View source for Fraction

''arithmetical''

A fraction is a number consisting of one or more equal parts of a unit. It is denoted by the symbol $a/b$, where $a$ and $b\ne 0$ are integers (cf.
[[Integer|Integer]]). The numerator $a$ of $a/b$ 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 $b$. A fraction may also be considered as the ratio produced by dividing $a$ by $b$.

The fraction $a/b$ remains unchanged if both the numerator and the denominator are multiplied by the same non-zero integer. Owing to this fact, any two fractions $a/b$ and $c/d$ may be brought to a common denominator, i.e. $a/b$ and $c/d$ may be replaced by fractions equal to $a/b$ and $c/d$, 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 $a/b$ and $c/b$ having a common denominator are given by

$$\frac{a}{b} \pm \frac{c}{b} = \frac{a\pm c}{b}$$
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|least common multiple]] of the numbers $b$ and $d$ is taken as the common denominator. Multiplication and division of fractions is given by the following rules:

$$\frac{a}{b}\cdot\frac{c}{d} = \frac{a\cdot c}{b\cdot d},\quad  \frac{a}{b} : \frac{c}{d} = \frac{a\cdot d}{b\cdot c},\quad (c\ne 0). $$
A fraction $a/b$ 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|Decimal fraction]]).

==Formal definition of fractions.==
Fractions may be represented as ordered pairs of integers $(a,b)$, $b\ne 0$, for which an equivalence relation has been specified (an equality relation of fractions), namely, it is considered that $(a,b) = (c,d)$ if $ad = bc$. The operations of addition, subtraction, multiplication, and division are defined in this set of fractions by the following rules:

$$(a,b)\pm (c,d) = (ad\pm bc,bd),$$

$$(a,b)\cdot (c,d) = (ac,bd),$$

$$(a,b): (c,d) = (ad,bc),$$
(thus, division is defined only if $c\ne 0$).

A similar definition of fractions is convenient in generalizations and is accepted in modern algebra (cf.
[[Fractions, ring of|Fractions, ring of]]).



====Comments====
The set of fractions (of the integers) is denoted by $\Q$. With the arithmetical operations and natural order defined in the main article above it is an
[[Ordered field|ordered field]]. The
[[Absolute value|absolute value]] gives a metric on $\Q$.
[[Completion|Completion]] of $\Q$ in this metric (e.g. by using Cauchy sequences, cf.
[[Fundamental sequence|Fundamental sequence]]) leads to $\R$, the ordered field of real numbers (cf.
[[Real number|Real number]]). In this connection, a fraction is also called a rational number, and a number from $\R$ that is not a fraction is called an irrational number, see, e.g.,
{{Cite|HeSt}}.

For a construction of $\R$ from $\Q$ using Dedekind cuts (cf. also
[[Dedekind cut|Dedekind cut]]) see, e.g.,
{{Cite|Ru}}.

For aliquot fractions (i.e. numbers of the form $1/n$, $n$ a positive
[[Integer|integer]]) see
[[Aliquot ratio|Aliquot ratio]].

====References====
{|
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|valign="top"|{{Ref|HeSt}}||valign="top"|  E. Hewitt,   K.R. Stromberg,   "Real and abstract analysis", Springer  (1965)  {{MR|0188387}}  {{ZBL|0137.03202}} 
|-
|valign="top"|{{Ref|Ru}}||valign="top"|  W. Rudin,   "Principles of mathematical analysis", McGraw-Hill  (1953)  {{MR|0055409}}  {{ZBL|0052.05301}} 
|-
|}