DECIMALS.
A decimal fraction is that which has for its denominator an unit (1), with as many ciphers annexed as the numerator has places; and it is usually expressed by setting down the numerator only, with a point before it, on the left hand. Thus, 5 10 is ·5; 25 100 is ·25; 25 1000 is ·025; ciphers being prefixed, to make up as many places as are required by the ciphers in the denominator.
A mixed number is made up of a whole number with some decimal fraction, the one being separated from the other by a point, thus 3·25 is the same as 3 25 100 or 325 100 .
Ciphers on the right hand of decimals make no alteration in their value; for ·5, ·50, ·500 are decimals having all the same value, each being = 5 10 . But when they are placed on the left hand, they decrease the value in a tenfold proportion; thus, ·5 is 5 10 ; but ·05 is 5 100 .
ADDITION OF DECIMALS.
Rule.—Set the numbers under each other, according to the value of their places, in which state the decimal separating points will all stand exactly under each other. Then beginning at the right hand, add up all the columns of numbers as in integers, and point off as many places for decimals as are in the greatest number of decimal places in any of the lines that are added; or place the point directly below all the other points.
Example.—Required the sum of 29·0146, 3146·5, 14·16, and 165.
| 29·0146 | |
| 3146·5 | |
| 14·16 | |
| 165· | |
| Answer | 3354·6746 |
SUBTRACTION OF DECIMALS.
Rule.—Place the numbers under each other according to the value of their places. Then, beginning at the right hand, subtract as in whole numbers, and point off the decimals, as in addition.
Example.—Subtract 4·90142 from 214·81.
| 214·81 | |
| 4·90142 | |
| Answer | 209·90858 |
MULTIPLICATION OF DECIMALS.
Rule.—Place the factors, and multiply them together, the same as if they were whole numbers. Then point off in the product just as many places of decimals as there are decimals in both the factors. But, if there be not so many figures in the product, prefix ciphers to supply the deficiency.[44]
Example.—Multiply 32·108 by 2·5.
| 32·108 | |
| 2·5 | |
| 160540 | |
| 64216 | |
| 80·2700 | Answer. |
DIVISION OF DECIMALS.
Rule.—Divide as in whole numbers, and point off in the quotient as many places for decimals as the decimal places in the dividend exceed those in the divisor. When the decimal places of the quotient are not so many as the above rule requires, the deficiency is to be supplied by prefixing ciphers. When there is a remainder after the division, or when the decimal places in the divisor are more than those in the dividend, then ciphers may be annexed to the dividend, and the quotient carried on as far as required.
Example.—Divide 234·7052 by 64·25.
| 64·25 ) | 234·7052 | ( 3·65 Answer. | |
| 19275 | |||
| 41955 | |||
| 38550 | |||
| 34052 | |||
| 32125 | |||
| 1927 | Remainder. | ||
REDUCTION OF DECIMALS.
To reduce a vulgar fraction to its equivalent decimal.
Rule.—Divide the numerator by the denominator, as in Division of Decimals, annexing ciphers to the numerator as far as necessary: and the quotient will be the decimal required.
Example.—Reduce 7 24 to a decimal.
| 24 = 4 × 6. | Then | 4 ) 7· | |
| 6 ) 1·75 | |||
| ·291666, | &c. |
To find the value of a decimal, in terms of the inferior denominations.
Rule.—Multiply the decimal by the number of parts in the next lower denomination, and cut off as many places to the right hand for a remainder, as there are places in the given decimal. Multiply that remainder by the parts in the next lower denomination, again cutting off for another remainder as before. Proceed in the same manner through all the parts of the integer; then the several denominations, separated on the left hand, will make up the answer.
Example.—What is the value of ·775 pounds sterling.
| ·775 | ||
| 20 | ||
| Shillings | 15·500 | |
| 12 | ||
| Pence | 6·000 | Answer 15s. 6d. |
To convert integers, or decimals to equivalent decimals of higher denominations.
Rule.—Divide by the number of parts in the next higher denomination, continuing the operation to as many higher denominations as may be necessary.
When there are several numbers, all to be converted to the decimal of the highest—
Set the given numbers directly under each other for dividends, proceeding from the lowest to the highest; opposite to each dividend, on the left hand, place such a number for a divisor as will bring it to the next higher name. Begin at the uppermost, and perform all the divisions, placing the quotient of each division, as decimal parts, on the right hand of the dividend next below it; so shall the last quotient be the decimal required.
Example.—Convert 15s. 9¾d. to the decimal of a pound sterling.
| 4 | 3· |
| 12 | 9·75 |
| 20 | 15·8125 |
| £·790625 Answer. |
Example.—Convert 1 dwt. to the decimal of a pound, Troy weight.
| 20 ) | 1 | |
| 12 ) | ·05 oz. | |
| ·004166 | lb., &c., Answer. |
RULE OF THREE IN DECIMALS.
Rule.—Prepare the terms, by reducing the fractions to decimals; compound numbers to decimals of the higher denominations, or integers of the lower; also the first, and second terms to the same name. Then multiply, and divide, as in the Rule of Three, in whole numbers.
Example.—If ⅜ of a yard of cloth cost £⅖, what will 5 16 of a yard cost?
| yd. | yd. | £. | s. d. | ||
| ⅜ = ·375 | As ·375 : | ·3125 | :: 4 : | ·333 &c. | or 6 8 |
| 4 | |||||
| ⅖ = ·4 | ·375) | ·12500 | (·3333 | &c. | |
| 1125 | 20 | ||||
| 5/16 = ·3215 | 1250s. | 6·666 | &c. | ||
| 1125 | 12 | ||||
| Answer, 6s. 8d. | 125d. | 7·999 | &c. nearly 8d. | ||
DUODECIMALS.
By Duodecimals, artificers, &c., compute the content of their works.
Rule.—Set down the two dimensions to be multiplied together one under the other, so that feet may stand under feet, inches under inches, &c.
Multiply each term in the multiplicand, beginning at the lowest, by the feet in the multiplier, and set the result of each straight under its corresponding term, observing to carry 1 for every 12, from the inches to the feet. In like manner multiply all the multiplicand by the inches, and parts of the multiplier, and set the result of each term one place removed to the right hand of those in the multiplicand: omitting, however, what is below parts of inches, only carrying to these the proper number of units from the lowest denominations. Or, instead of multiplying by the inches, take such part of the multiplicand as those are of a foot.
Then add the two lines together for the content required.
Example.—Multiply 14 feet 9 inches, by 4 feet 6 inches.
| ft. | in. | |
| 14 | 9 | |
| 4 | 6 | |
| 59 | 0 | |
| 7 | 4½ | |
| 66 | 4½ | Answer. |