APPENDIX IV.ON THE DEFINITION OF FRACTIONS.
The definition of a fraction given in the text shews that ⁷/₉, for instance, is theninthpart ofseven, which is shewn to be the same thing asseven-ninthsof 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. In⁷/₉we 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 turnsninesintosevens.6. Theratioof 7 to 9, or theproportionof 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.
1. In⁷/₉we 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 turnsninesintosevens.
6. Theratioof 7 to 9, or theproportionof 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: Fora timeis 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 apart 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 ofa= 7 timesa/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 asa/bmost frequently require thataandbshould 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,
of a unit; indeed, we coin a new species of adjective when we talk of the (4³/₅)th part of anything. But we can readily imagine that 2½ is some fraction of 4³/₅; that the first issomepart of a time the second; that there must besomemultiplier which turns every 4³/₅ in a number into 2½; and so on. Let us now see whether we can invent a distinct mode of applying the first and second views to such a compound fraction as the above.
We can easily imagine a fourth part of a length, and a fifth part, meaning the lines of which 4 and 5 make up the length in question; and there is also in existence a length of which four lengths and two-fifths of a length make up the original length in question. For instance, we might say that 6, 6, 2 is a division of 14 into 2⅓ equal parts—2 equal parts, 6, 6, and a third of a part, 2. So we might agree to say, that the (2⅓)th, or (2⅓)rd, or (2⅓)st (the reader may coin the adjective as he pleases) part of 14 is 6. If we divide the linea binto eleven equal parts inc, d, e, &c., we must then say thata cis the 11th part,
a dthe (5½)th,a ethe (3⅔)th,a fthe (2¾)th,a gthe (2⅕)th,a hthe (1⅚)th,a ithe (1⁴/₇)th,a kthe (1⅜)th,a lthe (1²/₉)th,a mthe (1⅒)th, anda bitself the 1st part ofa b. The reader may refuse the language if he likes (though it is not so much in defiance of etymology as talking ofmultiplyingby ½); but whena bis called 1, he must either calla f1/(2¾), or make one definition of one class of fractions and another of another. Whatever abbreviations they may choose, all persons will agree thata/bis a direction to find such a fraction as, repeatedbtimes, will give 1, and then to take that fractionatimes.
So, to get 2½/4⅗, the simplest way is to divide the whole unit into 46 parts; 10 of these parts, repeated 4⅗ times, give the whole. The
4⅗th is then ¹⁰/₄₆, and 2½ such parts is ²⁵/₄₆, ora c. The student should try several examples of this mode of interpreting complex fractions.
But what are we to say when the denominator itself is less than unity, as in 3¼/⅖? Are we to have a (⅖)th part of a unit? and what is it? Hadthere been a 5 in the denominator, we should have taken the part of which 5 will make a unit. As there is ⅖ in the denominator, we must take the part of which ⅖ will be a unit. That part is larger than a unit; it is 2½ units; 2½ is that of which ⅖ is 1. The above fraction then directs us to repeat 2½ units 3¼ times. By extending our word ‘multiplication’ to the taking of a part of a time, all multiplications are also divisions, and all divisions multiplications, and all the terms connected with either are subject to be applied to the results of the other.
If 2⅓ yards cost 3½ shillings, how much does one yard cost? In such a case as this, the student looks at a more simple question. If 5 yards cost 10 shillings, he sees that each yard costs ¹⁰/₅, or 2 shillings, and, concluding that the same process will give the true result when the data are fractional, he forms 3½/2⅓, reduces it by rules to ³/₂ or 1½, and concludes that 1 yard costs 18 pence. The answer happens to be correct; but he is not to suppose that this rule of copying for fractions whatever is seen to be true of integers is one which requires no demonstration. In the above question we want money which, repeated 2⅓ times, shall give 3½ shillings. If we divide the shilling into 14 equal parts, 6 of these parts repeated 2⅓ times give the shilling. To get 3½ times as much by the same repetition, we must take 3½ of these 6 parts at each step, or 21 parts. Hence, ²¹/₁₄, or 1½, is the number of shillings in the price.