Numbers

In preparing this monograph I had three objectives. First, I wanted to introduce the reader to some topics in mathematics that seldom receive coverage in typical high school and college math programs. The topics include axioms, sets, logic, truth tables and plausible reasoning. In the sections on logic and plausible reasoning, I wanted the reader to see how to transition from formal (mathematical) logic to plausible logic when analyzing the reliability of a source and the credibility of its information content. Readers whose formal education did not cover these topics were not given the opportunity to develop the skills necessary to compete successfully in the world of finance, business and management. These readers will find the information on sets, logic, truth tables and plausible reasoning especially useful. Included are examples that show how the new analysis skills can help analysts draw conclusions and make important decisions from subjective information supplied by less than reliable sources. Second, I wanted the reader to see how subjects in the foundations area of mathematics are used to develop the real number system and its extension through transfinite cardinal numbers. The development of the number system starts with a description of the history of numbers. Readers will find the history both interesting and understandable. The real number continuum is identified as consisting of seven sets of numbers. Each set of numbers can stand alone. The number sets include the simple to understand natural numbers to the more abstract transcendental numbers. Each set is defined and included in a vocabulary consisting of the natural numbers N, integers Z, the rational numbers F, the algebraic numbers A, transcendental numbers T, irrational numbers I, and real numbers R. Venn diagrams are used to explain the relationships existing among the seven sets. The relationships allow the reader to understand the role played by sets and logic in the development of the number system. Included In the development of the real number system are examples of base2 numbers and the algorithms used to convert between base 2 and base 10 numbers. Power Sets are introduced to show how the size of sets can be increased exponentially beyond the cardinal numbers N0 and c. Finally, through exponentiation, cardinal numbers are generated beyond the N0< c < f sequence. Third, I wanted this monograph to appeal to those adults, and their children, who have an anti-math bias. This bias is exhibited as innumeracy or an aversion to math. In either case, those afflicted find it difficult to compete against the mathematically literate in the world of business, finance and technology. Through this monograph I attempt to address this anti-math bias. The reader is introduced to the language of sets, logic and plausible reasoning. While these subjects are part of the foundations of mathematic, they are also subjects taught in Philosophy departments without math prerequisites. The reader is then shown how axioms, sets and logic are applied in the development of the number system. The subject of numbers is made intelligible for a broader spectrum of readers through the use of verbal descriptions and graphics, rather than equations, wherever possible. To achieve a continuous flow of understandable subject matter, the more tenuous procedures and methods of mathematics are explained in 6 appendices. Through understanding how math works at the fundamental level, and not having to work at math, the intelligent readers anti- math bias should be reduced, if not, eliminated.