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To prove the operation, multiply the multiplier by the multiplicand, and mark off decimals in the product as before; or cast out the mines, as in simple multiplication.
1. Multiply 12.34567 by 3.5847.
Thus, 12.34567 3.5847 multiplier.
37037Ol 107541 Prod. 44.255593249 71694
44.255593249 Prod. .
Here are 5 decimals in one factor, and 4 in the other, that is, 9 in both ; I
therefore count 9 places from the right of the product, and put the decimal mark to the left of the ninth place.
2. Multiply 7,38142 by .000078.
OPERATION. Thus, 7.38142 .000078 Proof- Earplanation. T5905.136 6 Here the factors contain 11 deci5166994 7 6 mals; 3 ciphers are therefore pre6 fixed to the product to make up 11. Prod. .00057575.076
= 'o. by Vulgar Fractions; also .5 × .3 = .15 by the rule: but 15. = . 15 100 100
by Decimal Notation; that is, the result obtained by this rule, and that obtained by Vulgar Fractions, are the same: the rule is therefore true.
If any doubt should remain respecting the truth of the rule when there are whole numbers concerned, let the factors in ex. 1. be turned into vulgar fractions and multiplied; thus 12.34567 = 1234567, and 3.5847 = 35847 5
1234567 × 35847 _44355523249_ 4 255523249 100000 10000 l O00000000 1000000000 = 44.255523249, as in ex. l.
wherefore by multiplication
. Multiply 4.82 by 3.53. Prod. 17,0146.
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. Prod. 1234,567. 10. Multiply .98765 by 100. 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 I 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 25,874856 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.
* 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 123456789 by 3697428, leaving only 4 places of decimals in the product.
I 11 110 under the 8, and therefore the highest figure 8642 being the first place of decimals must fall under
14. Maltiply 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 4 - . - - - - 10000 and To” therefore, inverting the divisor, and multiplying, we shall have ExAMPLEs. 1. Divide 44.80515 by 3.45. OPERATION. o 3.45)44.80515(12,987 quotient, 345 1030 690 Erplanation. 3405 The division being performed in the same 3105 manner as in whole numbers, I find there are 5 -:- decimals in the dividend and 2 in the divisor; 3001 5 exceeds 2 by 3, I therefore mark off 3 from 2760 the right of the quotient for decimals. 24.15 24.15
OPERATION. .01234).00006400(.00518638 &c. quotient. 6170 Earplanati Too a planation. 23OO Beginning at the first significant figure 1934 6, I find that two ciphers must be added 10660 on to the dividend; I afterwards bring 98.72 down ciphers, and continue the operation - as far as is thought necessary. Then 5 788O ciphers brought down added to 8 decimals 7404 in the dividend make 13: now there are —- 5 decimals in the divisor, therefore 5 from 47.60 13 and 8 remains to be marked off in the
3702 quotient; but there are only 6 figures; I 105so therefore add on 2 ciphers to the left, and 9§7 2 prefix the decimal mark.
* x *-* –-ol" – .616, by Art.211; but .216 divided by 10000 4 40000 1000 .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.