If we divide 1 by 5⁄7 we obtain, by this rule, 7⁄5. Thus the reciprocal of a number may be defined as the number obtained by dividing 1 by it. This definition applies whether the original number is integral or fractional.
By means of the present and the preceding sections the rule given in § 63 can be extended to the statement that a fractional number is equal to the number obtained by multiplying its numerator and its denominator by any fractional number.
67. Negative Fractional Numbers.—We can obtain negative fractional numbers in the same way that we obtain negative integral numbers; thus − 5⁄7 or − 5⁄7A means that 5⁄7 or 5⁄7A is taken negatively.
68. Genesis of Fractional Numbers.—A fractional number may be regarded as the result of a measuring division (§ 39) which cannot be performed exactly. Thus we cannot divide 3 in. by 11 in. exactly, i.e. we cannot express 3 in. as an integral multiple of 11 in.; but, by extending the meaning of “times” as in § 62, we can say that 3 in. is 3⁄11 times 11 in., and therefore call 3⁄11 the quotient when 3 in. is divided by 11 in. Hence, if p and n are numbers, p/n is sometimes regarded as denoting the result of dividing p by n, whether p and n are integral or fractional (mixed numbers being included in fractional).
The idea and properties of a fractional number having been explained, we may now call it, for brevity, a fraction. Thus “2⁄3 of A” no longer means two of the units, three of which make up A; it means that A is multiplied by the fraction 2⁄3, i.e. it means the same thing as “2⁄3 times A.”
69. Percentage.—In order to deal, by way of comparison or addition or subtraction, with fractions which have different denominators, it is necessary to reduce them to a common denominator. To avoid this difficulty, in practical life, it is usual to confine our operations to fractions which have a certain standard denominator. Thus (§ 79) the Romans reckoned in twelfths, and the Babylonians in sixtieths; the former method supplied a basis for division by 2, 3, 4, 6 or 12, and the latter for division by 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, or 60. The modern method is to deal with fractions which have 100 as denominator; such fractions are called percentages. They only apply accurately to divisions by 2, 4, 5, 10, 20, 25 or 50; but they have the convenience of fitting in with the denary scale of notation, and they can be extended to other divisions by using a mixed number as numerator. One-fortieth, for instance, can be expressed as 2½/100, which is called 2½ per cent., and usually written 2½%. Similarly 31⁄3% is equal to one-thirtieth.
If the numerator is a multiple of 5, the fraction represents twentieths. This is convenient, e.g. for expressing rates in the pound; thus 15% denotes the process of taking 3s. for every £1, i.e. a rate of 3s. in the £.
In applications to money “per cent.” sometimes means “per £100.” Thus “£3, 17s. 6d. per cent.” is really the complex fraction
| 3 | 176⁄12 |
| 20 | |
| 100 | |
70. Decimal Notation of Percentage.—An integral percentage, i.e. a simple fraction with 100 for denominator, can be expressed by writing the two figures of the numerator (or, if there is only one figure, this figure preceded by 0) with a dot or “point” before them; thus .76 means 76%, or 76⁄100. If there is an integral number to be taken as well as a percentage, this number is written in front of the point; thus 23.76 × A means 23 times A, with 76% of A. We might therefore denote 76% by 0.76.