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Explanation of the Characters made use of in
this Treatise. THERE
HERE are various characters or marks used throughout the work, to denote several of the operations and propofitions, the chief of which are as follows : + Signifies plus, or addition ;
minus, or subtraction; Х
square root ; 31 cube root. Thus,
5+3 denotes that 3 is to be added to s ; 6—2
2 is to be subtracted from 6; 7X3
is to be multiplied by 3 ; 8
8 is to be divided by 4 ; 4 5X4X2
=16 5 is to be multiplied by4, their pro2.5
duct again multiplied by 2, and that product divided by 2.5 ; the quotient
is equal to 16: 2 : 3 :: 4 : 6
2 is to 3, as 4
is to 6
; 6+4=10 6 and 4 added is equal to 10; 9--5=4 5 fubtracted from 9, remainder equal to 4 ;
square root of 3 ;
IT is indispensably required of every one who would
learn this, or any other science, to acquire some knowledge of decimal fractions, as all the instruments used in making calculations are decimally divided. I therefore apprehend it will not be improper to give, by way of introduction, a succinct account of decimal fractions.
Numeration of Decimals. A Decimal fraction is such, whose denominator is not expressed, but understood ; as is an unit with as many cyphers annexed as there are places in the numerator. 5
125 So will be expressed thus
and - thus .25 ; and
thus .125, &c. They have a point or comma prefixed, to distinguish them from an integer.
A cypher placed to the left hand of an integer or to the right hand of a decimal, neither increaseth nor decreaseth the value : but placed to the right hand of an integer, increaseth the value ten-fold ; and to the left hand of a decimal, decreaseth the value ten-fold : Two cyphers placed to the right hand of an integer, increaseth the value one hundred fold ; and to the left hand of a decimal, decrealeth the value one hundred fold, &c. Observe the following table.
The Table of Numeration.
o Hundreds of Millions.
co Hundred parts.
Thousand parts. ö Tenth parts.
Hundred Thousand parts. e Ten Thousand parts. - Millions of parts. 6 Hundred Millions of parts. c. Ten Millions of parts.
In the preceding table you may observe, that as integers increase in a ten-fold proportion to the left hand, so deci. mal fractions decrease in a ten-fold proportion to the right hand.
Addition of Decimals. Addition of decimals is performed in the same way as addition of integers, only observe the following
In writing down the numbers, take great care to place units under units in integers, and tenths under tenths în decimal parts : Add 2s in addition of integers ; point off decimal places equal to the greatest number of places in any of the lines that are to be added.
EXAMPLE, Required the fum of the following mixed numbers, viz. 26.489-2.05--18,-.5632--,82 and .076.
47.9982 the sum. Ex. 2. Required the sum of 123.45-17325.146.66667 2.5 and 27.00625.
Ans. 17484-7,2292. Subtraction of Decimals. Subtraction of Decimals is also performed in the same way as fubtraction of integers.
Place the numbers under each other according to the value of their places, as in the last rule : Subtract as in whole numbers, and point off the decimal places as in addition.
Ans. 89.591428 difference,
Multiplication of Decimals. By multiplication of decimals, the product of any two given decimal fractions or mixed numbers are found.
Place the factors, * and multiply them together as in
* The multiplicand and multiplier are called factors, because they constitute the product,
whole numbers. Then point off in the product, just as. many places of decimals as there are decimal places in both the factors. But if there be not so many figures in the product as there are in both the factors, then supply the defect by prefixing cyphers to the left hand.
369 Ans. .0771955 product. Ans. .0004132 products, Ex. 3. Multiply .764 by .28
Ans. .21392 Ex. 4. Multiply 79.25 by .459
Ans. 36.37575 Ex. s. Multiply 20.0291 by 35.45
Ans. 710.031595 Ex. 6. Multiply 32.0752 by .0325
Ans. 1.04244400 Ex. 70 Multiply ..01472 by 1045 Ans. .0001538240
Division of Decimals. By division of decimals the quotient arising from dividing one decimal fraction by another, also from dividing a mixed number by a mixed number, decimal fraction, or whole number, &c. &c. the method of operation being the same as in whole numbers--the only difficulty lies in determining the true value of the quotient, or in pointing off the right number of decimal places. To effect which, observe the following general
Point off as many figures in the quotient, as will make those pointed off in the divifor equal to those in the dividend ; taking notice, if there are not so many in the quotient, to prefix cyphers to the left hand. Allo, if the divis dend be an integer, or have less pointed off than in the divisor, add cyphers to the right hand of the dividend, until they be equal—but more will in general be found convenient.
Divide 742.633 by 41.
Divide .48 by 173a