is the greater. Then it is easy to see (or to prove) that
will be greater than
and less than
. Thus the series of ratios is one in which no two terms are consecutive, but there are always other terms between any two. Since there are other terms between these others, and so on ad infinitum, it is obvious that there are an infinite number of ratios between any two, however nearly equal these two may be.[16] A series having the property that there are always other terms between any two, so that no two are consecutive, is called "compact." Thus the ratios in order of magnitude form a "compact" series. Such series have many important properties, and it is important to observe that ratios afford an instance of a compact series generated purely logically, without any appeal to space or time or any other empirical datum.
[16]Strictly speaking, this statement, as well as those following to the end of the paragraph, involves what is called the "axiom of infinity," which will be discussed in a later chapter.
Positive and negative ratios can be defined in a way analogous to that in which we defined positive and negative integers. Having first defined the sum of two ratios