Mathematics

Algebra

• Artin, Michael. Algebra. Prentice Hall, New Jersey, 1991.

  A comprehensive yet chatty introduction suitable for seniors. Implicitly adheres to the the proposition that abstraction must be justified by concrete needs.

Analysis

• Bressoud, David. A Radical Approach to Real Analysis. Mathematical Association of America, Washington D.C. 1994.

  A genetic development of the basic notions of convergence.

• Courant, Richard and Fritz, John. Introduction to Calculus and Analysis I. Springer 1998.

  A reissue of one of the classic calculus texts.

• Steele, Michael. The Cauchy-Schwarz Master Class. Cambridge University Press, New York 2004.

  An engaging introduction to inequalities.

Combinatorics

• Wilf, Herbert. Generating Functionology, 3rd ed.. Academic Press 1994.

  Treats the application of power series to combinatorial problems. The second edition of the book in pdf format is available for free at: http://www.math.upenn.edu/~wilf/DownldGF.html.

Number Theory

• Apostol, T.M.. Introduction to Analytic Number Theory. Springer-Verlag 1976.

  A comprehensive introduction to number theory aimed at advanced undergraduates and graduate students.

• Hardy, G.H. and Wright, E.M.. An Intorduction to the Theory of Numbers, 5th ed.. Oxford Science Publications, Oxford U.K., 1979.

  Comprehensive intorduction to all aspects of elementary number theory.

Popularization (Math and science)

• Conway, J.H. and Guy, R.K.. Book of Numbers. Copernicus, 1996.

  Discusses the many amazing properties of the integers and integer sequences. Graphics in this book are striking.

Probability

• Grimmett, G.R. and Stirzaker, D.R.. Probability and Random Processes. Oxford University Press, Oxford UK 2001.

  Comprehensive introduction to probability good for upper level undergranduates and beginning graduate students.

• Klenke, A.. Probability Theory: A Comprehensive Course. Springer-Verlag, 2008.

  This book is appropriate for a graduate student. It is very thorough and has non-trivial examples.

Non-mathematical Books

(Auto)biography

• Hoffman, Paul. The Man Who Loved Only Numbers. Hyerpion, New York 1998.

  About Paul Erdos.

• Nasar, Sylvia. Beautiful Mind. Simon and Schuster, New York 1998.

  About John Nash.