Numbers and Number Systems

Abstract

Every intelligent person has a background which includes experiences with numbers. These experiences go back to preschool days. A person learns to count before he learns to read. His concept of number grows from a concept which includes only the counting numbers to one which includes fractions, decimals, irrationals, and perhaps negative numbers and complex numbers. These are the numbers of elementary mathematics. These are the numbers of science and of business. These are the numbers of a world equipped with electronic computers, long-distance power-transmission lines, dial telephones, guided missiles, and space satellites. Everyone takes numbers for granted. Everyone uses numbers as tools. Everyone has a number sense, an intuition about numbers and their relationships. But not everyone has spent time thinking about the nature of numbers. Our purpose in this paper is to focus primary attention upon numbers as objects of deliberate mature thought. We outline briefly the development of the number concept and the organization of numbers into number systems. We hope to stimulate the reader, to help him organize his past number experiences and perhaps help him to see something new and exciting about numbers. A system of numbers consists of a set or collection of numbers and one or more operations for combining the numbers of the system. The numbers of the system sometimes are called the elements of the system. The system of counting numbers consists of the elements 1, 2, 3, * ?, and the fundamental operations (addition, subtraction, multiplication, and division) for combining tnem. The symbol 1, 2, 3, .., suggests that there are an infinite number of elements; it also suggests that they have a natural order. This is the order in which they are used as counters, Following the natural number 1 is the natural number 2; 2 is sometimes called the successorofi. Then 2 is followed by 3, 3 is followed by 4, and so on; 9isfollowed by 10, 10 by 11, and so on; 1957 by 1958, and so on, ad infinitum. Although all of the natural numbers have never been written in symbols, they have been conceived, at least in a collective sense, as objects of thought. Natural numbers developed as a sequence of distinguishable grunts or marks which the prehistoric man employed to see if his prehistoric dog had brought in all of his prehistoric sheep. These are the natural numbers