often as we please in the numerator of a fraction, provided it is 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, multiply both the numerator and denominator successively by 10, 100, 1000, &c., which will give a series of 70 700 7000 160' 1600' 16000' 70000 &c. The denominator of each of these fractions can be divided 160000' 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 is 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 7 of the series 10, 100, 1000, &c., and which is equal in value to 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. fractions, each of which is equal to 7(108), viz. 16 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, 16 × 4375 4375 which is is the fraction required. 16 x 10000 70000 160000' or 10000 Therefore, to reduce a fraction to a decimal fraction, annex ciphers to the numerator, and divide by the denominator until there is no remainder. The quotient will be the numerator of the required fraction, and the denominator will be unity, followed by as many ciphers as were used in obtaining the quotient. 129. It will happen in most cases that the annexing of ciphers to the numerator will never make it divisible by the denominator without remainder. For example, try to reduce to a decimal fraction. 7 7) 10000000000oooooooo, &c. 142857142857142857, &c. I 7 The quotient here is a continual repetition of the figures 1, 4, 2, 8, 5, 7, in the same order; therefore cannot be reduced to a decimal fraction. But, nevertheless, if we take as a numerator any number of figures from the quotient 142857142857, &c., and as a denominator, I followed by as many ciphers as were used in making that part of the quotient, we shall get a fraction which differs very little from and which will differ still less from it if we put more figures in the numerator, and more ciphers in the denominator. 3 which is not so I 702 I much as IO In the first column is a series of decimal fractions, which come nearer, and nearer to as the third column shews. Therefore, though I 7 we cannot find a decimal fraction which is exactly which differs from it as little as we please. 130. The reason for the recurrence of the figures of the quotient in the same order is as follows: If 1000, &c. be divided by the number 247, the remainder at each step of the division is less than 247, being either o, or one of the first 246 numbers. If, then, the remainder never becomes nothing, by carrying the division far enough, one remainder will occur a second time. If possible, let the first 246 remainders be all different, that is, let them be 1, 2, 3, &c. up to 246, variously distributed. As the 247th remainder cannot be so great as 247, it must be one of these which have preceded. From the step where the remainder becomes the same as a former remainder, it is evident that former figures of the quotient must be repeated in the same order. 131. You will here naturally ask, What is the use of decimal fractions, if the greater number of fractions cannot be reduced at all to decimals? The answer is this: The addition, subtraction, multiplication, and division, of decimal fractions, are much easier than those of common fractions; and though we cannot reduce all common fractions to decimals, yet we can find decimal fractions so near to each of them, that the error arising from using the decimal instead of the common fraction will not be perceptible. For example, if we suppose an inch to be divided into ten million of equal parts, one of those parts by itself will not be visible to the eye. Therefore, in finding a length, an error of a ten-millionth part of an inch is of no consequence, even where the 10000000 I ; finest measurement is necessary. Now, by carrying on the table in 1428571 (129), we shall see that does not differ from by 7 10000000 and if these fractions represented parts of an inch, the first might be used for the second, since the difference is not perceptible. In applying arithmetic to practice, nothing can be measured so accurately as to be represented in numbers without any error whatever, whether it be length, weight, or any other species of magnitude. It is therefore unnecessary to use any other than decimal fractions; since, by means of them, any quantity may be represented with as much correctness as by any other method. EXERCISES. Find decimal fractions which do not differ from the following frac 132. Every decimal may be immediately reduced to a quantity consisting either of a whole number and more simple decimals, or of more simple decimals alone, having one figure only in each of the numerators. 326 is made up of 300, and 20, and 6; by (112) Therefore 147326 is 1000 1000 example, 147326, and form a number of fractions, having for their numerators this number, and for their denominators 1, 10, 100, 1000, 10000, &c., and reduce these fractions into numbers and more simple decimals, in the foregoing manner, which will give the table on the following page. |