ABSTRACT
Modern algebra has exposed for the first time the full variety and richness of possible mathematical systems. This chapter constructs and examines many such systems, but the most fundamental of them all is the oldest mathematical system—that consisting of all the positive integers (whole numbers). It assumes eight postulates for addition and multiplication. These postulates hold not only for the integers, but for many other systems of numbers, such as that of all rational numbers (fractions), all real numbers (unlimited decimals), and all complex numbers. They are also satisfied by polynomials, and by continuous real functions on any given interval. When these eight postulates hold for a system R, we shall say that R is a commutative ring. It is a familiar fact that the set Z of all integers satisfies these postulates.
