RULE.-Divide as in integers. Point off as many decimal places in the quotient, as the dividend has more than the divisor; if necessary place cyphers to the left of the quotient. If the divisor has more digits than the dividend, add cyphers to the right of the dividend. When there is a remainder, the quotient may be carried to any degree of exactness, by annexing cyphers to the remainder. In order to multiply a decimal by 10, remove the point one digit to the right; if by 100 remove it two places, and so on. Tc divide by 10, 100, &c., remove the decimal place of the divi leud as many places to the left as there are cyphers. F REDUCTION. Case L.-To reduce a vulgar fraction to a decimal. RULE. Divide the numerator by the denominator; annexing as many cyphers to the numerator as may be necessary. off as many decimal places in the quotient, as there were cyphera annexed to the numerator. Reduce to a decimal. Point Reduce to a decimal. 4)300 2)10 '75 Ans. Case. II. To reduce a decimal to a vulgar fraction. RULE.-Make the given decimal the numerator, and place under it, for a denominator, a unit with as many cyphers as there are places in the decimal. Case III.-To reduce numbers of a lower denomination to the decimal of a higher. RULE. Write the given numbers, if more than one, directly under each other, beginning with the lowest, and divide by as many of the lower as make one of the higher, annexing cyphers if necessary. Reduce 12s. 3d. to the deci- | mal of a pound. 12) 3·00 20) 12.250 6125 Ans. Here the shillings and pence are placed under each other, beginning with the lower; and each divided by as many of the lower as make one of the higher. Reduce 168. 6d. to the decimal of a pound. 4) 3·00 12) 6.7500 20) 16.56250 828125 Ans. Here the farthings, pence, and shillings are placed under each other, beginning with the lowest; each is then divided by as many of the lower as make one of the higher. 1. Reduce 198. 5d. to the decimal of a pound. 5. Reduce 3 cwt. 2 qrs. S Ib to the decimal of a cwt. 10. Reduce 5 dwt. 12 grs. to the decimal of an ounce. 11. Reduce 2 roods, 12 perches, to the decimal of an acre. 12. Reduce 17 yards, 1 foot, 6 inches, to the decimal of a mile. Case IV. To find the value of a decimal. RULE.-Multiply the decimal by as many of the next lower denomination as make one of the given denomination. Point off, from the product, as many decimal places as are in the given decimal. Proceed thus to the lowest denomination. The digits on the left of the points are the value of the decimal. 1. What is the value of £7634? 2. What is the value of £•3412? 3. What is the value of £0076? 4. What is the value of 764 cwt.? 5. What is the value of 6. What is the value of 7. What is the value of 8. What is the value of 9. What is the value of 10. What is the value of 11. What is the value of 12. What is the value of 13. What is the value of 936 lb avoirdupois ? 14. What is the value of 987 oz. troy? 15. What is the value of 779 lb avoirdupois? INVOLUTION. When a number is multiplied by itself, the product is called a power, and the number multiplied the root. Thus 2X2-4: here 4 is the square or second power of the root 2. Again, 2×2×2=8: here 8 is the cube or third power of the root 2. Again, 2×2×2×2=16, here 16 is the fourth power of the root 2. 1. Find the second power of 8. 2. Required the third power of 13. EVOLUTION. EVOLUTION is the method of finding the roots of numbers. EXTRACTION OF THE SECOND OR SQUARE ROOT. To extract the square root of any given number, is to find a number, which, when multiplied by itself, will produce the given number. 106929 (327 9 62) 169 124 What is the square root of 106929? RULE WITH EXAMPLE.-Divide the given number into periods of two digits each, by placing a point over the unit digit, and over every alternate digit towards the left. Find the square root, 3, of the first period, 10, and place it in the quotient. Subtract the square of it, 9, from the first period, 647) 4529 and to the remainder annex the next period, 69, for a dividend. Double 3, the root already found, for a divisor, and supposing the unit digit, 9, omitted, find how often it, viz. 6, is contained in the dividend. It is contained 2 times; place the 2 then both in the quotient and the divisor. Multiply the divisor, 62, by the 2, and subtract the product, 124, from the dividend. Bring down another period, and proceed thus till all the periods are brought down. 4529 If there be a remainder after all the periods are used, periods of cyphers may be annexed; when the result will be decimals. Should there be decimals in the given number, still the pointing is to begin from the unit's place of the integers, and a point to be placed over every alternate figure both right and left. The square root of a fraction is found by extracting the square rcot of the numerator for a new numerator, and the square root of the denominator for a new denominator; if, however, this cannot be done, let the fraction be reduced to a decimal and the root extracted as before. 1. What is the square root of 30976? 8. What is the square root of 169! EXTRACTION OF THE THIRD, OR CUBE ROOT. To extract the Cube Root of any given number is to find a number which, when multiplied twice by itself, will produce the given number. Find the Cube Root of 12812904. RULE WITH ExAMPLE.-Divide the given number into periods of three places, beginning at the place of units. Place the cube root of the first period 2, in the quotient, and subtract its cube 8, from the first period, and bring down the next period for a dividend, which makes 4812, to find a divisor, multiply the square of the digit placed in the quotient by 300,= 1200; find how often this is contained in the dividend, viz. 3 times; place the 3 in the quotient for the second digit of the root. Multiply the part of the root formerly found, viz., 2, by the last digit placed in the root, viz., 3, and the product by 30, = 180; add this and the square of the last digit placed in the root to the divisor, viz., 1200; multiply the sum of these, 1389, by the last digit placed in the root, 3, and subtract the product, 4167, from the dividend, 4812; bring down another period for a new dividend, and proceed in the same manner. In order to extract the cube root of a vulgar fraction, reduce it to a decimal, and then extract the root. |