Decimal Fractions

Decimal Fractions (Lat. decem, 'ten') are such as have for their denominator any of the numbers 10, 100, 1000, &c.—i.e. any power of ten (see FRACTION). Thus, $\backslash frac\{1\}\{10\}$, $\backslash frac\{2\}\{10\}$, $\backslash frac\{3\}\{10\}$, $\backslash frac\{4\}\{10\}$, $\backslash frac\{5\}\{10\}$, $\backslash frac\{6\}\{10\}$, $\backslash frac\{7\}\{10\}$, $\backslash frac\{8\}\{10\}$, $\backslash frac\{9\}\{10\}$, are decimal fractions. In writing these the denominator is conventionally omitted, and the fractions expressed 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. In the third example a cipher is prefixed to 19, because otherwise it would read as if it stood for $\backslash frac\{19\}\{10\}$. The expression £5·647 is read, Five pounds and 647-thousandths of a pound; or, Five pounds, and six-tenths, four-hundredths, and seven-thousandths of a pound. That these two readings are equivalent appears from this, that $\backslash frac\{647\}\{1000\}\; =\; \backslash frac\{6\}\{10\}\; +\; \backslash frac\{4\}\{100\}\; +\; \backslash frac\{7\}\{1000\}\; =\; \backslash frac\{6\}\{10\}\; +\; \backslash frac\{4\}\{100\}\; +\; \backslash frac\{7\}\{1000\}$. It thus appears that the first figure of a decimal to the right of the point expresses tenths of the unit; the second, hundredths; the third, thousandths, &c. 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, &c., 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 fractions cannot be reduced, as it is called, to decimal fractions, without leaving a remainder. Common fractions, such as $\backslash frac\{1\}\{2\}$, $\backslash frac\{2\}\{3\}$, $\backslash frac\{1\}\{4\}$, $\backslash frac\{2\}\{5\}$, $\backslash frac\{3\}\{5\}$, 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, &c. (The common process is merely an abridgment of this.) But that is possible only where the denominator divides 10, or 100, &c., without remainder. Thus, of the above denominators, 2 is contained in 10, 5 times; 4 in 100, 25 times; and 25 in 100, 4 times; therefore,

7 will divide 10 or any power of 10; and therefore these numbers cannot produce powers of 10 by multiplication. In such cases, therefore, and in fact in the case of any vulgar fraction (in its lowest terms) whose denominator contains any other prime factor than 2 or 5, an equivalent decimal cannot be found. If we try to find it, the result is an infinite series, which is called a repeating, recurring, or circulating decimal. Thus $\backslash frac\{2\}\{3\}\; =\; .6666$, &c., where the 6 repeats for ever—i.e. $\backslash frac\{2\}\{3\}$ cannot be expressed as a decimal. The non-terminating result, '.666, &c., is written '.6.