathematics curriculum,

although a daunting pursuit, has proved to be immensely satisfying. I

have enjoyed the opportunity to convey through the writing of this text,

hopefully successfully, a continued sense of joy and delight in what math-

ematics can offer.

Sadly, mathematics suffers from the ingrained perception that primary

and secondary education of the subject should consist almost exclusively

of an acquisition of a set of skills that will prove to be useful to students in

their later careers. With the push for standardized testing in the public

school system, this mind-set is only reinforced, and I personally fear that

the joy of deep understanding of the subject and the sense of play with the

ideas it contains is diminishing. For example, it may seem exciting that we

can produce students who can compute 584 × 379 in a flash, but I am sad-

dened with the idea that such a student is not encouraged to consider why

we are sure that 379 × 584 will produce the same answer. For those stu-

dents that may be naturally inclined to pause to consider this, I also worry

about the response an educator would give upon receiving such a query. Is

every teacher able to provide for a student an example of a system of arith-

metic for which it is no longer possible to assume that a×band b×aare

always the same and lead a student through a path of creative discovery in

the study of such a system? (As physicists and mathematicians have discov-

ered, such systems do exist.) By exploring fundamental questions that chal-

lenge basic assumptions, one discovers deeper understanding of concepts

and finds a level of creative play that is far more satisfying than the perfor-

mance of rote computation. Students encouraged to think this way have

learned to be adaptable, not only to understand and apply the principles of

a concept to the topic at hand, but also to apply those foundations and

habits of mind to new situations that may arise. After all, with the current