The sixteen chapters of this book provide the core material for the Elements of Mathematics Program, a secondary sequence developed for highly motivated students with strong verbal abilities. The sequence is based on a functional-relational approach to mathematics teaching, and emphasizes teaching by analysis of real-life situations. This text is intended to provide students with a general background in the analytical methods of mathematics and practice in reasoning with mathematical objects; other texts (Books 1-12) prepared for use with book O explore particular areas in depth. Chapter 1 begins with a discussion of simple codes; more complex codes are developed by using modular arithmetic. Addition, subtraction, and multiplication are developed for clocks and applied to time zone problems. Operations are then presented in a "machine" setting; discussion of operations includes not only those of standard arithmetic but also explicit consideration of minimum, GCD, and others as operations. Finally, the notion of operations is presented formally and properties (e.g. commutativity, invertibility) are discussed in an abstract setting. In each section the student is provided with a broad range of examples and problems. (SD)