pence, and then perform the subtraction. Generally, to find the sum or difference of two or more fractional numbers, we must replace them by other fractional numbers having the same denominator; it is usually most convenient to take as this denominator the L.C.M. of the original fractional numbers (cf. § 53).

64. *Complex Fractions*.—A fraction (or fractional number), the numerator or denominator of which is a fractional number, is called a *complex* fraction (or fractional number), to distinguish it from a *simple* fraction, which is a fraction having integers for numerator and denominator. Thus 5 2⁄3 / 11 1⁄3 of A means that we take a unit X such that 11 1⁄3 times X is equal to A, and then take 5 2⁄3 times X. To simplify this, we take a new unit Y, which is 1⁄3 of X. Then A is 34 times Y, and 5 2⁄3 / 11 1⁄3 of A is 17 times Y, *i.e.* it is ½ of A.

65. *Multiplication of Fractional Numbers*.—To multiply 8⁄3 by 5⁄7 is to take 5⁄7 times 8⁄3. It has already been explained (§ 62) that 5⁄7 times is an operation such that 5⁄7 times 7 times is equal to 5 times. Hence we must express 8⁄3, which itself means 8⁄3 times, as being 7 times something. This is done by multiplying both numerator and denominator by 7; *i.e.* 8⁄3 is equal to 7·8⁄7·3, which is the same thing as 7 times 8⁄7·3. Hence 5⁄7 times 8⁄3 = 5⁄7 times 7 times 8⁄7·3 = 5 times 8⁄7·3 = 5·8⁄7·3. The rule for multiplying a fractional number by a fractional number is therefore the same as the rule for finding a fraction of a fraction.

66. *Division of Fractional Numbers*.—To divide 8⁄3 by 5⁄7 is to find a number (*i.e.* a fractional number) x such that 5⁄7 times x is equal to 8⁄3. But 7⁄5 times 5⁄7 times x is, by the last section, equal to x. Hence x is equal to 7⁄5 times 8⁄3. Thus to divide by a fractional number we must multiply by the number obtained by interchanging the numerator and the denominator, *i.e.* by the *reciprocal* of the original number.

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 2 ½%. Similarly 3 1⁄3% 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

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.

If as our unit we take X = 1⁄100 of A = 1% of A, the above quantity might equally be written 2376 X = 2376⁄100 of A; *i.e.* 23.76 × A is equal to 2376% of A.

71. *Approximate Expression by Percentage*.—When a fraction cannot be expressed by an integral percentage, it can be so expressed approximately, by taking the *nearest* integer to the numerator of an equal fraction having 100 for its denominator. Thus 1⁄7 = 14 2⁄7 / 100, so that 1⁄7 is approximately equal to 14%; and 2⁄7 = (28 4⁄7)/100, which is approximately equal to 29%. The difference between this approximate percentage and the true value is less than ½%, *i.e.* is less than 1⁄200.

If the numerator of the fraction consists of an integer and ½—*e.g.* in the case of 3⁄8 = (37 ½)/100—it is uncertain whether we should take the next lowest or the next highest integer. It is best in such cases to retain the ½; thus we can write 3⁄8 = 37 ½% = .37 ½.

72. *Addition and Subtraction of Percentages*.—The sum or difference of two percentages is expressed by the sum or difference of the numbers expressing the two percentages.

73. *Percentage of a Percentage*.—Since 37% of 1 is expressed by 0.37, 37% of 1% (*i.e.* of 0.01) might similarly be expressed by 0.00.37. The second point, however, is omitted, so that we write it 0.0037 or .0037, this expression meaning 37⁄100 of 1⁄100 = 37⁄10000.

On the same principle, since 37% of 45% is equal to 37⁄100 of 45⁄100 = 1665⁄10000 = 16⁄100 + (65⁄100 of 1⁄100), we can express it by .1665; and 3% of 2% can be expressed by .0006. Hence, to find a percentage of a percentage, we multiply the two numbers, put 0’s in front if necessary to make up four figures (not counting fractions), and prefix the point.

74. *Decimal Fractions*.—The percentage-notation can be extended to any fraction which has any power of 10 for its denominator. Thus 153⁄1000 can be written .153 and 15300⁄100000 can be written .15300. These two fractions are equal to each other, and also to .1530. A fraction written in this way is called a *decimal fraction*; or we might define a decimal fraction as a fraction having a power of 10 for its denominator, there being a special notation for writing such fractions.

A mixed number, the fractional part of which is a decimal fraction, is expressed by writing the integral part in front of the point, which is called the *decimal point*. Thus 27 1530⁄10000} can be written 27.1530. This number, expressed in terms of the fraction 1⁄10000 or .0001, would be 271530. Hence the successive figures after the decimal point have the same relation to each other and to the figures before the point as if the point did not exist. The point merely indicates the *denomination* in which the number is expressed: the above number, expressed in terms