Basis of the Number System

BASIS OF THE NUMBER SYSTEM The Integers as are now in a position to indicate in what sense the integers alone may be regarded as basis for these extensions. Our first step is to define a fraction as a pair of associated integers — , and to define addition and multiplication of fractions by the usual algebraic formulae. It is then readily verified on the basis of the properties of the integers that such fractions obey the five general laws. We regard two fractions Pa — and as "equal" if pq'=qp'. Furthermore we write - =a, and find that the first special law also holds. We may next intro duce the inverse operations and obtain the usual corresponding laws from the five fundamental laws.

The System of

the course of the preceding development certain general and special laws were found to be characteristic of various types of number systems. It is of decided interest to investigate the question as to whether these laws are sufficient to characterize such systems. This can be accomplished by means of the postulational method. Here certain postulates are formulated in terms of certain elements and operations, and it is then proved that the elements and operations are essentially the numbers and the operations of addition and multiplication.

A natural set of postulates with which to begin is the following.

The set of elements and ±, X obey the fundamental laws /— V ; there are two special elements o, i which obey the special laws. The equations a+x=b, aXx=b have a unique solution x for all values of a and b, except for a=o in the second case.

These postulates however do not suffice. In fact they hold for the set of positive and negative fractions, for the real number system, and for the complex number system. They also hold for certain modular number systems referred to later in which there are only a finite number of elements ; the simplest such system contains only three marks o, i, 2 with addition and multiplication defined by the following tables: The complete treatment of the ordinary number system which Huntington has given introduces the further relation < (less than).

The advantage of such a postulational analysis is that we are led to see what are the essentially independent properties of an assigned number system. By the modification or removal of one or more of the postulates other types of number are obtained; e.g., if to the above set we add the postulate that there are three distinct elements, it may be proved immediately that the modular system specified is obtained.