DECIMAL FRACTIONS (Lat. damn, ten) are such as have for their denominator any of the numbers 10, 100, 1000, etc., i.e., any power of 10. See FRACTION. Thus, are, decimal fractions. In writing such fractions, the denominator is omitted, and the above stand thus: 0.7, or .7, .23, .019. That these numbers do not express integers is intimated by the point to the left; and the denominator is always 1, with as many ciphers annexed as there are figures in the decimal. A cipher is prefixed to 19, because otherwise it would read as If it stood for The expression E5.647 is read, five pounds and 647-thousandths of a pound; or, five pounds, and six-tenths, four-hun dredths, and seven-thousandths of a pound. That these two readings are equivalent appears from this, that VA -r = It thus appears that the first figure of a decimal to the left expresses tenths of the unit; the second, hundredths; the third, thousandths, etc. In this property lies the great advantage of decimal fractions; they form merely a continuation of the system of notation for integers, and undergo the common operations of addition, multiplication, etc., exactly as integers do. To explain the principles which determine the position of the decimal point after these operations, belongs to a treatise on arithmetic.
The disadvantage attending decimal fractions is, that comparatively few fractional quantities or remainders can be exactly expressed by them; in other words, the greater number of common fractious cannot be reduced, as it is called, to decimal fractions, without leaving a remainder. Common fractions, such as it, /, f, s, for instance, can
be reduced to decimal fractions only by multiplying the numerator and denominator of each by such a number as will convert the denominator into 10, or 100, 1000, etc. (The common process is merely an abridgment of this.) But that is possible only when the denominator divides 10, or 100, etc., without remainder. Thus, of the above denomi nators, 2 is contained in 10, 5 times; 4 in 100, 25 times; and 25 in 100, 4 times; there 1 1 x . 5 1 1 25 9 9 X4_ 36 fore, 2 - 2 x 5 10 - ' 4 - 4 X 25 - 25X 4 - 100 - 36. But neither 3 nor 7 will divide 10 or any power of 10; and therefore these numbers cannot produce powers of 10 by multiplication. In such cases, we can only approximate to the value of the fraction. Thus, 10, 100, 1000, etc.; divided by 3, give. 3, 33, 333, with a remainder in each case; and 2 2 X 333 = = 666 As this denominator is nearly 3 3 x 333 999' equal to 1000, / = or .666 nearly. As 10, and therefore its powers, are composed of the two factors 2 and 5, it is obvious that any fraction whose denominator contains any other factor than these, cannot be reduced exactly to a decimal fraction.