cover -- copyright page -- title page -- Introduction -- Contents -- I THE REALS -- Axioms for the Reals -- How to Build a Number System -- The Field Axioms -- The Order Axioms -- The Completeness Axiom -- Construction of the Reals -- Cantor's Construction -- Dedekind's Construction of the Reals -- Uniqueness of the Reals -- The Differential Calculus -- A Final Word about the Reals -- II MULTI- DIMENSIONAL NUMBERS -- The Complex Numbers -- Two-Dimensional Algebra and Geometry -- The Polar Form of a Complex Number -- Uniqueness of the Complex Numbers -- Complex Calculus -- A Final Word about the Complexes -- The Quaternions -- Four-Dimensional Algebra and Geometry -- The Polar Form of a Quaternion -- Complex Quaternions and the Quaternion Calculus -- A Final Word about the Quaternions -- III ALTERNATIVE LINES -- The Constructive Reals -- Constructivist Criticism of Classical Mathematics -- The Constructivization of Mathematics -- The Definition of the Constructive Reals -- The Geometry of the Constructive Reals -- Completeness of the Constructive Reals -- The Constructive Calculus -- A Final Word about the Constructive Reals -- The Hyperreals -- Formal Languages -- A Language for the Hyperreals -- Construction of the Hyperreals -- The Transfer Principle -- The Nature of the Hyperreal Line -- The Hyperreal Calculus -- Construction of an Ultrafilter -- A Final Word about the Hyperreals -- The Surreals -- Combinatorial Games -- The Preferential Ordering of Games -- The Arithmetic of Games -- The Surreal Numbers -- The Nature of the Surreal Line -- More Surreal Numbers -- Analyzing Games with Numbers -- A Final Word about the Surreals -- Bibliography -- Index -- About the Author.