Fraction.

Fraction. In Arithmetic, when a unit of any kind is divided into any number of parts, each is termed a fraction of the whole unit—e.g. one foot (lineal measure) is divided into inches; one inch is thus a fraction of one foot. The usual notation employed to denote the value of a fractional quantity is to place under a horizontal line the number of equal parts into which the whole unit has been divided; above the line is placed the number of these parts actually contained by the fraction. The former number is known as the denominator, the latter as the numerator of the fraction. Thus, 7 inches is expressed as the fraction of one foot by $\backslash frac\{7\}\{12\}$. Quantities expressed in this way are termed vulgar fractions; they are proper or improper according as the numerator is less or greater than the denominator; and when the numerator and denominator have no common factor the fraction is said to be in its lowest terms. When the denominator is 10, or a power of 10, the quantity is termed a Decimal Fraction (q.v.). In Algebra the term fraction, while including the sense of the arithmetical definition, is generally used to mean that any quantity affected by it is to be multiplied by the numerator and divided by the denominator. The addition, subtraction, multiplication, and division of fractions are performed according to rules which are practically the same both in arithmetic and algebra. Such rules will be found in any competent text-book on these subjects.

Continued Fractions.—Any expression of the form $a\; +\; \backslash frac\{1\}\{b\; +\; \backslash frac\{1\}\{c\; +\; \backslash frac\{1\}\{d\; +\; \backslash dots\}\}\}$ is termed a continued fraction. This expression is usually for convenience abbreviated to $a\; +\; \backslash frac\{1\}\{b\; +\; \backslash frac\{1\}\{c\; +\; \backslash frac\{1\}\{d\; +\; \backslash dots\}\}\}$, &c. Such fractions may be terminating or non-terminating. A series of quantities which successively approach towards the actual value of such a quantity are termed successive convergents to the value of the fraction; they are alternately smaller and greater than its actual value. Such a series for the fraction above given is: $a,\; a\; +\; \backslash frac\{1\}\{b\},\; a\; +\; \backslash frac\{1\}\{b\; +\; \backslash frac\{1\}\{c\}\},\; \backslash dots$

Vanishing Fractions.—When, by giving to one of the terms in a fractional algebraical expression a particular value, both the numerator and denominator become zero, the expression is said to be a vanishing fraction. Such is the case in the quantity $\backslash frac\{x^2\; -\; 1\}\{x\; -\; 1\}$, when $x\; =\; 1$; and in $\backslash frac\{x\}\{\backslash sqrt\{a\; +\; x\}\; -\; \backslash sqrt\{a\; -\; x\}\}$, when $x\; =\; 0$. But in the first case, by dividing both numerator and denominator by $x\; -\; 1$, the true value of the expression is $x\; +\; 1$, which is equal to 2 when $x\; =\; 1$. In the second example, by multiplying above and below by the complementary surd—viz. $\backslash sqrt\{a\; +\; x\}\; +\; \backslash sqrt\{a\; -\; x\}$, the fraction becomes equal to $\backslash sqrt\{a\}$, when $x\; =\; 0$. Such methods for finding the true value of vanishing fractions are all more or less tentative. For a general process by which their value may be found, reference may be made to Williamson's Differential Calculus, chap. iv. See also article CALCULUS.