In mathematics, a fraction (from the Latin fractus, broken) is a concept of a proportional relation between an object part and the object whole.

A fraction is an example of a specific type of ratio, in which the two numbers are related in a part-to-whole relationship, rather than as a comparative relation between two separate quantities.

A fraction is a quotient of numbers, the quantity obtained when the numerator is divided by the denominator. Thus represents three divided by four, in decimals 0.75, as a percentage 75%. The three equal parts of the cake are 75% of the whole cake.

Each fraction consists of a denominator (bottom) and a numerator (top), representing (respectively) the number of equal parts that an object is divided into, and the number of those parts indicated for the particular fraction. Fractions are rational numbers, which means that the denominator and the numerator are integers.

For example, the fraction could be used to represent three equal parts of a whole object, were it divided into four equal parts. Because it is impossible to divide something into zero equal parts, zero can never be the denominator of a fraction (see division by zero). A fraction with equal numerator and denominator is equal to one (e.g. = 1) and the fraction form is rarely, if ever, given as a final result.

In higher mathematics, a fraction is viewed as an element of a field of fractions.

Terminology

Historically, any number that did not represent a whole was called a "fraction". The numbers that we now call "decimals" were originally called "decimal fractions"; the numbers we now call "fractions" were called "vulgar fractions", the word "vulgar" meaning "commonplace".

Writing fractions

The numerator and denominator of a fraction may be separated by a slanting line called a solidus or slash, for example , or may be written above and below a horizontal line called a vinculum, thus: $\backslash tfrac\{3\}\{4\}$.

The solidus may be omitted from the slanting style (e.g. ³₄) where space is short and the meaning is obvious from context, for example in road signs in some countries.

Usage

Fractions are used most often when the denominator is relatively small. It is easier to multiply 32 by than to do the same calculation using the fraction's decimal equivalent (0.1875). It is also more accurate to multiply 15 by , for example, than it is to multiply 15 by a decimal approximation of one third. To change a fraction to a decimal, divide the numerator by the denominator, and round off to the desired accuracy.

The word is also used in related expressions, such as continued fraction and ''algebraic fraction—see Special cases]] below.

Vulgar, proper, and improper fractions