[15]Of course in practice we shall continue to speak of a fraction as (say) greater or less than 1, meaning greater or less than the ratio . So long as it is understood that the ratio and the cardinal number 1 are different, it is not necessary to be always pedantic in emphasising the difference. It will be observed that zero and infinity, alone among ratios, are not one-one. Zero is one-many, and infinity is many-one. There is not any difficulty in defining greater and less among ratios (or fractions). Given two ratios and , we shall say that is less than if is less than . There is no difficulty in proving that the relation "less than," so defined, is serial, so that the ratios form a series in order of magnitude. In this series, zero is the smallest term and infinity is the largest. If we omit zero and infinity from our series, there is no longer any smallest or largest ratio; it is obvious that if is any ratio other than zero and infinity, is smaller and is larger, though neither is zero or infinity, so that is neither the smallest [Pg 65] nor the largest ratio, and therefore (when zero and infinity are omitted) there is no smallest or largest, since was chosen arbitrarily. In like manner we can prove that however nearly equal two fractions may be, there are always other fractions between them. For, let and be two fractions, of which 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 and as , we define as the relation of to , where is any ratio; and is of course the converse of . This is not the only possible way of defining positive and negative ratios, but it is a way which, for our purpose, has the merit of being an obvious adaptation of the way we adopted in the case of integers.