FRACTION, a part of any unit, as a fraction of a pound, of an acre, of an inch, of an hour, or of a group.
If we take any two positive integers, say 2 and 3, the quotient is a fraction. More generally speaking, if we take any two positive finite integers, a and b, such that b is not zero, the quotient , is integral, if and only if, a is exactly divisible by b. If a is not so divisible and is less than b, speaking in the primitive sense b is called a fraction or, according to relatively late usage, a proper fraction. In this expression a and b are called the terms of the fraction, a being the numerator (numberer) and b the denominator (namer). We may also con sider this fraction as representing a of the b equal parts of some unit, a being then the "numberer" of the parts and b the "namer" of the parts, as in three-fourths (4) of a pound. We may also look upon the fraction as representing one bth of a units; for example, 3 may be thought of as representing one-third of two units. These various ideas of a fraction are consistent with one another, as also with the idea that, say, 4 means the ratio of 3 to 4, or the number which, multiplied by 4, becomes 3.
These elementary ideas of fraction have been extended from time to time so as to permit of terms that are fractional, irrational, or imaginary, or that rep resent still other types of number, the denominator b never being zero.
The tendency to generalize has led to allowing the numerator to be any multiple of the denominator, the fraction then becom ing I or some other integer ; or to be any number whatsoever, not only less than but equal to or greater than the denominator. Since these two latter generalized cases do not properly represent frac tions in the primitive sense ("proper" fractions), they are called improper fractions.
The fraction b may be expressed as a/b, this being a more con venient form for printing or typewriting. Since the slanting bar resembles the old symbol for solidus (later used in the British 2/6 for 2s. 6d.), the form a/b is sometimes called the solidus form. If a and b are both rational, a/b is called a rational frac tion; if both are integers, it is called a simple fraction; if either a or b is fractional, a/b is called a complex fraction. Questions relating to the least (in algebra, lowest) common denominator are sufficiently discussed in any elementary text book, where it is also shown that any rational complex fraction can be reduced to a simple fraction.
For the study of such continued fractions as Reduction.—With respect to the reduction of fractions to higher or lower terms, the following laws are valid for any value of in except zero and infinity: a ma a a b mb b b-:m When all common factors have been removed from both terms of a fraction, the fraction is said to be reduced to lowest terms, or to be irreducible.
It is often possible to reduce a proper fraction a/b to the sum of two or more simpler fractions. For example, to perform the following reduction to partial fractions:
—
a+
_
a-6 The methods of procedure are given in most intermediate or higher algebras, but the importance of the subject is not seen until the student reaches the calculus.
The col loquial use of a fraction like i is so extended as to cause con siderable difficulty on the part of a learner. He hears of half of an object, and can easily visualize it ; half of a group is a little more difficult ; half as large is still more so; and the idea becomes still less distinct when he hears such expressions as "half as long a time," "half as dark," and "half as beautiful." The simplest fraction thus comes, through such colloquial expressions, to be unusually difficult. Certain other fractions—notably
and 1 —are used in the same loose way.
Since "fraction" (from
frangere, to break) means "broken," it was natural for the
writers of the i 6th century to speak of "fractions or broken num bers" and of a "broken of broken" (Baker, 1568), meaning a fraction of a fraction. The word "fragment" was also occasionally used for "fraction." In the middle ages the word minuciae was used as the equivalent of fractiones, and not solely with the mean ing of "minutes" as in sexagesimal fractions. The fact that a fraction is a broken number led to such expressions in the early printed books as Ein gebrochene zal (Riese, German, 1522), and Die ghebroken ghetalen (Raets, Dutch, 158o). Since the Latin ruptus also means "broken," such names as rotto (Italian), rocto (Spanish), and roupt and hombre rompu (French) appear in the early printed books.
We speak of proper fractions, mixed numbers, irrational fractions, and the like ; but in view of the gradual extension of meaning, it is legitimate to say that any expression a/b is a fraction. It would even be allowable to admit the case of b=o, giving an interpretation accordingly, if this were necessary or advisable. Moreover, while we speak of common or vulgar fractions, of decimals, and of sexagesimals, these are chiefly differences in symbolism and have little bearing upon the nature of such a fraction as 1, O.5 (or o-5),
(I° being the unit), or
all of which have the same value.
