From Counting to Calculus: A Complete Course in Mathematics

A unique approach to mathematics covering all of the major topics from simple counting through calculus, including an introduction to differential equations. Starting with counting, all of the operations of arithmetic and the corresponding systems of numbers are developed as a single, interconnected framework. This framework is then used as a foundation for the construction of algebra and calculus. Each new topic is introduced as a logical extension of the topics that came before it, and is developed thoroughly and rigorously with the reader as if it was being invented for the first time. Although it is assumed that the reader is familiar with arithmetic and has had some exposure to algebra, proficiency with mathematics is not required. The conversational style and step-by-step approach make it easy to follow the flow of ideas, and numerous exercises sprinkled throughout allow readers to test their understanding before proceeding to the next topic. Among the topics covered are the additive and positional number systems, the operations of arithmetic, integer and non-integer exponents, fractions, rational and irrational numbers, real and complex numbers, algebraic solutions of equations, simultaneous equations, graphs and graphical solutions of equations, constructing polynomial equations from data, finding roots of polynomial equations, functions and inverse functions, differential calculus including the sum, product, and chain rules, integral calculus including proper and improper integrals, and an introduction to ordinary and partial differential equations, with applications to the physical sciences. Problems at the ends of the chapters, along with their solutions, provide the opportunity to practice methods discussed in the text, and explore important topics in more depth. The choice of subject matter and method of presentation makes this an ideal text for a high school or college level course, or as a self-teaching guide for the general reader interested in developing a deeper understanding of mathematics.