Appendix IV. On the Definition of Fractions.

The definition of a fraction given in the text shews that ⁷/₉, for instance, is the ninth part of seven, which is shewn to be the same thing as seven-ninths of a unit. But there are various modes of speech under which a fraction may be signified, all of which are more or less in use.

1. In ⁷/₉ we have the 9th part of 7.

2. 7-9ths of a unit.

3. The fraction which 7 is of 9.

4. The times and parts of a time (in this case part of a time only) which 7 contains 9.

5. The multiplier which turns nines into sevens.

6. The ratio of 7 to 9, or the proportion of 7 to 9.

7. The multiplier which alters a number in the ratio of 9 to 7.

8. The 4th proportional to 9, 1, and 7.

The first two views are in the text. The third is deduced thus: If we divide 9 into 9 equal parts, each is 1, and 7 of the parts are 7; consequently the fraction which 7 is of 9 is ⁷/₉. The fourth view follows immediately: For a time is only a word used to express one of the repetitions which take place in multiplication, and we allow ourselves, by an easy extension of language, to speak of a portion of a number as being that number taken a part of a time. The fifth view is nothing more than a change of words: A number reduced to ⁷/₉ of its amount has every 9 converted into a 7, and any fraction of a 9 which may remain over into the corresponding fraction of 7. This is completely proved when we prove the equation ⁷/₉ of a = 7 times a/9. The sixth, seventh, and eighth views are illustrated in the chapter on proportion.

When the student comes to algebra, he will find that, in all the applications of that science, fractions such as a/b most frequently require that a and b should be themselves supposed to be fractions. It is, therefore, of importance that he should learn to accommodate his views of a fraction to this more complicated case.

We shall find that we have, in this case, a better idea of the views from and after the third inclusive, than of the first and second, which are certainly the most simple ways of conceiving ⁷/₉. We have no notion of the (4³/₅)th part of 2½,