Ed Pegg Jr., January 4, 2006

"Gentle Reader: This is a handbook about TeX, a new typesetting system intended
for the creation of beautiful books — and especially for books that contain
a lot of mathematics." (Donald Knuth, Preface for *The
TeXBook*.)

At the start of the last century, typography required long hours of work with either lead blocks or gas-powered linotype machines. The average life expectancy of a printer was 28 years. About half of them contracted tuberculosis. Even as late as the 1970's, the Environmental Protection Agency had to increase the acceptable amount of lead in a worker's bloodstream due to every Linotype operator exceeding the suggested limit. The 1970's also marked a huge change in printing as publishers replaced linotype experts with computer typography.

Unfortunately, for mathematics, the new printing methods looked
terrible. Donald Knuth took a look at the galleys of *The Art of Computer
Programming* and decided they were unpublishable. He figured he could fix computer
typography in about six months. The actual time was ten years, but his creation of TeX allowed mathematical
books to be beautiful again.

For this month's column, I thought I'd look at some mathematical references that I've found particularly useful — the books on my closest shelf that I wind up using regularly. I wear a lot of mathematical hats, and need references to answer dozens of questions every day. Typically, I can find a reference which has the definitive answer, and that will settle things. I'll start with the first beautiful reference of the modern era.

My favorite volume here is vol. 2, Seminumerical Algorithms, with lots of great info on random numbers, how computer numerics work for addition and multiplication, different number systems, factorization, and polynomials. The sorting algorithms of vol. 3 are also fascinating. Knuth's books
are meticulously indexed and famously accurate (if you can find an error, Knuth
will pay you $2.56). TAoCP vol 4 Fascicles
2,
3, and
4 are also available. |

That's not to say that all good references are modern. Dover
Publications regularly
brings back the classic works of the past at reasonable prices. Such Dover titles as *Amusements in Mathematics* (Dudeney), *Pearls in Graph Theory* (Hartsfield and Ringel), and *The Four-Color Problem * (Saaty and Kainen) are all marvelous, inexpensive books, but I don't use them very much. Often, the indices of
these older books aren't the best, and I have to find the index amongst the usually
outdated Dover catalog. Hopefully, Dover will one day offer modern indices
for older classics. One set of books stands above the others as a beautiful,
valuable reference.

Originally published in 1919-1923, these three books cover almost every number
theory idea from the beginning of mathematics until 1920. Vol 1, Divisibility and Primality, covers prime numbers, Farey sequences, perfect numbers, Fermat numbers, and much more. Vol 2, Diophantine Analysis, covers right triangles, sums of powers, and Fermat's last theorem. Vol 3, Quadratic and Higher Forms, covers exactly that. Each page lists roughly five interesting facts, all referenced by footnotes. |

The reference I use the most is the website *MathWorld*.
Full disclosure: I'm now the associate editor. Eric originally designed his Treasure Troves to be a website with occasional printed snapshots. CRC was interested, and printed the first book. Unfortunately, CRC
then shut *MathWorld* down, which led to immediately to the creation of *PlanetMath*
(a good site for long proofs), and spurred greater development on *Wikipedia*.
More than a year later, the courts allowed *MathWorld* to return. I am personally boycotting all CRC books until they make amends. The *MathWorld* website
is much better than the book, but I'll still recommend it:

An encyclopedia on everything in mathematics. Back in 1998, I was drawn to Eric's work by his commitment to making good
diagrams of recreational math concepts. I soon started contributing things, like pentagon tiling and snarks. More lazily, whenever there was a concept I wanted to mention on a webpage but not explain in detail, such as Heron triangles, I would just make a |

Martin Gardner wrote the books and columns that got me started on mathematics.
Back in the seventh grade, I searched through the microfiche of *Scientific American* back issues in the school library to find and read all his columns. The task took weeks. At the end of it, I had an unshakable love for mathematics. Now that his 25 years of columns are available on a single CD, I cannot recommend
this next reference enough:

30 years of math columns from the world's foremost popular mathematics
writer. This CD-rom contains all the |

A recent addition to my nearest bookshelf has long been a classic in Europe. The *Oxford Users' Guide to Mathematics* is also based on the the original Bronshtein book, and I have both — I prefer the *Handbook of Mathematics*. I also consider it superior to the *Mathematics Handbook* (Råde & Westergren), the *Handbook of Mathematics and Computational Science* (Harris & Stocker), and the *CRC Standard Mathematical Tables and Formulae* (Zwillinger).

With a nice small size, high quality assembly, beautiful layout, and an extensive 50 page index, this is now the first book I turn to after |

For a classic, dry, book with long lists of lots of functions, I generally recommend
the *Wolfram Functions* site. The formula
search is particularly useful when someone believes they have a new formula.
For an actual book of function tables, Abramowitz is one of the better ones.

This book is inexpensively priced from Dover, and is also in the public
domain. Several websites feature free copies of it, such as http://www.math.sfu.ca/~cbm/aands/. Take a look at it (perhaps via convertit.com) before considering a purchase. The NIST plans to replace
Abramowitz (as it is usually called) with a new version. I perhaps sound slightly hostile towards the book here, but I do wind up referencing it frequently. |

A high-level, nearly diagram-free reference produced by the Mathematical Society of Japan is another book I frequently pull down.

This book aims at high level of mathematical knowledge. For example, if I looked up Lattices, I would need to find it in the index first, leading to section 243. "Definition: When |

Richard Guy has helped to produce three books I use regularly. One he wrote with
John Conway is *The Book of Numbers*, which uses an easy, friendly tone
to describe scads of good mathematics.

This book seems deceivingly simple and breezy. Lots of illustrations, gentle sentences, and very readable commentary. However, there is a lot of good math in the book, enough that I wind up referencing it frequently. The core topic of the book is numbers, in all their varieties. From a random page: "Delos is an island in the Greek archipelago, once famous as the reputed birthplace of Apollo and Artemis. The story is told that when a plague was raging at Athens, the inhabitants sent an emissary to ask the oracle of Apollo at Delos what to do. The oracle replied that the plague would cease if the altar to Apollo were exactly doubled in size." A rapt discussion of algebraic numbers follows. |

Guy's other inescapable books involves famous unsolved problems. I also adore
the *Winning Ways* books he wrote with Conway and Berlekamp, but I don't often get
a reason to reference them.

An interesting fraction of the mail I receive comes from people desiring a discussion of various unsolved problems. The Unsolved Problems series can usually bring these correspondents up to speed on what is known. I've been guilty of that myself— pondering some problem on my site, only to get a friendly letter from Guy to see a certain section which explains all that is known on the topic. As new revisions come out, he records the progress that has been done for all these interesting problems.Recommended for: Amateur mathematicians, unsolved problem solvers. Hardcover, 438 pages, Springer, 2004, 9.7×6.4×1.0 inches. |

Michael Trott is mostly known for making lots of gorgeous images. Less known are
his recently published *Mathematica GuideBooks*, which fully explain, extensively
reference, and provide commented code for thousands of mathematical images.

A truly gorgeous set of books, with about 5000 pictures, a thousand exercises, 11000 hyperlinked references, and commented code for producing everything. For each image, there is a leading discussion, and a list of related references with more information. Also, here's a small secret: if you buy any one of the books, the included DVD has all four books on it within a fully searchable set of Mathematica notebooks (updated versions in |

The *Mathematical
Constants* website is much smaller than it used to be (due to a court order, sigh), but all of the richness is now
available in book form.

Some of the most popular math books have been about single constants, such as |

One small book does an admirable job of containing all of the must-know mathematics in one compact text.

This is a textbook, and a slightly pricey one at that. From the preface: "This book arose from discussions about the undergraduate mathematics curriculum. We asked several questions. Why do students find it difficult to write proofs? What is the role of discrete mathematics? How can the curriculum better integrate diverse topics? Perhaps most important, why don't students enjoy and appreciate mathematics as much as we might hope?" Then there is a preface for the student, with 37 classic math problems, many from the rec.puzzles FAQ. The problems serve as a springboard to all the important concepts of college math, all of which are concisely explained, along with almost a thousand exercises. When I need a definition, I often find the simplest and best definition here. |

I regularly get asked for the history of a word. For that, one gigantic book proves worthwhile about twice weekly.

"If there is any truth in the old Greek maxim that a large book is a
great evil, English dictionaries have been steadily growing worse ever since
their inception...."— From the |

Here's a beautiful book that I'll actually discourage. Through the assistance of thousands of professional and amateur mathematicians, the Online Encyclopedia of Integer Sequences website has expanded vastly beyond the original book. I use OEIS every day.

"In spite of the large number of published mathematical tables, until the appearance
of the first authors |

The following book has long been out of print, but it's been one of my favorites for more than a decade. It seems to be readily obtainable (Amazon currently shows 30 copies) for around $30, which is a great price for a reference of this quality.

An amazing book spanning almost all of mathematics. Can be described as a textbook without any exercises. In addition to the beautiful layout, color is discretely and extensively used to highlight key points. Simpler concepts, such as addition, get written at an easier level than more difficult concepts, like spherical trigonometry. Profuse illustrations are shown throughout the book. In the back, pictures of many famous mathematicians are shown. |

I personally found advanced Analysis brutally difficult — I doubt I'd have gotten through the following text without lots of good instruction from my professor, Rinaldo Schinazi. Extremely difficult courses can truly be appreciated only when one is standing atop the conquered mountain.

This is a hard book, not very suitable for self-study. If you fully understand the book, you're a mathematician. It has a lot of excellent examples, discussions, definitions, and proofs. When I need to get the wording of an analysis definition exactly right, I'll consult Rudin first. Recommended for: Graduate students struggling with analysis. Hardcover, 342 pages, McGraw-Hill, 1976, 9.3×6.2×0.8 inches. |

I frequently get asked for how to solve math problems. The following
text by Engel is superb. It's now considered a mandatory text for students competing seriously in math olympiads. Another good book of this type is *Mathematical Circles* (Fomin).

Most math texts provide a grounding in mathematics, but not in problem solving. This book provides lots of good instruction on how to tackle seemingly impossible problems. The chapters outline the various strategies: the invariance principle, coloring proofs, the extremal principle, the box principle, enumerative combinatorics, number theory, inequalities, the induction principle, sequences, polynomials, functional equations, geometry, games, and further strategies. Many of the listed problems have deep theory behind them. For example, find Euler's proof for the following: If n ≥ 3, then 2 can be represented in the form 2^{n} = 7 ^{n}x^{2} + y^{2} with odd integers x and y. I often need to explain these same concepts.Recommended for: Students, amateur mathematicians, math olympians. Paperback, 403 pages, Springer, 1999, 9.2×6.2×0.9 inches. |

I'll finish with a list of some of my favorite online references, some already mentioned.

MathWorld — A huge encyclopedia of mathematics by Eric Weisstein. |
Functions — The Wolfram Functions site. |

To some extent, I hope to answer many common math questions with this column. If you are a student and have a question about a math, I recommend the
team at Ask Dr. Math. For help with algebra and calculus problems, I recommend calc101.com. An excellent computer excursion into mathematics is Stan Wagon's *The Mathematical Explorer*.

Thus ends my list of beautiful references, hopefully with enough caveats that I'll lead no-one astray. A list of beautiful math books would be considerably longer. Here is one of my older book lists, for example, to which I need to add such beautiful books as *Set Theory* (Jech), *A New Kind of Science* (Wolfram), *Triangle Centers and Central Triangles* (Kimberling), and *Puzzle Cyclopedia * (Nikoli). I'm always on the look out for beautiful reference books and websites, so feel free to recommend something to me.

Gene Gable, "Heavy Metal Madness," http://www.creativepro.com/story/feature/19578.html.

Tug.org, "Just What is TeX?" http://www.tug.org/whatis.html.

Woodsidepress.com, "The Linotype," http://www.woodsidepress.com/LINOTYPE.HTML.

Comments are welcome. Please send comments to Ed Pegg Jr. at ed@mathpuzzle.com.

Ed Pegg Jr. is the webmaster for mathpuzzle.com.
He works at Wolfram Research, Inc. as an associate editor of *MathWorld*.