Fraction (mathematics)

In mathematics, a fraction is a concept used to convey a proportional relation between a part and the whole. It consists of a numerator (an integer - the part) and a denominator (a natural number - the whole). For instance, the fraction $\scriptstyle {\frac {3}{5}}$ can represent three equal parts of a whole object, if the object is divided into five equal parts. A fraction with equal numerator and denominator is equal to one (e.g., $\scriptstyle {\frac {5}{5}}=1$). We can represent all rational numbers with fractions.

Fractions are a special case of ratios. For instance, $\scriptstyle {\frac {e}{\pi }}$ is a valid ratio, but it is not a fraction since we cannot compute an equivalent fraction with integer numerator and integer denominator.

Since we can compute the quotient from a fraction, we can represent any fraction with a decimal number (e.g., $\scriptstyle {\frac {3}{5}}=0.6$). However, because the division by zero is undefined, zero should never be the denominator of a fraction.

Due to tradition and conventions, there are at least two ways to write a fraction. The numerator and the denominator may be separated by a slash (a slanted line : 3/4), or by a vinculum (an horizontal line : $\scriptstyle {\frac {3}{4}}$).

Basic operations

The most common operations done on fractions are addition, substraction, multiplication, and division. In order to perform the addition and the substraction, we must frequently compute the equivalent fractions. We may need the multiplicative inverse when dividing.

The end result must be an irreducible fraction.

In this section, $a,b,c,d\in \mathbb {Z}$ and $b\neq 0\,$, $d\neq 0\,$.

Equivalent fractions

A fraction where the numerator and the denominator do not have any common factor, 1 excepted, is said irreducible (or in its lowest terms). If it is not the case, then we divide its numerator and its denominator by their gcd.

Multiplying (or integer dividing) the numerator and the denominator of a fraction by the same non-zero integer results in a new fraction that is said to be equivalent to the original fraction. For instance, $\scriptstyle {\tfrac {4}{20}}$ is not in lowest terms because both 4 and 20 can be exactly divided by 4, giving $\scriptstyle {\tfrac {1}{5}}$ (the quotient of both fractions is 0.2). In contrast, ${\tfrac {3}{5}}$ is in lowest terms.

Inverses

The additive inverse of a fraction is :

$-\left({\frac {a}{b}}\right)={\frac {-a}{b}}={\frac {a}{-b}}$

The multiplicative inverse of a fraction is :

$\left({\frac {a}{b}}\right)^{-1}={\frac {b}{a}}{\mbox{ if }}a\neq 0$.