148. We have thus far assumed both numerator and
denominator to be positive integers, and have shown in Art. 140 that a fraction itself is the quotient resulting from the division of numerator by denominator. But in algebra division is a process not restricted to positive integers,
and we shall now extend this definition as follows: The
algebraic fraction {a}{b} is the quotient resulting from the division
of a by b, where a and b may have any values whatever.
149. By the preceding article {-a}{-b} is the quotient resulting from the division of — a by —b; and this is obtained by dividing a by b, and, by the rule of signs, prefixing + .
Therefore :{-a}{-b} =+ {a}{b} = {a}{b}(1).
Again, {-a}{b} is the quotient resulting from the division of — a by b ; and this is obtained by dividing a by b, and, by the rule of signs, prefixing — .
Therefore {-a}{b} = - {a}{b} (2). Similarly, {a}{-b} = - {a}{b} (3).
These results may be enunciated as follows :
(1) If the signs of both numerator and, denominator of a fraction be changed, the sign of the whole fraction ivill be unchanged.
(2) If the sign of either numerator or denominator alone be changed, the sign of the whole fraction ivill be changed.
The principles here involved are so useful in certain cases of reduction of fractions that we quote them in another form, which will sometimes be found more easy of application.
1. We may change the sign of every term in the numerator and denominator of a fraction without altering its value.
2. We may change the sign of a fraction by simply changing the sign of every term in either the numerator or denominator.
