Elements of Rational Number Theory

Abstract

For the time being the objects of arithmetic are the whole numbers, 0, ± 1, ± 2,... which can be combined by addition, subtraction, multiplication and division (not always) to form integers. Higher arithmetic uses methods of investigation analogous to those of real or complex numbers. Moreover it also uses analytic methods which belong to other areas of mathematics, such as infinitesimal calculus and complex function theory, in the derivation of its theorems. Since these will also be discussed in the latter part of this book, we will assume as known the totality of complex numbers, a number domain, in which the four types of operations (except division by 0) can be carried out unrestrictedly. The complex domain is usually developed more precisely in the elements of algebra or of differential calculus. In this domain the number 1 is distinguished as the one which satisfies the equation

$$1 \cdot a = a$$

for each number a. All successive integers are obtained by the process of addition and subtraction from the number 1, and if the process of division is then carried out the set of rational numbers is obtained as the totality of quotients of integers. Later, from §21 on, the concept of “integer” will be subjected to an essential extension.