FRACTIONAL EQUIVALENTS OF REPEATING DECIMALS. 149 tor 9 in the numerator will cancel the same factor in the denominator, the fractional equivalent, in its lowest terms, will have 11 for its denominator; that is to say, will be a number of elevenths. Thus, ,27 3 = Now the figure 2, in the place of tens, is 2 (9+1;) that is, it is 2 nines plus 2; and this remainder 2, with the 7 units, makes one nine more. Therefore, when the sum of two figures, which repeat, is 9, add a unit to the left-hand figure, and the sum will be the number of elevenths equivalent to the repeating decimal fraction. Hence also, to have the decimal value of a proper fraction, having 11 for its denominator, we have only to multiply its numerator by 9, and the product will be the decimal period. Thus, to find the value of 5, we say 5 X 9=45; therefore, 11 =,45. ;=,09; &=‚18; &=‚27; =,36; ‚§;=,45 11 ;=,81; 10,90. 256. If a fraction be given in its lowest terms, its decimal value will be finite; which means exact or definite, when its denominator is a measure of 10 or of some power of 10; that is, of one of the numbers in the scale 10, 100, 1000, &c. When the denominator is prime to 10, and consequently, to every number in the scale 10, 100, 1000, &c., the decimal value will be infinite; that is, (251,) it will be a repeating decimal. For, in placing a cipher continually on the right of each remainder, we do the same as to multiply the dividend by 10, 100, 1000, &c.; consequently, if the denominator is a measure of any number in this scale, it must be a measure of the numerator when multiplied by that number, and hence the quotient will be exact. Also, when the denominator is prime to the numbers 10, 100, 1000, &c., as it is prime to its numerator, it will (180) be prime to the product of the numerator multiplied by any of these numbers, and the quotient (251) will, consequently, be a repeating decimal. Now the denominator cannot measure any of the numbers 10, 100, 1000, &c. if it contains any prime factor which is not common to those numbers; that is, if it contains any prime factor other than 2 or 5; we may therefore (181) ascertain the prime factors of the denominator, which will determine whether the decimal equivalent will be finite or infinite. Hence, the possibility or impossibility of expressing in decimals the value of a vulgar frac tion, depends entirely upon its denominator; for, if this contains any prime factor other than 2 or 5, it is impossible; if not, it is possible. Thus, we may have, in decimals, the exact value of 16' 32' 64' 20' 25' 50' 125' 400, &c., but not of 7 11 9 13 19 1 1 13 29 111 231 4, 3, 14, 14, 12, 33, &c. 39 257. When the exact value of a fraction cannot be obtained in decimals, we can, by the following examination of its denominator, find how many figures its decimal period will contain, and where that period will begin. Before the fractional equivalent of a repeating decimal is reduced to lower terms, its denominator (254) consists of as many nines as there are places in the period, terminated by as many ciphers as there are places between the period and the comma. Wherefore, as any of the numbers 9, 99, 999, &c. is prime to 10, if the denominator of the given fraction terminates with ciphers, there must be, in its decimal equivalent, as many places between the comma and the period as there are terminating ciphers. These ciphers may therefore be suppressed and counted as so many places. Now, as the denominator of the period is prime to 2 or 5, and as either of these is only once factor in 10, neither 2 nor 5 could be factor in the denominator of the given fraction, after the suppression of its terminating ciphers, unless 10 were as often factor in the denominator of the fractional equivalent first found, after the suppression of the same number of its terminating ciphers. Therefore, the remaining part of the given denominator, after the suppression of its terminating ciphers, may be divided by 2 and by 5 as often as possible; and the number of divisors employed will show the remaining number of times that 10 is factor in the fractional equivalent, found as in art. 254; consequently, the number of ciphers suppressed, if any, together with the number of divisors, will show the number of times that 10 is factor in the denominator of the first fractional equivalent; that is, the number of ciphers in this denominator, and finally (254) the number of places between the comma and the period. Again, if we divide a unit by any number prime to 10, the quotient figures will repeat, as soon as the remainder is 1; therefore, as any of the numbers 9, 99, 999, &c. is 1 less than the corresponding one in the series (succession) 10, 100, 1000 &c., if we divide a series of nines by the number prime to 10, the quotient will be exact, when we have reached the limit of the period. But the decimal value of a fraction is the same (165) in whatever terms it is expressed. Therefore, in dividing a series of nines by the remaining quotient, after having exhausted the factors 10, 5, and 2, found in the denominator of the given fraction, the division will be exact, when we reach the limit of the period; consequently, the number of nines divided will show the number of figures in that period. 1040 To find whether 1033 is finite or infinite, and if infinite, of how many places of figures its period consists, as well as where it begins, we proceed thus: 2) 104,0 2)26 13) 99 (76923 89 119 29 39 ** Having found a factor 13, which is prime to 10, the decimal will be infinite, (256.) Again, as we have suppressed one cipher and employed three divisors, there will be four places between the comma and the period, and hence the period will begin at the fifth place. Finally, as we have employed six nines, in dividing the series to find an exact quotient, there will be six decimals in the period. Proof. The value of 1838 in decimals is ,9990384615. 104 For practice, the scholar may examine the following fractions, and prove as above: 258. A unit, divided by any of the numbers 9, 99, 999, &c., can give no other figure (253) in the quotient or remainder. Therefore, if we except the case of 1, any figure whatever divided by 9, 99, 999, &c., will always give the same figure in the quotient and remainder. Now, if by 9, we divide a figure in the unit place, which is less than 9, as we must first reduce it to tenths, the first quotient figure will be tenths, and the repeater will, of course, begin at the comma. If we divide any figure whatever by 99, as we must first reduce it to hundredths, the first quotient figure will be hundredths; and, of course, we shall have 0 in the place of tenths: but, as the remainder is the same figure, which, when reduced to the next lower order, does not contain 99, the next quotient figure will be a cipher; consequently, in this case also, the period begins at the comma. In the same manner we might show that, in dividing by any other number of nines, the period will begin at the comma. 259. Excepting an equal number of nines, if by 9, 99, 999, &c., we divide any number of figures, not exceeding the number of nines divided by, these figures, in the order in which they are given, will be the only figures in the quotient and remainder; and the period will begin at the comma and consist of as many places as there are nines in the divisor. For example, if we would divide 678 by 9999, we may consider the division of each figure separately, thus: As any of the numbers 10, 100, 1000, &c., is greater than the corresponding one of the numbers 9, 99, &c., if any figure has as many ciphers on the right as there are nines in the divisor, it will contain that divisor. Now, the figure 6, which is 600 units, must be reduced to 60000 hundredths, before it will contain 9999; consequently, the first figure 6, in the quotient and remainder must be 6 hundredths. The next figure 7, being 70 units, must, before we can divide, be reduced to 70000 thousandths; consequently, the figure 7 will take its place as a number of thousandths, in both quotient and remainder, on the right of the 6 hundredths already written in each. The last figure 8, being units, we reduce to 80000 ten-thousandths; consequently, the figure 8, as quotient and remainder, takes its place as 8 ten-thousandths, on the right of the 7 thousandths already written. Now, as the first figure 6 is hundredths, there must be one cipher between it and the comma; the quotient found is, therefore, ,0678, which begins at the comma. Also, When the number of figures in the divisor and dividend is the same, as we must, before we can divide, reduce the latter to tenths, the first quotient figure will be tenths: the period, therefore, in this case, evidently begins at the comma. from the above, it is evident that, when the number of figures in both is not the same, the difference will show the number of ciphers between the first significant figure of the period and the comma. Wherefore, the period always begins at the comma, and contains a number of figures equal to the number of nines in the divisor. 260. A proper fraction, the denominator of which is prime to 10, is equal (257) to a fraction, the denominator of which is a number of nines, equal to the number of figures in its decimal period. Now, each of these two fractions must give the same decimal quotient: but the numerator does not contain more figures than the denominator; and we have seen above that, in this case, the quotient will be a repeating decimal, beginning at the comma. Wherefore, when the denominator of a fraction is prime to 10, its decimal period begins at the comma. For practice, the student may find the periodical decimal equivalents of the following fractions, performing the divisions as in Art. 243; namely, of Also, if he wishes to obtain the fractions again from the decimal periods, the last of which will consist of 35 figures, he may find (254) the fractional equivalent of each, and divide both terms by their greatest common measure. 261. If we repeat a decimal period any number of times, it will assume for a new denominator, the number of nines in the denominator of its fractional equivalent, repeated the same number of times. Thus, if we repeat the period, 0678 three times, the denominator of the fractional equivalent will be 999999999999: and this is evident, because the new numerator ,067806780678 is the product of ,000100010001678, and the new denominator the product of ,000100010001X 9999; and because, as we have seen, (259,) any number of figures, divided by an equal number of nines, will repeat in their exact order, beginning at the comma. Hence, also, a single repeater may be repeated any number of times, and will take, for a denominator, as many nines as there are places on on the right of the comma. Thus : 262. Also, either of the above may (254) be represented in mixed form, thus: ,06780;,067806;,0678067;,06780678, &c.; and Here we must observe, that although any period may be so expressed as to have any number of figures at pleasure between the comma and the period, it is only with a single repeater |