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The discovery-based Constructing Numbers: An Inquiry-Based Capstone Mathematics Course helps students synthesize material from elementary analysis, linear algebra, algebraic structures, and geometry while providing perspective on how mathematical thought has developed. The book opens with the history of numbers. It moves on to cover axiomatic systems, real, complex, and constructible numbers, and an algebraic formulation of field construction. The final chapters are devoted to finite fields, probability, and permutation groups. By actively participating in the exercises and proofs, students learn how the material progresses in a logical way. They master the content by doing the work and presenting their results. Working through Constructing Numbers gives students a deeper understanding of the discipline of mathematics. They understand the importance of moving beyond a strictly mechanical approach to "doing" mathematics and to instead ask "why" mathematical processes are as they are. Intended for one-semester upper-division classes, Constructing Numbers can be used in capstone courses in the major, or classes for pre-service secondary mathematics teachers.