The rules that have been proved to hold good for all numbers may be applied when the numbers are represented by letters.
SECTION VI.
DECIMAL FRACTIONS.
126. We have seen (112) (121) the necessity of reducing fractions to a common denominator, in order to compare their magnitudes. We have seen also how much more readily operations are performed upon fractions which have the same, than upon those which have different, denominators. On this account it has long been customary, in all those parts of mathematics where fractions are often required, to use none but such as either have, or can be easily reduced to others having, the same denominators. Now, of all numbers, those which can be most easily managed are such as 10, 100, 1000, &c., where 1 is followed by ciphers. These are called decimal numbers; and a fraction whose denominator is any one of them, is called a decimal fraction, or more commonly, a decimal.
127. A whole number may be reduced to a decimal fraction, or one decimal fraction to another, with the greatest ease. For example,
| 94 is | 940 | , or | 9400 | , or | 94000 | (106); | |
| 10 | 100 | 1000 | |||||
| 3 | is | 30 | , or | 300 | , or | 3000 | (108). |
| 30 | 100 | 1000 | 10000 | ||||
The placing of a cipher on the right hand of any number is the same thing as multiplying that number by 10 (57), and this may be done as often as we please in the numerator of a fraction, provided it be done as often in the denominator (108).
128. The next question is, How can we reduce a fraction which is not decimal to another which is, without altering its value? Take, for example, the fraction ⁷/₁₆, multiply both the numerator and denominator successively by 10, 100, 1000, &c., which will give a series of fractions, each of which is equal to ⁷/₁₆ (108), viz. ⁷⁰/₁₆₀, ⁷⁰⁰/₁₆₀₀, ⁷⁰⁰⁰/₁₆₀₀₀, ⁷⁰⁰⁰⁰/₁₆₀₀₀₀, &c. The denominator of each of these fractions can be divided without remainder by 16, the quotients of which divisions form the series of decimal numbers 10, 100, 1000, 10000, &c. If, therefore, one of the numerators be divisible by 16, the fraction to which that numerator belongs has a numerator and denominator both divisible by 16. When that division has been made, which (108) does not alter the value of the fraction, we shall have a fraction whose denominator is one of the series 10, 100, 1000, &c., and which is equal in value to ⁷/₁₆. The question is then reduced to finding the first of the numbers 70, 700, 7000, 70000, &c., which can be divided by 16 without remainder.
Divide these numbers, one after the other, by 16, as follows:
| 16) | 70 | (4 | 16) | 700 | (43 | 16) | 7000 | (437 | 16) | 70000 | (4375 | |||
| 64 | 64 | 64 | 64 | |||||||||||
| 6 | 60 | 60 | 60 | |||||||||||
| 48 | 48 | 48 | ||||||||||||
| 12 | 120 | 120 | ||||||||||||
| 112 | 112 | |||||||||||||
| 8 | 80 | |||||||||||||
| 80 | ||||||||||||||
| 0 | ||||||||||||||
It appears, then, that 70000 is the first of the numerators which is divisible by 16. But it is not necessary to write down each of these divisions, since it is plain that the last contains all which came before. It will do, then, to proceed at once as if the number of ciphers were without end, to stop when the remainder is nothing, and then count the number of ciphers which have been used. In this case, since 70000 is 16 × 4375,