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The famous twentieth century mathematician Paul Erdős spoke of “The Book,” a tome existing in heaven where God collects the most elegant mathematical theorems. The theme of beauty in mathematics is expounded at some length in the prominent Oxford mathematician G. H. Hardy’s autobiography of 1940, “A Mathematician’s Apology.” Hardy contended that only beautiful mathematics has a right to exist and, moreover, the mathematics that falls into this category is almost exclusively of the “pure” variety, existing for its own sake and devoid of any real-world applications.1 Hardy characterizes what he considers the best mathematical proofs as something akin to great music and poetry, exhibiting a seemingly paradoxical combination of inevitability and surprise. He cites as examples Euclid’s proof of the infinitude of the prime numbers and the Pythaoreans’ proof of the irrationality of √2. Hardy notwithstanding, mathematics is valued today both for its esoteric qualities and as a tool that has played a fundamental role in creating the modern world. Viewed in this way, the discipline appears as one of the most magnificent creations of the human mind. But mathematics also has a human dimension. The men and women who created this vast body of work were just that, men and women. They had wives, husbands, children, or in some cases, they didn’t. They were gamblers, philanderers, judges, clerics, soldiers, poets, writers, and rakes. They suffered, they triumphed, they starved, they courted the favor of nobles, they lorded it over each other at times, some were even killed. It is the purpose of this book is to present mathematics in both its humanistic and scientific aspects, although being mathematicians rather than historians, our focus is largely on the latter. So, welcome to our tour! There are indeed beautiful vistas. If the terrain gets a little rocky in places, don’t worry, just stay close and let us be your guide. The landscape is divided into subject areas. The first two chapters are on Geometry and Number Theory. Chapter 3, on Medieval and Renaissance Mathematics, is transitionary in nature, bridging the gap between “old mathematics” and the invention of the subject’s greatest tool, calculus. Chapter 4 concerns Algebra, told through the history of polynomial equations. Chapter 5 provides an introduction to Calculus proper, with discussions of the main ideas and theorems. Chapter 6 is an account of Complex Variables, the calculus of functions defined using complex numbers. Chapter 7 addresses Graph Theory, a subject that originated with Euler’s solution to the bridges of Königsberg problem, and which has grown into a major branch of mathematics, with numerous applications in areas ranging from economics to computer networks. Chapter 8 is concerned with Probability and traces the development of the subject from its ancient roots in gambling to its role as the foundation of mathematical statistics. Finally, Chapter 9 deals with the subject of the infinite, mathematical logic, and elements of computer science, from the theory of countability originated by Cantor, through Gödel’s revolutionary work on undecidable propositions, to Turing’s work on computing machines and computable numbers.