A number consisting of one or more equal parts of a unit. It is denoted by the symbol
,
where
and
are integers (cf.
Integer).
The
numerator
of
denotes the number of parts taken of the unit;
this is divided by the number of parts equal to the number appearing as the
denominator
.
A fraction may also be considered as the ratio produced by dividing
by
.
The fraction
remains unchanged if both the numerator and the denominator
are multiplied by the same non-zero integer. Owing to this fact, any two fractions
and
may be brought to a common denominator, i.e.
and
may be replaced by fractions equal to
and
,
respectively, both of which have the same denominator. Moreover, fractions may be
reduced by dividing their numerator and denominator by the same
number; accordingly, any fraction may be represented as an
irreducible fraction,
i.e. a fraction the numerator and denominator of which have no common factors.
The sum and the difference of two fractions
and
having a common denominator are given by
In order to add or to subtract fractions with different denominators they
must first be reduced to fractions with a common denominator. As a rule, the
least common multiple
of the numbers
and
is taken as the common denominator. Multiplication and division
of fractions is given by the following rules:
A fraction
is said to be a
proper fraction
if its numerator is smaller than its denominator; otherwise it is an
improper fraction.
A fraction is said to be a
decimal fraction
if its denominator is a power of the number 10 (cf.
Decimal fraction).
Formal definition of fractions.
Fractions may be represented as ordered pairs of integers
,
,
for which an equivalence relation has been specified (an
equality relation of fractions),
namely, it is considered that
if
.
The operations of addition, subtraction, multiplication, and division
are defined in this set of fractions by the following rules:
(thus, division is defined only if
).
A similar definition of fractions is convenient in
generalizations and is accepted in modern algebra (cf.
Fractions, ring of).