ARITHMETIC.
237
right hand of a <
l decimal, have no signification ; but when placedon the left hand diminish the value ten times each.
The notation and numeration of the decimals will be still moreObvious from the following examples:
4.7 Four and seven-tenth parts.
.47 Four-tenth parts, and seven hundredth parts, or 47
hundredth parts.
Four hundredth parts, and seven thousandth parts, or47 thousandth parts.
Four, and seven hundredth parts.
Four, and seven thousandth parts.
The column next the decimal point is sometimes called deci-mal primes, the next decimal seconds; and the next decimalthirds, and so on.
.047
4.07
4.007
Addition and Subtraction of Decimals.
(44.) Write the proposed numbers under each other, accordingto the value of their places, as in the whole numbers; in whichorder, the decimal points will stand directly below each other;then add or subtract as in whole numbers, placing a decimal pointin the sum or difference, straight below the other points.
Examples.
169-3
27-48
213-97 from 321-02
•04 sub. 26-481
Sum 410-79 Dif. 294-539
Multiplication of Decimals.
(45.) Write down the factors, and multiply exactly as in in-tegers, placing the dec imal point to the product, so as to makejust as many decimals in it, as there are in both factors; and ifthere be not as many ligures in the product, as there ought tobe decimals, prefix cyphers to supply the deiiciency,
Examples.
52-3 5-12
2-41 -0098
Contracted.
Common Hay.
27-14986
27-14986
53014-29
• 92-41035
24434874
131574930
542997
81144958
108599
27141980
2715
108599144
81
* 542997 2
14
24434874|
2508-9280
2508-9280|650510
Division
(47.) Divide as in integers; and to know how many decimalsmust be in the quotient, observe either of the following rules:
Rule I. The first figure of the quotient must possess the sameplace of decimals or integers, as doth that figure of the dividendwhich stands over the units place of the first product.
Rule If. The decimal places of the divisor and quotients to-gether, must be equal in number to those of the dividend.Whence, if the number of decimals in the divisor be taken fromthe number in the dividend, the remainder will be the number inthe quotient, if, in any case, there be a remainder after all thedividend ligures are used; the quotient may be continued to anynumber of decimals by subjoining a cypher continually to the lastremainder. And whenever the number of figures in the quotientis less than the required number of decimals, prefix cyphers t*supply the defect.
Examples.
52.3)73.743(1.41 13.5)-243(-01S
2144 1080
..523 _
.016)5.04(315
24
.80
523 4096
2092 4608
1046 --
Contractions.
(46.) When decimals are to be multiplied by 1, with any num-ber of cyphers, it is done by only removing the decimal point somany places farther to the right hand, as there are cyphers in themultiplier, and subjoining cyphers, if need he.
Example.
The product of 51-3 and 1000 is 51300.0.
When the product will contain many more decimals than arenecessary for the present purpose, the work may he contractedthus: write the units figure of the multiplier straight under suchdecimal place of the multiplicand, as the last of the product isintended to be, writing the other figures of the multiplier in aninverted order; then, in multiplying, rejeet all the figures in themulti; licand, which are on the right of the multiplying figures;writing the products down so, that their right hand figures fallstraight below each other ; and carrying to such right hand figuresfrom the product of the two preceding figures in the multiplicandthus, viz. 1 horn 5 to 15, 2 from 15 to 25, 3 from 25 to 35, &c.that is, carry as usual when the units place is less than 5, andcarry one more when it is greater than 5, ami the sum of the lineswill be the product to the number of decimals required, and willbe seldom wrong in the last figure.
Example.
Multiply 27-149S6 by 92-41035, so as to retain only four placesof decimals in the product,voi. i.— no. 15.
(4S.) Contractions.
1. To divide decimals by 1, w.th any number of cyphers, re-move the decimal point in the dividend as many places further tothe left as there are cyphers in the divisor, prefixing cyphers, ifneed be, the result is the quotient required.
217.3 — 100 = 2.173.
2. If the divisor be any integer with cyphers at the end, cutoff the cyphers, and remove the decimal point in the dividendas many places to the lett, and divide by the integer as before.
953-1-21000 = -953-f- 21.
When the number of figures in the divisor is great, the divisionat large will be very troublesome, but may be contracted thus:having by the first general rule found what place of decimals orintegers the first figure of the quotient will possess; considerhow many figures of the quotient will serve the present purpose ;then take the same number of the left hand figures of the divisor,and as many of the dividend figures as will contain them (less thanten times); by these find the first figure of the quotient; and foreach following figure, divide the last remainder by the divisor,wanting one figure to the right, more than before, but observingwlrat must be carried to the first product, for such omitted figures,as in the second contraction of multiplication ; and continue theoperation till the divisor is exhausted. When there are not asmany figures in the divisor as are required to be in the quotient,begin the division with all the figures as usual, and continue it tillthe number of figures in the divisor, and those remaining to befound in the quotient, be equal, after which, use the contraction.
Example.
Divide 2508-92806 by 92-41035, so as to have four decimalsin the quotient, in which case, the quotient will contain sixligures.
3 P
Common