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.
    Recommended for: Algorithm designers, math historians.
     Hardcover, 650 / 762 / 780 pages, Addison-Wesley, 1998, 9.5×6.6×1.6 inches.

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.
   Recommended for: Amateur mathematicians, math historians.
    Paperback, 486 / 803 / 313 pages, Dover, 2005, 8.5 x 5.5 x 1.7 inches.

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 MathWorld link. The book itself, particularly the first edition, is beautiful to leaf through. Do feel free to join the contributors. If you use the browser Firefox, there is a plugin for MathWorld searches, and also for Wolfram Functions searches.
   Recommended for: People that cannot use the MathWorld website.
   Hardcover, 3242 pages, Chapman & Hall/CRC, 2002, 11.4×8.5×3.2 inches.

30 years of math columns from the world's foremost popular mathematics writer. This CD-rom contains all the Mathematical Games columns written by Martin Gardner (Scientific American, 1956 to 1986) on one CD in a searchable database. Type a word in the Search box, press Go and a list of columns will appear.
In these columns, Gardner eloquently describes the delights of mathematics, puzzles and problem solving. His column broke such stories as Rivest, Shamir, and Adelman on public-key cryptography, Mandelbrot on fractals, Conway on Life, and Penrose on tilings. He enlivened classic geometry and number theory and introduced readers to new areas such as combinatorics and graph theory. The search feature is fantastically useful for researchers.
   Recommended for: Everybody. Mandatory for recreational mathematicians. Excellent for young adults.
   CD-ROM PDF, 4500 pages, Mathematical Association of America, 2005, 7.8×5.0×0.8 inches

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 MathWorld. The pages have wonderfully concise definitions and are packed with important information on each subject. Many useful diagrams help to explain the material. Note that although it is a good handbook for all areas of college math, it isn't a textbook. For example, pages 299-302 on Group theory would fill 150 pages of an introductory textbook on the subject, and Galois isn't mentioned in those pages. It's the mathematical facts — just as a handbook should be.
   Recommended for: Everybody. Useful from high school to college and beyond.
   Paperback, 1160 pages, Springer, 2003, 8.2×5.5×1.7 inches.

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.
   Recommended for: Function historians and Engineers.
   Paperback, 1046 pages, Dover Publications, 1974, 10.4×8.0×1.6 inches.

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 x and y are elements of an ordered set L, the supremum and infimum of {x,y}, whenever they exist, are called the join and meet of x, y and denoted by x∪y and x∩y, respectively. L is called a lattice (or lattice-ordered set) when every pair of its elements has a join and a meet." (Most of the book is even more abstract.) An in-depth discussion follows, with a bibliography of classic texts at the end. An excellent set of books, but not for beginners.
   Recommended for: High-level mathematicians.
   Paperback, 2148 pages, MIT Press, 1993, 12.6×6.3×4.0 inches.

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.
   Recommended for: Students, amateur mathematicians, math historians.
   Hardcover, 310 pages, Springer, 1996, 9.5×6.4×0.8 inches.

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 Numerics & Symbolics). For example, I looked up LATTICES, and got links to Brillouin zones of ~, charged ~, counting points in ~, cubic ~ in 3D, cubic and hexagonal ~ in 2D, morphing of ~, points visible in ~, superpositions of ~, vortex ~, ~ Green’s function and Mathematica function LatticesSuperposition. With Eric Weisstein, I wrote a column on the GuideBooks, which shows many sample graphics. The GuideBooks website mathematicaguidebooks.org contains a detailed table of contents, sample chapters, a full index, and considerable additional material. The additional material alone contains 50 detailed notebooks. To sample the additional material, or a DVD from the book itself, you can download a trial version of Mathematica, which serves as a fully functional reader after the trial expires.
   Recommended for: Mathematica users, illustrators, math graphic fans, physicists.
   Hardcover, 1340 / 1209 / 904 / 1454 pages, Springer, 2004 / 2005, 9.4×7.5×2.1 inches.

Some of the most popular math books have been about single constants, such as e, π, i, and φ. There are hundreds of other mathematical constants. They are all lovingly described by Finch in either his book and at his supporting site. Finch has also contributed tremendously to MathWorld Constants section. Charles Ashbacher: "I consider this book to be an essential component of all mathematical libraries. I have placed it on my 'within the grasp' shelf and have strongly recommended to the college library that it be added to the reference collection." Myself, I wind up needing to look up obscure constants quite often, and always consult Finch.
   Recommended for: Anyone that likes constants.
   Hardcover, 622 pages, Cambridge University Press, 2003, 1.8×6.5×9.5 inches.

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.
   Recommended for: Students, amateur mathematicians, math historians.
   Hardcover, 412 pages, Prentice Hall, 1999, 9.3×6.3×0.8 inches.

"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 OED Historical Introduction. It's amazing how often I need to look up the history of a particular word, and OED provides. For example, Proclus, who wrote the commentaries for Euclid's Elements, used the greek form of the word TRAPEZIUM for quadrilaterals with exactly two sides parallel, and the word TRAPEZOID for a quadrilateral with no sides parallel. In 1795, Hutton's Mathematical Dictionary accidentally swapped the definitions, and these became the standard definition in the United States and some other countries. Thus, there is no good universal definition for TRAPEZOID and TRAPEZIUM. The definition depends on the country you live in. Indeed, it depends on what a given teacher expects. I had no idea this raging 200+ year educational war even existed before looking it up in the OED.
   Recommended for: Math historians, anyone that likes good references.
   Hardcover, 2416 pages, Oxford University Press, 1991, 19.0×13.0×5.0 inches.

"In spite of the large number of published mathematical tables, until the appearance of the first authors A Handbook of Integer Sequences in 1974 there was no table of sequences of integers. The 1974 book remedied this situation to a certain extent, and the present work is a greatly expanded version of that book. The main table contains 5488 sequences of integers (compared with 2372 in the first book), collected from all branches of mathematics and science. The sequences are arranged in numerical order, and for each one a brief description and a reference is given. An invaluable tool. I shall say no more about this marvelous reference except that every recreational mathematician should buy a copy forthwith." — Martin Gardner in Scientific American.
   Recommended for: Fans of the OEIS website and reference collectors.
   Hardcover, 587 pages, Academic Press, 1995, 9.3×6.3×1.2 inches.

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.
   Recommended for: Everyone, especially students.
   Hardcover, 790 pages, Van Nostrand Reinhold, 1975, 9.4×6.6×1.9 inches.

MathWorld — A huge encyclopedia of mathematics by Eric Weisstein.
History of Mathematics — Jeff Miller's "earliest usages" list.
OEIS — Over 100000 important sequences, maintained by NJA Sloane.
Free Online Math Journals — by combinatorics.org.
MacTutor History of Mathematics — Math biographies.
Free Online Math Textbooks — Maintained by Alex Stef.
Cut the Knot — by Alex Bogonolmy. Lots of good mathematics.

Functions — The Wolfram Functions site.
Geometry Junkyard — by David Eppstein.
Mathematical Constants — by Steven Finch.
Math Symbols — by metamath.org.
Math ArXiv — thousands of online math papers.
Math Pronunciations Guide — by Kent Kromarek
Google — For everything else.