Explanation of the Characters made use of in this Treatife. THERE are various characters or marks ufed throughout the work, to denote feveral of the operations and propofitions, the chief of which are as follows: +Signifies plus, or addition; -: : :: : 3√ Thus, minus, or fubtraction; cube root. 5+3 denotes that 3 is to be added to 5 ; 2.5 =16 2:34:6 2 is to be fubtracted from 6; 8 is to be divided by 4 ; 5 is to be multiplied by4, their product again multiplied by 2, and that product divided by 2.5; the quotient is equal to 16: 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 ; √3 fquare root of 3; cube root of 5 ; 7 is to be fquared; 9 is to be cubed, &c. Introduction. DECIMAL FRACTIONS. IT is indifpenfably required of every one who would learn this, or any other fcience, to acquire fome knowledge of decimal fractions, as all the inftruments used in making calculations are decimally divided. I therefore apprehend it will not be improper to give, by way of introduction, a fuccinct account of decimal fractions. Numeration of Decimals. A Decimal fraction is fuch, whofe denominator is not expreffed, but understood; as is an unit with as many cyphers annexed as there are places in the numerator. thus .125, &c. They have a point or comma prefixed, to diftinguish 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, increafeth the value one hundred fold; and to the left hand of a decimal, decreafeth the value one hundred fold, &c. Obferve the following table. The Table of Numeration. Hundreds of Millions. Tens of Millions. Hundreds of Thousands. Tens of Thousands. Thousands. Hundreds. Tens. Hundred Thousand parts. Ten Thousand parts. Hundred parts. Tenth parts. Thousand parts. Hundred Millions of parts. ∞ Ten Millions of parts. Millions of parts. In the preceding table you may obferve, that as integers increase in a ten-fold proportion to the left hand, so decimal fractions decrease in a ten-fold proportion to the right hand. parts. Addition of decimals is performed in the fame way as addition of integers, only obferve the following RULE. In writing down the numbers, take great care to place units under units in integers, and tenths under tenths in decimal parts: Add as 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. 26.489 18. .5632 .82 Ans. 6076 47.9982 the fum. Ex. 2. Required the fum of 123-45-17325.1-6.66667 2.5 and 27.00625. Ans. 17484.72292. Subtraction of Decimals. Subtraction of Decimals is alfo performed in the fame way as fubtraction of integers. RULE. 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. EXAMPLE. To find the difference between 91.73 and 2.138572. 95.73 Ans. 89.591428 difference. Ex. 2. Subtract 1.9185 from 2.73. Ans. .8115 Ans. 209.90858 Ans. 2713.084 Multiplication of Decimals. By multiplication of decimals, the product of any two given decimal fractions or mixed numbers are found. RULE. Place the factors,* and multiply them together as in The multiplicand and multiplier are called factors, because they constitute the product. B 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 fo many figures in the product as there are in both the factors, then fupply the defect by prefixing cyphers to the left hand. Divifion of Decimals. By divifion of decimals the quotient arifing from dividing one decimal fraction by another, alfo from dividing a mixed number by a mixed number, decimal fraction, or whole number, &c. &c. the method of operation being the fame 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, obferve the following general RULE. Point off as many figures in the quotient, as will make those pointed off in the divifor equal to thofe in the dividend; taking notice, if there are not fo many in the quotient, to prefix cyphers to the left hand. Alfo, if the dividend be an integer, or have lefs pointed off than in the divifor, add cyphers to the right hand of the dividend, until they be equal-but more will in general be found convenient. EXAMPLES. Divide 742.633 by 41. Divide .48 by 173 |