than the divisor: Reduce them both to the same denomina-. tion, and divide as in whole numbers. N. B. There are two ways of reducing them to the same denomination. First, the divisor may be reduced to the same denomination as the dividend, by annexing zeros, and taking away the points from both. Secondly, the dividend may be reduced to the same denomination as the divisor, by taking away the point from the divisor, and removing it in the dividend towards the right as many places as there are in the divisor. The second method is preferable. The same result may be produced by another mode of reasoning. The quotient must be such a number, that be- * ing multiplied with the divisor will reproduce the dividend. Now a product must have as many decimal places as there are in the multiplier and multiplicand both. Consequently the decimal places in the divisor and quotient together must be equal to those in the dividend. In the last example there were four decimal places in the dividend and two in the divisor ; this would give two places in the quotient. Then a zero was annexed in the course of the division, which made three places in the quotient. The rule may be expressed as follows: Divide as in whole numbers, and in the result point off as many places for. decimals as those in the dividend exceed those in the divisor. If zeros åre 'annexed to the dividend, count them as so many decimals in the dividend. If there are not so many places in the result as are required, they must be supplied by writing zeros on the left. Division in decimals, as well as in whole numbers, may be expressed in the form of common fractions. What part of .5 is .3? Ans.g. To answer this, .3 must be reduced to hundredths. .3 is .30, the answer therefore is zo What part of 14.035 is 3.8 ? TT0356 In fine, to express the division of one nimber by another, when either or both contain decimais, reduce them both to the lowest denomination mentioned in either, and then write the divisor under the dividend, as if they were whole numbers. Circulating Decimals. XXIX. There are some common fractions which cannot be expressed exactly in decimals. If we attempt to change to decimals for example, we find .3333, &c. there is always å remainder 1, and the same figure 3 will always be repeated however far we may continue it. At each division we approximate ten times nearer to the true value, and yet we can never obtain it. & 5.1666, &c.; this begins to repeat at the second figure. i .545454, &c.; this repeats two figures. In the division the remainders are alternately 6 and 5. = .168168, &c. ; this repeats three figures, and the remainders are alternately 56, 227, and 272. Some do not begin to repeat until after two or three or more places. It is evident that whenever the same remainder recurs a second time, the quotient figures and the same remainders will repeat over again in the same order. In the last example for instance, the number with which we commenced was 56; we annexed a zero and divided ; this gave a quotient 1, and a remaiņder 227 ; we annexed another zero, and the quotient was 6, and the remainder 272; we annexed another zero, and the quotient was 8, and the remainder 56, the number we commenced with. If we annex a zero to this, it is evident that we shall obtain the same quotient and the same remainder as at first, and that it will continue to repeat the same three figures for ever. It is evident that the number of these remainders, and consequently the number of figures which repeat, must be one less than the number of units in the divisor. If the fraction is , there can be only six different remainders ; after this number, one of them must necessarily recur again, and then the figures will be repeated again in the same order. 30 1 (7 It commences with 1 for the dividend, then annexing zeros, 28 the remainders are 3, 2, 6, 4, 5, which are all the numbers below 20 7; then comes 1 again, the num14 ber with which it commenced, and it is evident the whole will be repeated again in the same order. Decimals which repeat in this way are called circulating deci 40 mals. 35 60 56 50 49 10 3 Whenever we find that a fraction begins to repeat, we may write out as many places as we wish to retain, without the trouble of dividing. As it is impossible to express the value of such a fraction by a decimal exactly, rules have been invented by which operations may be performed on them, with nearly as much accuracy as if they could be expressed ; but as they are long and tedious, and seldom used, I shall not notice them. Sufficient accuracy may always be attained without them. I shall show, however, how the true value of them may always be found in common fractions. The fraction ţ reduced to a decimal, is .1111 &c. Therefore, if we wish to change this fraction to a common fraction, instead of calling it i'o, jóo, or joto, which will be a value too small, whatever number of figures we take, we must call it ì. This is exact, because it is the fraction which produces the decimal. If we have the fraction .2222.. &c. it is plain that this is twice as much the other, and must be called $. If be reduced to a decimal, it produces .2222 &c. If we have .3333 .. &c. this being three times as 8 much as the first, is 3 = . If} be reduced to a decimal, it produces .3333 .. &c. It is plain, that whenever a single figure repeats, it is so many ninths. Charge .4444 &c. to a common fraction. Ans. . Change .5555 &c. to a common fraction. Change .6666 &c. to a common fraction. Change .7777 &c. to a common fraction. Change .9999 &c. to a common fraction. Change .5333 &c. to a common fraction. This begins to repeat at the second figure or hundredths. The first figure 5 is ; and the remaining part of the fraction is of that is, a=; these must be added together.io is j, and makes o The answer is 1. If this be changed to a decimal, it will be found to be : 5333 &c. If a decimal begins to repeat at the third place, the two first figures will be so many hundredths, and the repeating figure will be so many ninths of another hundredth. Change .4666 &c. to a common fraction. Change .3888 &c. to a common fraction. Change .3744 &c. to a common fraction. Change .46355 &c. to a common fraction. . If to be changed to a decimal, it produces .010101 &c The decimal .030303 &c. is three times as much, therefore it must be =s. The decimal .363636 &c. is thirty-six times as much, therefore it must be =: If be changed to a decimal, it produces .001001001 &c. "The decimal .006006 &c. is 6 times as much, therefore it must be go=z3h: The fraction .027027 &c. is twenty-seven times as much, and must be itThe fraction .354354 &c. is 354 times as much, and must be 3=;l. This principle is true for any number of places. Hence we derive the following rule for changing a circulating decimal to a common fraction: Make the repeating figures the numerator, and the denominator will be as mamy 9s as there are repeating figures. If they do not begin to repeat at the first place, the preceding figures must be called so many tenths, hundredths, fc. according to their number, then the repeating part must be changed in the above manner, but instead of being the fraction of an unit, it will be the fraction of a tenth, hundredth, f'c. according to the place in which it commences. Instead of writing the repeating figures over several times, they are sometimes written with a point over the first and last to show which figures repeat. 'Thus .333 &c. is written .3. 2525 &c. is written 25. .387387 &c. is written .387. :57346346 &c. is written .57346. Change 24 to a common fraction. Note. To know whether you have found the right answer, change the common fraction, which you have found, to a decimal again. If it produces the same, it is right. 3 Proof of Multiplication and Division by casting out İs. If either the multiplicand or the multiplier be divisible by 9, it is evident the product must be so. Multiply 437 by 35. 437 81 times 437 = 35397 85 4 times 432 = 1728 4 times 20 2185 3496 37145 5 Ans. 37145 85 = 81 + 4, and 437 = 432 + 5. 81 is divisible by 9, and 85 being divided by 9 leaves a remainder 4. 432 is divisible by 9, and 437 leaves a remainder 5. 81 times 437,and 4 times 432, and times 5, added together, are equal to 85 times 437. 81 times 437 is divisible by 9, because 81 is so, and 4 times 432 is divisible by. 9, because 432 is so. The only part of the product which is not divisible by 9, is the product of the two remainders 4 and 5. This product, 20, divided by 9, teaves a remainder 2. It is plain, therefore, that if the whole product, 37145, be divided by 9, the remainder must be 2, the same as that of the product of the remainder. Therefore to prove multiplication, divide the divisor and the dividend by 9, and multiply the remainders together, and |