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). 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 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).

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).

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. The absolute value gives a metric on $\Q$. Completion of $\Q$ in this metric (e.g. by using Cauchy sequences, cf. Fundamental sequence) leads to $\R$, the ordered field of real numbers (cf. 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., [HeSt].

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

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

References

[HeSt]	E. Hewitt, K.R. Stromberg, "Real and abstract analysis", Springer (1965) MR0188387 Zbl 0137.03202
[Ru]	W. Rudin, "Principles of mathematical analysis", McGraw-Hill (1953) MR0055409 Zbl 0052.05301