By a comparison of the common with the eastern method, it will appear upon inspection that the difference is only in the arrangement of the work: and by a comparison of the eastern with the contracted method, it will be seen that the only difference is to leave out that portion of the multiplication which does not contribute to the figures within the limits prescribed for the contraction. In this contraction a correction column is kept to the right of the vertical line, which, in fact, is computing one decimal place more than was required, in order to insure accuracy in the required number of places. It is always desirable to do this, as otherwise the last figure cannot be depended on; and the more so as such a correction column must always be kept in almost the only place where the method is of constant occurrence; viz. in the solution of equations, and its subordinate class of operations, the extraction of roots. Here, in the contracted way, we have multiplied first by the left-hand figure, 9; then by the 2, omitting the product of 2 × 6, but regarding the 1 carried on; then by the 4, omitting the product of 4 x 86, but regarding the 3 carried on. The rest of the process is, in like manner, conformable to the rule; and it is much easier than the usual method of contracted multiplication by inverting the multiplier. 2. Multiply 480-14936 by 2.72416, retaining only four decimals in the product. 3. Multiply 2490-3048 by 573286, retaining only five decimals in the product. 4. Multiply 325-701428 by 7218393, retaining only three decimals in the product. DIVISION OF DECIMALS. DIVIDE as in whole numbers; and point off in the quotient as many places for decimals as the decimal places in the dividend exceed those in the divisor *. * The reason of this rule is evident; for, since the divisor multiplied by the quotient gives the dividend, therefore the number of decimal places in the dividend is equal to those in the divisor and quotient, taken together, by the nature of multiplication; and consequently the quotient itself must contain as many as the dividend exceeds the divisor. The investigation may, as in the last case, be here given in a general form. N and be the divisor and dividend respectively, which designates = When m is greater than n, there will be a removal of the decimal point m -n places to the right, or, in other words, if the division be complete, m - n ciphers must be added; but when n is greater than m, there will be n-m decimal places. Another way to know the place for the decimal point is this: The first figure of the quotient must be made to occupy the same place of integers or decimals, as that figure of the dividend which stands over the unit's figure of the first product. When the places of the quotient are not so many as the rule requires, the defect is to be supplied by prefixing ciphers. When there happens to be a remainder after the division; or when the decimal places in the divisor are more than those in the dividend; then ciphers may be annexed to the dividend, and the quotient carried on as far as required. WHEN the divisor is an integer, with any number of ciphers annexed; cut off those ciphers, and remove the decimal point in the dividend as many places farther to the left as there are ciphers cut off, prefixing ciphers, if need be; then proceed as before. Hence, if the divisor be one with ciphers, as 10, 100, or 1000; then the quotient will be found by merely moving the decimal point in the dividend so many places farther to the left, as the divisor has ciphers; prefixing ciphers, if need be. WHEN there are many figures in the divisor; or when only a certain number of decimals are necessary to be retained in the quotient; then take only as many figures of the divisor as will be equal to the number of figures, both integers and decimals, to be in the quotient, and find how many times they may be contained in the first figures of the dividend, as usual. Let each remainder be a new dividend; and for every such dividend, leave out one figure more on the right-hand side of the divisor : remembering to carry for the increase of the figures cut off, as in the second contraction in multiplication. Note. When there are not so many figures in the divisor as are required to be in the quotient, begin the operation with all the figures, and continue it as usual till the number of figures in the divisor be equal to those remaining to be found in the quotient; after which begin the contraction. EXAMPLES. 1. Divide 2508-92806 by 92-41035, so as to have only four decimals in the quotient, in which case the quotient will contain six figures. In this operation, as in multiplication, a correction column to the right of the vertical line should be kept. The method itself is obviously only a rejection of the figures which do not contribute to the result within the prescribed limits. 2. Divide 4109-2351 by 230·409, so that the quotient may contain only four decimals. Ans. 17.8345. 3. Divide 37.10438 by 5713.96, that the quotient may contain only five decimals. Ans. 00649. 4. Divide 913.08 by 2137.2, that the quotient may contain only three decimals. REDUCTION OF DECIMALS. CASE I. To reduce a vulgar fraction to its equivalent decimal. DIVIDE the numerator by the denominator, as in division of decimals, annexing ciphers to the numerator, as far as necessary; so shall the quotient be the decimal required *. * It will frequently happen (indeed always when the fraction in its lowest terms has in its denominator any factors besides 2 and 5, or powers and products of these,) that the division will EXAMPLES. 1. Reduce to a decimal. 244 x 6. Then 47 61.750000 never terminate. In this case, it is commonly sufficient to carry it to some specific number of quotient places, and neglect the remaining ones as of comparatively no value. If, however, the question be one in which the accurate result instead of the approximation is required, it will be necessary to work by vulgar fractions instead of decimals. In this case any other decimal quantities that have presented themselves amongst the data of the question can be readily thrown into the usual fractional form for greater facility of combination with the fraction above mentioned. A rapid and elegant method of throwing a vulgar fraction, whose denominator is a prime number, into a decimal consisting of a great number of figures, is given by Mr. Colson, in page 162 of Sir Isaac Newton's Fluxions. It will be readily understood from the following example : Let be the fraction which is to be converted into an equivalent decimal. Then, by dividing in the common way till the remainder becomes a single figure, we shall have = '034485 for the complete quotient, and this equation being multiplied by the numerator 8, will give = 2758403, or rather = 275868: and if this be substituted instead of the fraction in the first equation, it will make = 03448275865. Again, let this equation be multiplied by six, and it will give $ = 2068965517; and then by substituting as before =034482758620689655177; and so on, as far as may be thought proper; each fresh multiplication doubling the number of figures in the decimal value of the fraction. In the present instance the decimal circulates in a complete period of 23 figures, i. e. one less than the denominator of the fraction. This, again, may be divided into equal periods, each of 14 figures, as below: 03448275862068 in which it will be found that each figure with the figure vertically below it makes 9;0+9=9; 3+6=9; and so on. This circulate also comprehends all the separate values of ..... in corresponding circulates of 28 figures, only each beginning in a distinct place, easily ascertainable. Thus, 2 = 06896...... beginning at the 12th place of the primitive circulate. 2 = 103448...... beginning at the 28th place. So that, in fact, this circle includes 28 complete circles. The property of circulation of some number of the figures in periods is not one peculiar to this or any other of the interminable decimals, but belongs to them all. Sometimes the period is composed of a single figure, as 333...... which is the decimal expansion of; sometimes of a considerable number, as in the example above given from Colson. It is, however, never composed of a number of places so great as is expressed by the denominator: as the expansion of is composed of one term, and one is less than the denominator, three; and the period of the expansion of is composed of 28 places, and less therefore than the denominator 29. Sometimes the circulating periods are composed of a half, or a fourth of the number of places that would be expressed by subtracting one from the denominator. Whenever in the division a remainder occurs that has occurred before, (the number brought. down from the dividend being necessarily the same in both cases, viz. 0,) then the same quotient figure will again occur, and the same remainders will occur after both; and hence the same quotient figures, and the same remainders, will continually succeed, till the first mentioned remainder again occurs. A continual circulation of the same process will thence take place without end. The same circumstances would obviously take place if, instead of bringing down ciphers, as above mentioned, we brought down any other figure from the dividend, or even the figures of any circulating dividend. That there must be a less number of figures in the circulating period, than is expressed by the denominator, appears from this. The only remainders that can exist are all less than the denominator, and being integers, their number is less than is expressed by it and hence the number of steps in the division that takes place without falling upon the same remainder must To find the value of a decimal in terms of the inferior denominations. MULTIPLY the decimal by the number of parts in the next lower denomination; and cut off as many places for a remainder, to the right-hand, as there are places in the given decimal. Multiply that remainder by the parts in the next lower denomination again, cutting off for another remainder as before. Proceed in the same manner through all the parts of the integer; then the several denominations separated on the left-hand will make up the answer. Note. This operation is the same as reduction descending in whole numbers. at the most be one less than the number expressed by the denominator; and hence again the number of quotient figures must be at most one less than the same number. That is, the quotient is composed of such repeating circles as we have stated. Any further examination of this subject (which, nevertheless, is a very curious and a very important one) would be incompatible with the limits of this work. The method of finding the value of such a decimal will, however, be found in the Chapter on Geometrical Progression : and we may refer the inquiring student also to Mr. Goodwin's Tables of Decimal Circles, and to the Ladies' Diary for 1824. A convenient notation has been used to designate the circulating period which is that of putting a dot over the first and last figures of the period. Thus 72-968625 signifies that 625 is the circulating period. When the period is at only one place, there is but one dot required, as in the value of, which is 3. The reverse problem, of finding the finite fractional value of an interminable decimal, will be found in Geometrical Progression, in the Algebra. |