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 :

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

Addition

Formally, apply this algorithm to add two fractions :

1. ${\frac {a}{b}}+{\frac {c}{d}}={\frac {ad+bc}{bd}}$
2. $e=gcd(ad+bc,bd)\,$
3. ${\frac {a}{b}}+{\frac {c}{d}}={\frac {(ad+bc)\div e}{bd\div e}}$

By hands, the addition is done like this.

1. Compute an equivalent fraction of $\scriptstyle {\frac {a}{b}}$ and $\scriptstyle {\frac {c}{d}}$, making sure both have the same denominator.
2. For the resulting fraction,
   1. Add the numerators to get its numerator.
   2. Set its denominator to the computed denominator (the three fractions have the same denominator).
3. Reduce the resulting fraction if you need to.

For instance, what is the result of $\scriptstyle {\frac {3}{4}}+{\frac {1}{3}}$ ?

Let's find a number that both denominators will divide : It is 12. We are ready to compute the equivalent fractions :

${\frac {3}{4}}+{\frac {1}{3}}$

$={\frac {3\times 3}{4\times 3}}+{\frac {1\times 4}{3\times 4}}$

$={\frac {9}{12}}+{\frac {4}{12}}$

$={\frac {13}{12}}$

This is the final answer since it is an irreducible fraction.

Substraction

Formally, apply this algorithm to substract two fractions :

1. ${\frac {a}{b}}-{\frac {c}{d}}={\frac {ad-bc}{bd}}$
2. $e=gcd(ad-bc,bd)\,$
3. ${\frac {a}{b}}-{\frac {c}{d}}={\frac {(ad-bc)\div e}{bd\div e}}$

By hands, the substraction is done like this.

1. Compute an equivalent fraction of $\scriptstyle {\frac {a}{b}}$ and $\scriptstyle {\frac {c}{d}}$, making sure both have the same denominator.
2. For the resulting fraction,
   1. Substract the numerators to get its numerator.
   2. Set its denominator to the computed denominator (the three fractions have the same denominator).
3. Reduce the resulting fraction if you need to.

Since this algorithm is very similar to the addition algorithm, we do not give any example.

Multiplication

Formally, apply this algorithm to multiply two fractions :

1. ${\frac {a}{b}}\times {\frac {c}{d}}={\frac {a\times c}{b\times d}}$
2. $e=gcd(ac,bd)\,$
3. ${\frac {a}{b}}\times {\frac {c}{d}}={\frac {(ac)\div e}{bd\div e}}$

By hands, the multiplication is done like this.

1. For the resulting fraction,
   1. Set its numerator to the product of both numerators.
   2. Set its numerator to the product of both denominators.
2. Reduce the resulting fraction if you need to.

For instance, what is the result of $\scriptstyle {\frac {3}{5}}\times {\frac {1}{3}}$ ?

${\frac {3}{5}}\times {\frac {1}{3}}$

$={\frac {3\times 1}{5\times 3}}$

$={\frac {3}{15}}$

Since the result is not an irreducible fraction, we must reduce it. We divide the numerator and the denominator by 3 :

${\frac {3\div 3}{15\div 3}}={\frac {1}{5}}$.