DE C

230

DEC

Example II. To find the decimal of 5 seconds and 4 thirds.

Now in this example, there are three places of duodecimals,therefore

0 5 412

12 ) 64 number of thirds in the whole.

12 ) 5.33333

12 ) .44444

.03703

And thus for any other number of denominations whatever.

If each foot of our measuring-rules for taking the linealdimensions were divided into ten parts, instead of twelve,and each of these ten parts again into ten others, we shouldhave no occasion for reduction of decimals, as the rule itselfwould give the decimal. In most cases we should not thenhave occasion to work with more than two places of decimals;the tenth part of the tenth part, that is, the hundredth partof a foot, is very nearly equal to the eighth part of an inch,or the ninety-sixth part of a foot, being only a small matterless than the eighth of an inch. Were measuring-rules thusdivided, the work by decimals would be much shorter thanany other denomination whatever. ■ The principal reason ofoperations in measuring by decimals being longer thanduodecimals, arises from the necessity of reducing the duode-cimals to decimals,, and this in many cases cannot be donewith the same accuracy, without having four or five denomi-nations. In practical cases, there are never more than threeplaces of duodecimals, and if rules were divided decimally,there would not be more than three places.

A decimal part of a foot being given, to find its equivalentin duodecimals.

Multiply the decimal by 12, cut off as many decimals fromthe product as there are places of figures in the multiplicand,from the right-hand to the left, and the figure or figuresremaining on the left, if any, will show the number of inches:multiply the number of decimals so cut off, if any, again by12, and cut off as many figures from the left-hand of the newproduct as there are decimal places in the multiplicand, andthe remaining figures on the left will show the seconds, ifany. Proceed in this manner as often as there is a remainder,or as often as may be thought necessary to obtain a sufficientdegree of accuracy, and the places cut off will be equivalentto the given decimal, if no remainder, and very nearly so ifthere is, but in this case something less.

Example. Reduce .44005 of a foot to a decimal.

.44005

12

5.28060

12

3.3672

12

4.4064

12

4.8768

To add decimals ; write down the several parts undertheir respective denominators, viz., all the points in a verticalline, the tenths in a succeeding vertical line, and so on ; addthe several columns, as in common addition, from right toleft; place a point under the column of points, or cut off as

many decimal parts from the right-hand figure towar Bleft, as there are columns, and the figure or figure 8 ^the left-hand side of the point, if any, will b eand those upon the right-hand side of the pointdecimals.

Example I. Add the following decimals togethe >.7854, .07958, .5236.

.7854

.07958

.5236

tbe

1.38858 the sum required., optTo subtract decimals ; place the numbers as m . 3 i3the less under the greater, and perform the opera®subtraction of integers.

Example I. Subtract .25 from .75.

.75

.25

T( . , -50 the number required. t j, e y

It the parts of a foot are given in feet, inches, ® ’’ ; 3 cemust be reduced to the farthest denomination from tb e Vof feet. . 0 i

To multiply in decimals, proceed as in multip licat ‘° g r gtintegers, and point as many figures, beginning withfigure from the right in the product, as the number of d ^in both factors ; and the remaining figures, if an f’ u fleft-hand, will be integers ; but if there are not a* ^

figures in the product as in both, the deficiency 10made up by the addition of one or more ciphers.

Example I.

Multiply .9087by .852

18174

45435

72696

Product .7742124

In large decimals, the work may be contract, , iC eWrite the units’ place of the multiplier under0 the decimals in the multiplicand, whose place y®reserve in the product; write the other figures in «*plier in a contrary order. . 0 riff?!

Begin with the figure of the multiplier nearest * r ig» lhand, and multiply by the next figure awardsof it, in the multiplicand, if any; and if this prod<f c '^Jtor above five, in the number of tens in the P r< f tb eone more than there are tens in the last product to fofproduct; then set down the overplus above the pro c ® tfirst figure, and carry the tens to the next P lace ^" ce ed ,vl .

through the line as in common muitipiicat.on.^-j.o^

pvpmr ntlipi. Imp in flip enmp manner, Obsef ; (1 ]* Ih®

■eft

every other line in the same manner, ou»c, ^j erto place the first figure of every line direct y^ ^ v ill cfigure of the last line, and the product so °

answer. jjiulPP 111 "

Example I.—What is the product of .9° ? , b gi-

.852, in order to retain four places of. de ^ ml pon ini° a ^

By contraction..9087258.0

Bytbe-^r

g52

7270 = 908 X 8 +454 = 90 X 5 +18 = 9x2 +

7742

I