Properties of Rational Numbers / Dedekind

The development of the arithmetic of rational numbers is here presupposed,

but still I think it worth while to call attention to certain important matters

without discussion, so as to show at the outset the standpoint assumed in what

follows. I regard the whole of arithmetic as a necessary, or at least natural,

consequence of the simplest arithmetic act, that of counting, and counting it-

self as nothing else than the successive creation of the infinite series of positive

integers in which each individual is defined by the one immediately preceding;

the simplest act is the passing from an already-formed individual to the con-

secutive new one to be formed. The chain of these numbers forms in itself an

exceedingly useful instrument for the human mind; it presents an inexhaustible

wealth of remarkable laws obtained by the introduction of the four fundamental

operations of arithmetic. Addition is the combination of any arbitrary repeti-

tions of the above-mentioned simplest act into a single act; from it in a similar

way arises multiplication. While the performance of these two operations is

always possible, that of the inverse operations, subtraction and division, proves

to be limited. Whatever the immediate occasion may have been, whatever com-

parisons or analogies with experience, or intuition, may have led thereto; it is

certainly true that just this limitation in performing the indirect operations has

in each case been the real motive for a new creative act; thus negative and

fractional numbers have been created by the human mind; and in the system of

all rational numbers there has been gained an instrument of infinitely greater

perfection. This system, which I shall denote by R, possesses first of all a com-

pleteness and self-containedness which I have designated in another place1 as

characteristic of a body of numbers [Zahlk¨ orper] and which consists in this that the four fundamental operations are always performable with any two individu-als in R, i. e., the result is always an individual of R, the single case of division by the number zero being excepted.

For our immediate purpose, however, another property of the system R is still more important; it may be expressed by saying that the system R forms a well-arranged domain of one dimension extending to infinity on two opposite sides. What is meant by this is su ciently indicated by my use of expressions borrowed from geometric ideas; but just for this reason it will be necessary to bring out clearly the corresponding purely arithmetic properties in order to avoid even the appearance as if arithmetic were in need of ideas foreign to it.

To express that the symbols a and b represent one and the same rational number

we put a = b as well as b = a. The fact that two rational numbers a, b are di erent appears in this that the di erence a - b has either a positive or negative value. In the former case a is said to be greater than b, b less than a; this is also indicated by the symbols a > b, b <> a, a <>

i. If a > b, and b > c, then a > c. Whenever a, c are two di erent (or unequal) numbers, and b is greater than the one and less than the other, we shall, without hesitation because of the suggestion of geometric ideas, express this brie y by saying: b lies between the two numbers a, c.

ii. If a, c are two di erent numbers, there are infinitely many di erent numbers lying between a, c.

iii. If a is any definite number, then all numbers of the system R fall into two classes, A_1 and A_2 , each of which contains infinitely many individuals; the first class A_1 comprises all numbers a1 that are <> all numbers a_2 that are > a; the number a itself may be assigned at pleasure to the first or second class, being respectively the greatest number of the first class or the least of the second. In every case the separation of the system R into the two classes A_1 , A_2 is such that every number of the first class A_1 is less than every number of the second class A_2 .