Recommended Reading: Math

This is a short list of books that I like. The links in the title go to Amazon, and help with the hosting of my website.

Number Theory

Carl F. Gauss, "Disquisitiones Arithmeticae", Yale University Press, 1965. This book could probably also fit under the historical section. I think the famous author Gauss needs no introduction. Gauss published this book in 1801, and in it he establishes the triple parallel bar notation for congruences, as well as quadratic reciprocity and the construction of regular n-gons using a ruler and compasses. A very engaging and surprisingly modern book by one of the greatest mathematicians in history. This is a well-done translation from A.A. Clarke from the original latin.

Mathematical Technique And Problems

Under this category I include books that are for sharpening your mathematical technique.

G.H. Hardy, J.E. Littlewood, G. Pólya, "Inequalities, 2nd Edition", Cambridge University Press, 2001. First published in 1934, this classic c. 300 page text treats inequalities from areas like infinite series, integrals, and the calculus of variations. This is not an easy book in the sense that the proofs require some sophistication to follow as many details are not present, and thus it forces the reader to go over every point. The notation used in the book is also quite antiquated (think Gothic letters), and I well advise any reader to rewrite the theorems and proofs in modern notation.

Analysis and Topology

Lynn Arthur Steen and J Arthur Seebach, Jr, "Counterexamples In Topology", Dover, 1995.Steen and Seebach write a very interesting book. The book falls into four main parts: definitions, counterexamples, metrization theory, and some appendicies with charts. The counterexamples section is the bulk of the book and it contains numerous spaces with odd properties.

Dmitry Kozlov, "Combinatorial Algebraic Topology", Springer, 2008.There's much detail here on topological spaces that have a cell structure, like simplicial complexes and CW Complexes.

Jean Dieudonné, " A History of Algebraic and Differential Topology 1900-1960", Birkhauser, 2009.A highly recommended book for anyone interested in topology. This heavily mathematical and historical book shows all the twists and turns taken by topologists to create the pristine subject we know today.

Charalambos D. Aliprantis and Kim C. Border, "Infinite Dimensional Analysis, A Hitchhiker's Guide, 3rd ed.", Spring Verlag, 2006.This book is a terse but readable roadmap on topics of analysis and topology. It manages to cram the elementary but important results from topology, measure theory, metrizable spaces, topological vector spaces, normed spaces, convexity, Riesz spaces, Banach lattices, charges and measures, integrals (including the Lebesgue, Riemann, Gelfand varieties), measures and topology, L_p spaces, the Riesz representation theorems, probability measures, spaces of sequences, correspondeces, measurable correspondences, Markov transitions, and ergodicity.

Godfrey Harold Hardy, "A Course Of Pure Mathematics, 10th ed.", Cambridge University Press, 2004.This is a classic text on analysis, written when there were no rigorous books in England on analysis. The notation is outdated but entirely rigorous. This is a superbly written text with many harder problems from the Tripos as exercises, often with hints.

Algbera

Pierre Antoine Grillet, "Abstract Algebra, 2nd Edition", Springer, 2007. This contains fairly typical first-year graduate material including chapters on semisimple rings and modules and some homological algebra. The not too infrequent humour is greatly appreciated.

Saunders Mac Lane, "Homology", Springer, 1963. Mac Lane's introduction to homology is a bit outdated now but nevertheless is well-written.

Linear Algebra

Kenneth Hoffman and Ray Kunze, "Linear Algebra, 2nd Edition", Prentice Hall, 1971. A good treatment of a typical honours linear algebra course, covering foundations along with canonical forms, symmetric operators, bilinear forms, and tensor products. It could definitely use more examples in its explanation of the Jordan normal form.

Foundations And Logic

Thomas Jech, "Set Theory, 3rd Mil. Ed.", Springer-Verlag, 2006.This is a great and thorough book for those really interested in set theory. The proofs are advanced and terse, and I can't comment on the more advanced sections.

Biographies

Goro Shimura, "The Map Of My Life", Springer, 2008.Although this isn't a complete autobiography, it's an entertaining and often candid selection of stories, and some of Shimura's thoughts about the mathematical community.

General and Historical Mathematics

W.W.R. Ball, "A Short Account of the History of Mathematics", Sterling Publishing Company, Inc., 2001. A comprehensive treatment of the history of math from the earliest known point until about 1900. It is a fairly compact account at 500 pages and focuses mainly on mathematical detail with little biographical information.

Heinrich Dorrie, "100 Great Problems of Elementary Mathematics", Dover, 1965. Another history book, this time covering 100 important problems in mathematics. The book is divided into the following sections: arithmetical problems, planimetric problems, problems concerning conic sections and cycloids, nautical and astronomical problems, stereometric problems, and extremes. Contains complete solutions, but the solutions aren't always elementary and occasionally cite well-known theorems.

Ian Stewart, "From Here To Infinity", Oxford University Press, 1996. I can't say enough about this book. I am indebted to Ian Stewart for writing this, because it was quite influential in my decision to pursue higher mathematics. This book is very well written, and has 19 chapters, each one focusing on a specific area or famous problem of mathematics. It's written for the layperson, and only requires some high school knowledge to enjoy. From Here To Infinity is entertaining and fun, and is a must-read for any budding mathematician. The chapters cover things like cryptography, the four-colour theorem, prime numbers, and fractals.