Fraction (mathematics)

In mathematics, a fraction is a quotient of numbers, like ³⁄₄, or more generally, an element of a quotient field.

The word is also used in related expressions, like continued fraction, see Special cases below.

For other, non-mathematical meanings of this word, see fraction.

Special cases

A vulgar fraction is a rational number written as one integer (the numerator) divided by a non-zero integer (the denominator). The line that separates the numerator and the denominator is called the vinculum. Rational numbers are the quotient field of integers.

irreducible fraction: a vulgar fraction "in lowest terms", where the numerator is an integer, the denominator is a positive integer, and the highest common factor of the numerator and the denominator is 1;
proper fraction: a vulgar fraction with (absolute) value between 0 and 1;
improper fraction: a vulgar fraction with a (absolute) value greater than 1;
unit fraction: a vulgar fraction with a numerator of 1;
Egyptian fraction: the sum of distinct unit fractions;
decimal fraction: a vulgar fraction where the denominator is a power of 10;
dyadic fraction: a vulgar fraction in which the denominator is a power of two.

Fractions which are rational numbers and could be written as vulgar fractions include:

A mixed fraction: A mixed fraction is an integer plus a proper fraction.
A compound fraction is a fraction where the numerator or denominator (or both) contain fractions.

Fractions which are not necessarily rational numbers include:

Let us end with the only example on this page where the "fraction" is not an element of a quotient field:

A continued fraction is an expression such as <math>a_0 + \frac{1}{a_1 + \frac{1}{a_2 + ...}} <math>, where the a_i are integers.

An irrational fraction is, if all fractions must be capable of being expressed as a vulgar fraction, a contradiction in terms. An irrational number is, by definition, not rational i.e. it cannot be expressed as a vulgar fraction.

In Primary Schools, fractions have been demonstrated through Cuisenaire rods.