# Fraction

*arithmetical*

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).

#### Comments

The set of fractions (of the integers) is denoted by . 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 . Completion of in this metric (e.g. by using Cauchy sequences, cf. Fundamental sequence) leads to , the ordered field of real numbers (cf. Real number). In this connection, a fraction is also called a rational number, and a number from that is not a fraction is called an irrational number, see, e.g., [a1].

For a construction of from using Dedekind cuts (cf. also Dedekind cut) see, e.g., [a2].

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

#### References

[a1] | E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965) |

[a2] | W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1953) |

**How to Cite This Entry:**

Fraction.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Fraction&oldid=16993