DECIMALS
, any thing proceeding by tens; as Decimal arithmetic, Decimal fractions, Decimal scales, &c.
Decimal Arithmetic, in a general sense, may be considered as the common arithmetical computation in use, in which the decimal scale of numbers is used, or in which the places of the figures change their value in a tenfold proportion, being 10 times as much for every place more towards the left hand, or 10 times less for every place more towards the right hand; the places being supposed indefinitely continued, both to the right and left. In this sense, the word includes both the arithmetic of integers, and decimal fractions. In a more restrained sense however, it means only.
Decimal Fractions, which are fractions whose denominator is always a 1 with some number of ciphers annexed, more or fewer according to the value of the fraction, the numerator of which may be any number whatever; as 1/10, 3/100, 7/1000, &c.
As the denominator of a decimal is always 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 distinguish them from integers, for the value of each place of figures will be known in decimals, as well as in integers, by their distance from the 1st or unit's place of integers, having similar names at equal distances, as appears by the following scale of places, both in decimals and integers:
| &c | 8 | 8 | 8 | 8 | 8 | 8 | 8 | . | 8 | 8 | 8 | 8 | 8 | 8 | &c |
| millions | hund. of thousands | tens of thousands | thousands | hundreds | tens | units | tenths | hundredths | thousandths | ten thousandths | hund. thousandths | millionths |
The mark of distinction for decimals, called the separatrix, has been various at different times, according to the fancy of different authors; sometimes a semiparenthesis, or a semicrotchet, or a perpendicular bar, or the same with a line drawn under the figures, or simply this line itself, &c; but it is usual now to write either a comma or a full point near the bottom of the figures; I place the point near the upper part of the figures, as was done also by Newton; a method which prevents the separatrix from being confounded with mere marks of punctuation.
In setting down a decimal fraction without its denominator, the numerator must consist of as many places as there are ciphers in the denominator; and if it has not so many figures, the defect must be supplied by setting before them as many ciphers as will make them up so many: thus 3/10 is .3; and 14/100 is .14; and 14/1000 is .014; and 3/1000 is .003; &c.
So that, as ciphers on the right-hand side of integers increase their value decimally, or in a tenfold proportion, as 2, 20, 200, &c; so, when set on the left-hand of decimal fractions, they decrease the value decimally, or in a tenfold proportion, as .2, .02, .002, &c. But ciphers set on the other sides of these numbers, make no alteration in their value, neither of increase nor decrease, viz, on the left-hand of integers, or on the righthand of decimals; so 2, or 02, or 002, &c, are all the same; as are also .2, or .20, or .200, &c.
Decimal fractions may be considered as having been introduced by Regiomontanus, about the year 1464, viz, when he transformed the tables of sines from a sexagesimal to a decimal scale. They were also used by Ramus, in his Arithmetic, written in 1550; and before his time by our countrymen Buckley and Recorde. But it was Stevinus who first wrote an express treatise on decimals, viz, about the year 1582, in La Practique d'Arithmetique; since which time, this has commonly made a part in most treatises on arithmetic.
To reduce any Vulgar fraction, or parts of any thing, as suppose 3/8, to a decimal fraction of the same value; add ciphers at pleasure to the numerator, and divide by the denominator: thus,
| 8)3.000 | |
| .375 | =3/8; |
Some vulgar fractions can never be reduced into decimals without defect; as 1/3, which by division is .33333 &c infinitely.
Such numbers are very properly called circulating decimals, and repetends, because of the continual return of the same figures. See Repetends and CIRCULATES.
The common arithmetical operations are performed the same way in decimals, as they are in integers; regard being had only to the particular notation, to distinguish the fractional from the integral part of a sum. Thus,
In Addition and Subtraction, all figures of the same place or denomination are set straight under each other, the separatrix, or decimal points, forming a straight column.
In Multiplication, set down the numbers, and mul- | tiply them as integers; and point off from the product as many places of decimals as there are in both factors; prefixing ciphers if there be any defect of figures.
In Division, set down the numbers and divide also as in integers; making as many decimals in the quotient, as those in the dividend are more than those in the divisor.
Examples are numerous and common in most books of arithmetic.
Decimal Scales, are any scales divided decimally, or by tens.
