Prod. 17.0146. 3. Multiply 4.82 by 3.53. 225. When the multiplier is a whole number, consisting of an unit with ciphers subjoined. RULE. Remove the decimal mark as many places to the right as there are ciphers in the multiplier *. 9. Multiply 123.4567 by 10. 10. Multiply .98765 by 100. Prod. 1234.567. Prod. 98.765. 11. Multiply .00001 by 100000. Prod. 1. 226. To contract the operation, so as to retain in the product as many decimals only as may be thought necessary. RULE I. Count off from the left hand of the decimals in the multiplicand, as many figures as are intended to be reserved in the product; and put a point over the last of these. II. Place the units figure of the multiplier under the pointed figure, then write down the rest of the multiplier so, that the whole may stand in an inverted order, viz. the last figure first, and the first last. III. In multiplying, always begin at that figure in the multiplicand which stands one place to the right of the multiplying figure, and carry 1 for all numbers from 5 to 15, 2 from 15 to 25, 3 from 25 to 35, &c.; but this mode of carrying is to be observed only in the first place, namely, to the figure over the multiplying figure, the product of which is the first you set down; for the rest, you are to set down and carry in the usual way. IV. Place the several products so that all the right hand figures may stand under each other in a line; add up the pro The value of any figure is increased tenfold, an hundredfold, a thousandfold, &c. by its being removed one, two, three, &c. places to the left, or (which is the same thing) by removing the decimal mark so many places to the right, as appears from Art. 18; wherefore the rule is plain. ducts, and mark off from the right hand as many decimals as were proposed to be reserved". Operations in this rule are proved by common multiplication. Art. 224. 12. Multiply 25.374856 by 5.35647, reserving only five decimal places in the product. Beginning at the decimal mark, I count 5 decimals; over the fifth I put a dot, and place the units figure 5 of the multiplier under the dotted figure, and dispose of the other figures so that the multiplier may stand backwards. I then begin, 5 times 6 are 30; put nothing down, but carry 3; then 5 times 5 are 25 and 3 are 28; put down 8, and, carrying the 2, proceed through the whole line as usual. For the second line, I begin 3 times 5 are 15; put nothing down, but carry 2; then I multiply the 8, carry 2, and set down throughout this line as in the first line. In the third line, I begin by multiplying the 8 for carrying, but set down the product of the 4; in the fourth, I begin at the 4, and set down at the 7; in the fifth, I begin at the 7, and set down at the 3; in the sixth line, I begin with the 3, and set down the product of the 5. u When the factors contain a great number of decimal places, and but few are required in the product, much labour and time will be saved by the application of this rule; but care should be taken to work for one or two figures more than are wanting, as the right hand decimal arising from the contracted operation will sometimes unavoidably be wrong. The reason of placing the units figure of the multiplier under the figure to be reserved is this, namely, that the right hand figure in every product is of the same denomination with that under which the said units figure of the multiplier stands. The reason for reversing the multiplier will appear by consulting the operation, and comparing it with the proof; it will be seen that the first, second, third, &c. lines from the top in the former, are respectively equal to the first, second, third, &c. from the bottom in the latter. The reason for the increase in carrying to the first figure in each line is, that the deficiency arising from the loss of what would be carried in the multiplication and addition of the figures omitted may be compensated as nearly as possible. 13. Multiply 1234.56789 by .3697428, leaving only 4 places of decimals in the product. 14. Multiply 2.38645 by 8.2175, retaining only 4 decimal places in the product. Prod. 19.6107. 15. Multiply 128.678 by 38.24, retaining one decimal place only in the product. Prod. 4920.5. 16. Multiply 325.1234567 by 23.987654, with three decimals only in the product. Prod. 7798.948. 17. Multiply .374853 by .0031245, with 7 decimals in the product. Prod. .0011713. 227. DIVISION OF DECIMALS. RULE I. Divide as in whole numbers, then count the decimal places in the dividend, and also in the divisor, and mark off as many decimals from the right hand of the quotient as the former exceeds the latter. II. If there are not figures enough in the quotient, add as many ciphers to the left hand as will make up the difference. III. When there is a remainder, the quotient may be carried to any length, by bringing down ciphers, and continuing the division; but the ciphers brought down must be considered as decimals belonging to the dividend, and must be counted with those which actually stand in the dividend, in order to estimate the number of decimals to be marked off in the quotient *. × The truth of this rule may be shewn by Division of Vulgar Fractions; thus, let .2464 be divided by .4; these numbers reduced to fractions are 2464 and ; therefore, inverting the divisor, and multiplying, we shall have 10000 4 10 .01234).00006400(.00518638 &c. quotient. 6170 2300 1234 9872 7880 7404 4760 3702 10580 9872 Remainder 708 Explanation. Beginning at the first significant figure 6, I find that two ciphers must be added on to the dividend; I afterwards bring down ciphers, and continue the operation as far as is thought necessary. Then 5 ciphers brought down added to 8 decimals in the dividend make 13: now there are 5 decimals in the divisor, therefore 5 from 13 and 8 remains to be marked off in the quotient; but there are only 6 figures; I therefore add on 2 ciphers to the left, and prefix the decimal mark. .4, according to the rule, gives also .616; wherefore this rule agreeing with one, the truth of which is established, is shewn to be right. But it is not necessary to have recourse to Vulgar Fractions; the rule is plain from the nature of simple Division, except the right placing of the decimal mark in the quotient, which may be thus explained: if the quotient be multiplied by the divisor, with the remainder added in, the result will be the dividend; whence, by multiplication, the dividend will have as many decimal places as there are in the divisor and quotient together; wherefore the quotient must contain as many decimal places as the number of decimals in the dividend exceeds that in the divisor; which was to be shewn. 3. Divide 17.0146 by 3.53. Quot. 4.82. 4. Divide .51731381 by 9.23. Quot. .056047. 5. Divide 123.4531579 by 2.5728. Quot. 47.98396, &c. 228. To contract the operation. RULE. Take as many of the left hand figures of the divisor as the quotient is intended to consist of; divide by these only, at the first step of the operation, and at each succeeding step cut off one figure (or more figures if necessary) from the divisor, instead of bringing down from the dividend, using only the figures not cut off; in multiplying these by the quotient figure, you must observe to carry from the product of the preceding figure cut off, in the same manner you did in contracted multiplication, viz. 1 from 5 to 15, 2 from 15 to 25, &c. but carry as usual after you begin to set down. To determine the place of the decimal mark in the quotient, observe under what figure of the dividend the units place of the product of the divisor by the quotient figure stands, and that will be the value of the first figure in the quotient; if it stand under units, the first figure only will be a whole number; if under tens, two figures will be whole numbers; if under primes, the first figure will be primes; if under seconds, a cipher must be prefixed to the quotient, &c.2 y Observing to put a cipher in the quotient whenever a figure is cut off, and the remaining figures will not go in the number to be divided. 2 The place of the decimal mark in the quotient may be known two ways, viz. either from the decimals, or from the whole numbers; the former has been explained, and the latter may be shewn as follows. Let 10.9 divide 1234.56; now it is plain that if we had only the integers to operate with, (namely, 1234 to divide by 10,) the quotient would be 123, that is, the highest denomination I is of the same denomination with the figure 2 of the dividend under which the units figure 0 of the divisor stands, namely, (in this instance,) hundreds; wherefore, in the above example, seeing the units place (0) in actual division stands under the hundreds place (2), the highest place in the quotient will evidently be hundreds. Again, let 99.9 divide 1.1000; now 99 will not divide 1 (unit), and therefore will not produce units in the quotient; it will not divide 11 tenths, and therefore will not produce tenths; but it will divide 110 hundredths, and therefore will produce hundredths, or, the first or |