DECIMAL arithmetic, the art of com puting by decimal fractions.
Mensal. fraction, that whose denomi nator is always 1, with one or more cy phers : thus an unit may be imagined to be equally divided into 10 parts, and each of these into 10 more ;.so that by a conti nual decimal subdivision the unit may be supposed to be divided into 10, 100, 1000, &c. equal parts, called tenth, hundredth, thousandth part of an unit. In decimal fractions, the figures of the numerator are only expressed, the denominator be ing omitted, because it is known to be al ways an unit with so many cyphers as there are places in the numerator. A decimal fraction is distinguished from an integer with a point prefixed, as .2 for .34 for 34 .567 for &c. The same is observed in mixed numbers, as 678.9 for 678 Tab' 67.89 for 67 I* 6.789 for 6 • Cyphers at the right hand of a decimal fraction alter not its value ; for .5 or .50 or .5000 is each of them of the same value equal to N ' or : but cyphers at the left 115 hand, in a decimal fraction, decrease the value in a tenfold proportion ; for .05 is , .005 is s .0005 is 5 &C 1-01" Tb orb' As the denominator of a decimal is al ways one of the numbers 10, 100, 1000, &c. the inconvenience of writing these denominators down may be saved, by placing a proper distinction before the figures of the numerator only to distin guish them from integers, for the value of each place of figures will be known in de cimals, as well as in integers, by their dis tance from the 1st, or unit's place of inte gers, having similar names at equal dis tances, as appears by the following scale of places, both in decimals and integers : Decimal fractions are easily reduced into a common denominator, by making, or even supposing, all of them to consist of the same number of places ; so .3, .45, .067, .0089, may be written thus, .3000, .4500, .0670, .0089; all which consisting of four places, their common denomina tor is an unit with four cyphers, namely, 10000.
Addition and subtraction of decimals are the same as in whole numbers, when the places of the same denomination are set under one another, as in the following examples : To 34.25 From 16.5 Add 3.026 Subtract .125 Sum 37276 Rem. 16.375 In multiplication the work is the same as in whole numbers, only in the product ; separate, with a point, so many figures to the right hand as there are fractional places both in the multiplicand and mul tiplier; then all the figures on the left hand of the point make the whole num ber, and those on the right a decimal frac. tion.
It is to be noted, that if there be not Sc many figures in the product, as ought to be separated by the preceding rule, then place cyphers at the left, to com plete the number, as may be seen in Lx. 5.
Ex. 1. Mult. 456 Ex. 2. Mult. 45.6 by 21.3 by 21.11368 Product 971.2£ 456 912 Product 9712.8 Ex. 3. Multiply 456 by 0.213 Product 97.128Er. 4. Multiply 45.6 by 0.213Product , 97.128 Ex. 5. Multiply 0.0456 by 0.213Product 0.0097128 in division the work is the same as in whole numbers, only in the quotient, se parate, with a point, so many figures to the right hand for a decimal fraction, as there are fractional places in the dividend, more than in the divisor, because there must be so many fractional places in the divisor and quotient together, as there are in the dividend.
As division of decimal fractions is ex tremely difficult, especially with regard to the value of the figures of the quotient, we shall here give a general rule for as certaining their values, viz.
Rule, place the first multiple of the divisor under the dividend, as in opera tions of common division ; then will the unit's place of this multiple stand under such a place of the dividend, as the first significant figure of the quotient is to be ; that is, the first significant figure of the quotient will be of the same name or value with the figure of the dividend which stands above the unit's place of the multiple.