62. Fractional Numbers.—According to the definition in § 50 the quantity denoted by 3⁄6 of A is made up of a number, 3, and a unit, which is one-sixth of A. Similarly p/n of A, q/n of A, r/n of A, ... mean quantities which are respectively p times, q times r times, ... the unit, n of which make up A. Thus any arithmetical processes which can be applied to the numbers p, q, r, ... can be applied to p/n, q/n, r/n, ... , the denominator n remaining unaltered.
If we denote the unit 1/n of A by X, then A is n times X, and p/n of n times X is p times X; i.e. p/n of n times is p times.
Hence, so long as the denominator remains unaltered, we can deal with p/n, q/n, r/n, ... exactly as if they were numbers, any operations being performed on the numerators. The expressions p/n, q/n, r/n, ... are then fractional numbers, their relation to ordinary or integral numbers being that p/n times n times is equal to p times.
This relation is of exactly the same kind as the relation of the successive digits in numbers expressed in a scale of notation whose base is n. Hence we can treat the fractional numbers which have any one denominator as constituting a number-series, as shown in the adjoining diagram. The result of taking 13 sixths of A is then seen to be the same as the result of taking twice A and one-sixth of A, so that we may regard 13⁄6 as being equal to 21⁄6. A fractional number is called a proper fraction or an improper fraction according as the numerator is or is not less than the denominator; and an expression such as 21⁄6 is called a mixed number. An improper fraction is therefore equal either to an integer or to a mixed number. It will be seen from § 17 that a mixed number corresponds with what is there called a mixed quantity. Thus £3, 17s. is a mixed quantity, being expressed in pounds and shillings; to express it in terms of pounds only we must write it £317⁄20.
63. Fractional Numbers with different Denominators.—If we divided the unit into halves, and these new units into thirds, we should get sixths of the original unit, as shown in A; while, if we divided the unit into thirds, and these new units into halves, we should again get sixths, but as shown in B. The series of halves in the one case, and of thirds in the other, are entirely different series of fractional numbers, but we can compare them by putting each in its proper position in relation to the series of sixths. Thus 3⁄2 is equal to 9⁄6, and 5⁄3 is equal to 10⁄6, and conversely; in other words, any fractional number is equivalent to the fractional number obtained by multiplying or dividing the numerator and denominator by any integer. We can thus find fractional numbers equivalent to the sum or difference of any two fractional numbers. The process is the same as that of finding the sum or difference of 3 sixpences and 5 fourpences; we cannot subtract 3 sixpenny-bits from 5 fourpenny-bits, but we can express each as an equivalent number of 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 52⁄3 / 111⁄3 of A means that we take a unit X such that 111⁄3 times X is equal to A, and then take 52⁄3 times X. To simplify this, we take a new unit Y, which is 1⁄3 of X. Then A is 34 times Y, and 52⁄3 / 111⁄3 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.