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nominator; and if we choose to take any number of such parts less than the whole, this is called the numerator of a fraction.
The numerator, in the vulgar form, is always written over the denominator, and these are separated by a small line thus 2, or ; the first of these is called three-fourths, and the latter five-eighths, of an inch, yard, &c., or of whatever the whole thing originally consisted: the 4 and the 8 are the denominators, showing into how many equal parts the unit is divided; and the three and the five are the numerators, showing how many those parts are under consideration.
Fractions are expressed in two forms, that is, either vulgarly or decimally.
All fractions whose denominators do not consist of a cipher or ciphers, set after unity, are called vulgar; and their denominators are always written under their numerators. The treatment of these, however, would be foreign to our present purpose. But fractions whose denominators consist of a, unit prefixed to one or more ciphers, are called decimal fractions; the numerators of which are written without their denominators, and are distinguished from integers by a point prefixed; thus *, 1o uio, in the decimal form, are expressed by .2, .42, .172.
The denominators of such fractions consisting always of a unit prefixed to as many ciphers as there are places of figures in the numerators, it follows, that any number of ciphers put after those numerators, will neither increase nor lessen their value: for io, ioo,
are all of the same value, and will stand in the decimal form thus .3, .30, .300; but a cipher or ciphers prefixed to those numerators lessen their value in a tenfold proportion: for i, o, and 100%, which in the decimal form we denote by .3, .03, and .003, are fractions, of which the first is ten times greater than the second ;. and the second, ten times greater than the third.
Hence it appears, that as the value and denomination of any figure, or number of figures, in common arithmetic is enlarged and becomes ten, or a hundred, or a thousand times greater, by placing one, or two, or three ciphers after it; so in decimal arithmetic, the value of any figure, or number of figures, decreases and becomes ten, or a hundred, or a thousand times less, while the denomination of it increases, and becomes so many times greater, by prefixing one, or two, or three ciphers to it: and that any number of ciphers before an integer, or after a decimal fraction, has no effect in changing its value.
ADDITION OF DECIMALS. Write the numbers under each other according to the value or denomination of their places; which position will bring all the decimal points into a column, or vertical line, by themselves. Then, beginning at the right-hand column of figures, add in the same manner as in whole numbers, and put the decimal point in the sum directly beneath the other points.
What is the sum of 6.57, 1.026, .75, 146.5, 8.7, 526., 3.97, and .0271 ?
What is the sum of 4.51, 146.071, .507, .0006, 132., 62.71, .507, 7.9, and .10712?
SUBTRACTION OF DECIMALS. Write the figures of the subtrahend beneath those of the minuend according to the denomination of their places, as directed in the rule of addition ; then, beginning at the right-hand, subtract as in whole numbers, and place the decimal point in the difference exactly under the other two points.
MULTIPLICATION OF DECIMALS. Set the multiplier under the multiplicand without any regard to the situation of the decimal point ; and having multiplied as in whole numbers, cut off as many places for decimals in the product, counting from the right-hand towards the left, as there are in both the multiplicand and multiplier: but if there be not a sufficient number of places in the product, the defect may be supplied by prefixing ciphers thereto.
For the denominator of the product being a unit, prefixed to as many ciphers as the denominators of the multiplier and multiplicand contain of ciphers, it follows that the places of decimals in the product will be as many as in the' numbers from whence it arose.
Multiply .0089789 by 1085.
Product .03379399401. DIVISION OF DECIMALS. Divide as in whole numbers; observing that the divisor and quotient together must contain as many decimal places as there are in the dividend. If, therefore, the dividend have just as many places of decimals as the divisor has, the quotient will be a whole number without any decimal figures. If there be more places of decimals in the dividend than there are in the . divisor, point off as many figures in the quotient for decimals, as the decimal places in the dividend exceed those in the divisor; the want of places in the quotient being supplied by prefixing ciphers. But if there be more decimal places in the divisor than in the dividend, annex ciphers to the dividend, so that the decimal places here may be equal in number to those in the divisor; and then the quotient will be a whole number, without fractions.
When there is a remainder, after the division has been thus performed, annex ciphers to this remainder, and continue the operation till nothing remains, or till a sufficient number of decimals shall be found in the quotient.
Divide 63.72413456922 by 2718.