Elementary Number Theory in Nine Chapters (Second Ed.) by James J. Tattersall
This is an excellent (and readable) intro text to number theory with proofs to many important number-theoretic formulas and theorems. Aside from theory, there is also an emphasis on applications such as calendrics, representations, and cryptography. From front to back, there are many interesting historical notes that connect the subject to several cultures (including Chinese, Muslim and Indian). I highly recommend this text to anyone who enjoys challenging and enlightening problems.
Explorations in Geometry by Bruce Shawyer
This book draws many problems from the IMO and other math contests, with supplementary information and commentary. Certainly, this book is great for expanding one's horizons to solving challenging geometry problems in the spirit of competitions.
The Works of Archimedes
Forget Euclid's unnecessarily long treatments on trifles, Archimedes is the greater teacher. This book is very readable to the modern mathematician (unlike Newton's Principia) despite being over 2000 years old. The math is rigorous and by no means elementary, though its archaic qualities will make one gasp at how brilliant this man was.
Ancient Puzzles: Classic Brainteasers and Other Timeless Mathematical Games of the Last 10 Centuries by Dominic Olivastro
This book is great for three reasons:
A) You learn a lot about problem solving in discrete math
B) It is a fun read because the problems are presented in recreational and novel ways
C) The book introduces tons of ancient and medieval math outside the Greco-Roman tradition (which is a breath of fresh air)
Three Must Read Books about e and i !!!
e: the Story of a Number by Eli Maor
An Imaginary Tale: The Story of by Paul J. Nahin
Dr. Euler's Fabulous Formula by Paul J. Nahin
These classic books on the number e, i and their pivotal importance to modern math will both captivate and inspire future generations. Any reader will learn a lot about math and the elegance of this long forsaken language.
How to Solve It by G.Polya
Anybody interested in the teaching of mathematics, or teaching in general should read this book! This classic book teaches people that anyone can do mathematics and problem solve, all you need is clear thinking, patience and the right type of guidance (nature and/or nuture).
A History of Chinese Mathematics by Jean-Claude Martzloff
For those interested in mathematics in other cultures, this book is a perfect window into Ancient Chinese Mathematics. Concise but very informative, this book will open new eyes to mathematics from the layman to the expert.
Janos Bolyai: Non-Euclidean Geometry and the Nature of Space by Jeremy J. Gray
This book goes into the details of how Janos Bolyai obsession with Euclid's Fifth Postulate ushered a new age of geometry. The book also comes with Bolyai's groundbreaking dissertation on the Fifth Postulate.