Stewart, From Here to Infinity: A Guide to Today's
Mathematics. A good view of modern
math, very accessible.
Hofstadter, Metamagical Themas: Questing for the Essence of
Mind and Pattern. Collection of articles
treating subjects from recursion to game theory.
Conway and Guy, The Book of Numbers
Martin Gardner, Any of his math books. They're all great.
Ball and Coxeter, Mathematical Recreations and Essays
Ian Stewart, Game, Set and Math: Enigmas and Conundrums
Dunham, Journey Through Genius. A history of important
theorems and their motivations. Accessible and well-written.
Dunham, The Mathematical Universe. An A to Z on all sorts
of topics in math. Chapter M, entitled "Mathematical Personality" is
Davis and Hirsh, The Mathematical Experience. One of the best and
most accessible books on the philosophy of mathematics -- deals with such
questions as "What constitutes a proof?" It's a lot less straightforward than
you might imagine!
Stillwell, Mathematics and its History
Hardy, A Mathematician's Apology
Stewart, Does God Play Dice?On non-linear dynamics
(more infamously known as chaos theory) and other things. Very accessible, no
equations, doesn't really tell you how to compute these things.
Méro, Laszlo, Moral Calculations. See below under Game
David Wells, The Penguin Dictionary of Curious and
Interesting Numbers and other books by the same author.
Polya, How to Solve it
Stephen Hawking, A Brief History of Time. A very famous popular
book on cosmology
Richard Feynman, Surely You're Joking, Mr Feynman, and other
books by him. This is a collection
of autobiographical stories, and it really great fun and inspiring.
Weisstein, The CRC Concise Encyclopedia of Mathematics. This is a fabulous
handbook of all sorts of mathematical
terms and suchlike, and is not too exorbitantly priced. This used to be available online, but has
been removed due to legalities. There
are bootleg copies available in various places if you can find them.
Daniel Solow, How to Read and Do Proofs
Escher, Visions of Symmetry: Notebooks, Periodic drawings and
Related Work of M.C.Escher
Gian-Carlo Rota, Indiscrete Thoughts. Stories about
Körner, The Pleasures of Counting. This is a very
readable book introducing one to some very interesting branches of applied
mathematics. It has a strong historical
flavour which also discusses the mathematics of such things as the Eureka
machine, competitive processes in biology and algorithms. It also contains information on the
mathematics of bombing U-boats during World War II, which would have formed the
basis of the Operational Research class.
Brualdi, Introductory Combinatorics. An elegant
presentation of introductory combinatorics
Graham, Knuth and Patashnik, Concrete Mathematics. An awesome book, full
of everything you ever wanted to know about how to add and a lot more
besides. Although it is quite advanced
in places, it is a fabulous book. It
also covers a significant amount of combinatorial material. The "How to Add" class was based on Chapter
2 of this book, and recurrence relations are also covered. And if you thought
Fermat wrote hot marginalia... Plus, a review from Meep
Koh and Chen, Principles and Techniques in Combinatorics. A harder textbook, but
with interesting material on many branches of combinatorics including design
theory, generating functions, coding theory and graph theory.
Cameron, Combinatorics. Covers a lot of
interesting material in many different areas of combinatorics ranging from
moderate to quite advanced. The book is
arranged in roughly increasing order of difficulty.
Stanley, Enumerative Combinatorics I and II. Again, an advanced
Bollobás, Graph Theory; Modern Graph Theory. Both of these books
are quite advanced, but they present a lot of information very nicely. There are a lot of good exercises, ranging
from elementary to highly challenging
West, Introduction to Graph Theory
Godsil and Royle, Algebraic Graph Theory. A graduate level
textbook -- not many examples, but lots of cool theory.
Schaum's Outline on Graph Theory
Herbert Wilf, Algorithms and Complexity. Reasonably advanced,
but covers the material we looked at in week four of the Discrete Track and
much more besides. Available online
Schneier, Applied Cryptography: protocols, algorithms and
source code in C. A big thick book, but
easy to read with little difficult mathematics. Lots of references, and a very
solid description of the basics of the subject.
Niven, Montgomery and Zuckerman, An Introduction to the
Theory of Numbers. Great book
Davenport, The Higher Arithmetic
Dudley, Elementary Number Theory
Ore, Number Theory and Its History
Silverman, A Friendly Introduction to Number Theory. A gentle book which
walks through some very interesting math, ending up with a discussion of
elliptic curves and some major theorems relating to them.
Baker, A Concise Introduction to the Theory of Numbers
Hartley and Hawkes, Rings, Modules and Linear Algebra. Although this little
book is out of print, it is worth tracking down in a library. It covers the ED => PID => UFD
material we studied, and goes on to discuss modules over PIDs, concluding with
some beautiful classical structure theorems such as the Jordan Canonical Form
for endomorphisms of a vector space.
Atiyah and Macdonald, An Introduction to Commutative Algebra. This is for scary
people! It's brilliant but very hard.
Downing, Calculus the Easy Way. Contains good stories.
Spivak, Michael. Calculus. This is an awesome
book to learn calculus from. It's one
of the best rigorous introductions to theoretical calculus out there. Look for
jokes in the index.
Apostol, Tom. Mathematical Analysis: A modern approach to
advanced calculus. Covers a lot of
material in a formal way, introducing the basics of analysis, differentiation,
Riemann-Stiltjes integration and Lebesgue integration
Bartle and Sherbert, Introduction to Real Analysis
Silverman and Tate, Rational Points on Elliptic Curves. A fabulous book on the
boundaries between algebraic geometry and number theory
Bix, Conics and Cubics
Reid, Undergraduate Algebraic Geometry
Kirwan, Complex Algebraic Curves. Uses a manifolds and
Cox et al, Ideals, Varieties, and Algorithms: An Introduction
to Computational Algebraic Geometry and Commutative Algebra. An introduction to
algebraic geometry via computational algorithms and Gröbner bases
Cofman, What to Solve? Problems and Suggestions for Young Mathematicians
P'olya, How to Solve it
Larson, Problem-Solving Through Problems
Gilbert, Larson and Mark Krusemeyer, The Wohascum County
Problem Book. That's our Mark!
Shklarsky et al, The USSR Olympiad Problem Book: Selected
Problems and Theorems of Elementary Mathematics
Yaglom and Yaglom, Challenging Mathematical Problems with
Elementary Solutions, vol. 1
Holton, Let's solve some math problems
Gardiner, The Mathematical Olympiad Handbook: An
Introduction to Problem Solving Based on the First 32 British Mathematical
Olympiads 1965--1996. Nice book, providing a
brief introduction with some methods and giving both hints and full solutions
for all of the problems. He's been
involved in Olympiad training and mathematics education in England for many years.
Garnier and Taylor, 100% Mathematical Proof. Good intro to logic in
math proofs and simple proof techniques. If you're shaky on how to prove things rigorously through logic, this is
a good place to start.
Solow, How to Read and Do Proofs
P'olya, Mathematics and Plausible Reasoning. A somewhat more
difficult book on logic and proofs
Hofstadter, Gödel, Escher, Bach: An Eternal Golden Braid. Strange loops, strange
isomorphisms (in a loose sense of the word) between music, biology, logic,
computers, and other similar stuff. Won the Pulitzer Prize. A review from Meep
Johnstone, Notes on Logic and Set Theory. Quite a terse
introduction to logic, ordinal and cardinal arithmetic, but mostly well-written
Cameron, Sets, Logic and Categories. Similar material but less terse and more intuitively presented
Smullyan, any of his books. If you're really into
insane and sane liars and truthtellers, he's your man.
Games Magazine (Or Games World of Puzzles),
Paint-by-numbers. The cutest logic
puzzles one ever did see. Interestingly,
the two-color puzzles are easier than the one-color ones (once you get in the
hang of things).
This is also a useful
source of information. It contains
links to CTAN (the Comprehensive TEX Archive Network), a repository
of hundreds of LATEX -related pieces of software and
complete TEX distributions.
Kernighan and Pike, Programming in the UNIX Environment. They are among
the group who created UNIX. While it's
an old book and doesn't have any of the up-to-date snazzy commands or anything
about the X Windows system, it teaches the philosophy of UNIX and how to make
the most out of it better than any other book I've seen without getting lost in
myriad details. It gives the basis
necessary for going onto anything else. The second part of the book talks about programming in C, which may or
may not be of interest.
Welsh, Running LinuxA nice book on
installing Linux and using it. (I was co-poobah of the computer club with Matt. He really knows his Linux -- Meep)
Fritsch & Fritsch, The Four-Color Theorem. Quite an advanced
book; it expends a significant amount of effort developing the problem from a
topological perspective, finally moving into graph theory and describing the
nature of the actual proof.
Lang, Challenges. A mathematician reads
the newspapers and gets into politics. A disturbing book by a very interesting person. (He was a visiting speaker at MC2K, and had
a great time with Sanjoy Mahajan.)
A CD with all of the
CoRT material is available; an order form can be found on http://www.edwdebono.com/.
Edward de Bono has also written dozens of books on the nature of
thinking, the skill of thinking and other related subjects ranging from very
original to somewhat repetitive; in particular, Six Thinking Hats presents a
very powerful technique for
running meetings and discussions highly effectively (assuming that all of the
participants are willing to follow the system).
Tony Buzan has written
several very good books on mind skills. In particular, Use Your Head, Use
Your Memory and The Mind Map Book contain amazingly powerful ideas. (The former appears to have a different
title in North America: Make the Most of Your Mind.)