DECIMAL ARITHMETIC, the art of computing by fractions whose denominator is 10, 100, &e... Decimal fractions differ from vulgar fractions in this ; that the denominator is not written ; instead of writing -A, or ye,„ the fraction would be written decimally, .4 or .15. The decimal point before it is used to distinguish it from whole numbers.
To reduce any vulgar fraction to a decimal. say, As the denominator of the vulgar fraction is to the denominator of the decimal, so is the numerator of the vulgar fraction ,to the numerator of the decimal.
Example. To reduce 3 to a decimal fraction, whose denominator is 10.
It will then be as 3 : I0 : : 2 10 3) 20 6 so that 6 is the numerator required : but then there is a remainder of 2, consequently the numerator is more than 6, but less than 7; therefore is the nearest decimal fraction, whose numerator consists of a single unit. In order to come nearer to the truth, we must then suppose the denominator of the decimal to be divided into more parts, say 100.
Then again 3 : 100 : : 2 3) 200 66 but here is still a remainder of 2, that is, 66 is too small and 67 too great, the decimal fraction is still too small ; 66 is, however, a greater portion of 100, than 6 is of 10 : we have therefore come nearer to the truth ill the latter operation than in the first. We shall thus find, that if the number arising by multiplying the denominator of the decimal fractiourby the numerator of the vulgar fraction, be not divisible by the denominator of the vulgar fraction, an increase of the denominator of the decimal will give a more exact portion of the unit, than when fewer figures are used ; and thus, if worth the trouble, the numerator of a decimal fraction may be found to any degree of exactness at pleasure, by augmenting the number of figures in the denominator, either till the division terminate, or till as many figures be found as will render the operation sufficiently exact for the intended purpose.
A decimal fraction may be sufficiently denoted, by throw ing away the denominator, and using any character or mark instead of it, since the denominator is always 1, followed by one, two. three, or a series of ciphers, which is the only thing
that is variable ; to ascertain this point, the number expressed by the numerator, is always less than the denominator, and always consists of as many figures as there are ciphers ; therefore, if a point be placed before the numerator of a decimal fraction. it will show that the number following it is a decimal fraction, and by reckoning a cipher for every figure, and supposing unity placed before them, the number thus expressed will show how many decimal parts the unit is divided into, and the figures themselves that portion of these parts taken.
Thus is represented by .6 66 c cc .G0 45 cc .785 too° R 5 cc .085 t.000 But instead of saving 66 hundredths, 785 thousandths, &c., say, as in the second, 6 tenths and 6 hundredths ; as in the third, 7 tenths, S hundredths and 5 thousandths ; and as in the fourth. S hundredths and 5 thousandths, as the cipher, 0, occupies the place of tenths ; 6 66 for 10 100 1F)() 5 7S5 and 10 100 + 8 5 08:5 also 100 1000 1000 The point is not only useful in marking the following number to be a decimal fraction, hut is likewise necessary in separating the decimal parts from integers, when both are concerned.
From what has been said, it is observable that decimal fractions decrease in the same order from unity towards the right hand, that integers increase towards the right.
Thus, in 348.5683, unity is the place where the numbering commences both for integers and for decimals ; going over the places of the integers, we have units, tens, hundreds, 348; then, numbering the decimals, we have units, tenths, hun dredths, thousandths, ten thousandths, which is 5 tenths, 6 hundredths, S thousandths, and 3 ten thousandths ; in this notation of the fractions, the unit's place was not reckoned, as being already counted into the whole numbers.