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. 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 an integer numerator and a natural number denominator. A fraction with equal numerator and denominator is equal to one (e.g., $\scriptstyle {\frac {5}{5}}=1$). 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 : $\scriptstyle {\frac {3}{4}}$). Since we can compute the quotient from a fraction, we can represent any fraction with a decimal number (e.g., $\scriptstyle {\frac {3}{5}}=3\div 5=0.6$).

Forms

A vulgar fraction (or common fraction) simply refers to a numerator divided by a denominator (e.g., $\scriptstyle {\frac {5}{11}}$ and $\scriptstyle {\frac {4}{3}}$). It is said to be a proper fraction if the absolute value of the numerator is less than the absolute value of the denominator (e.g. $\scriptstyle {\frac {5}{11}}$). An improper fraction (top-heavy fraction in Great Britain) is said if the absolute value of the numerator is greater than or equal to the absolute value of the denominator (e.g. $\scriptstyle {\frac {4}{3}}$). All non-zero integers can be represented by an improper fraction, since for example $\scriptstyle 7\div 1={\frac {7}{1}}$. The 1 at the denominator is sometimes called an "invisible denominator".

A mixed number is the sum of an integer and a proper fraction (e.g., $\scriptstyle 2{\frac {3}{4}}=2+{\frac {3}{4}}$). An improper fraction can be transformed into a mixed number and vice-versa.

Arithmetic operations

The most common arithmetic operations on fractions are addition, substraction, multiplication, and division. When adding and substracting, we must often compute the equivalent fractions. When dividing, we usually compute the multiplicative inverse.

After any computation, the end result should be an irreducible fraction.

In this section, it is understood that $a,b,c,d\in \mathbb {Z}$ 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.

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 :

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

Multiplicative inverse

The multiplicative inverse of a fraction is :

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

Division

Dividing by a fraction is the same as multiplying by its inverse.

Formally, apply this algorithm to divide two fractions :

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

By hands, the division is done like this.

1. Compute the multiplicative inverse of the second fraction (exchange the numerator and the denominator).
2. For the resulting fraction,
   1. Set its numerator to the product of both numerators.
   2. Set its numerator to the product of both denominators.
3. Reduce the resulting fraction if you need to.

For instance, what is the result of $\scriptstyle {\frac {3}{5}}\div {\frac {1}{4}}$ ?

${\frac {3}{5}}\div {\frac {1}{4}}={\frac {3}{5}}\times {\frac {4}{1}}={\frac {3\times 4}{5\times 1}}$

$={\frac {12}{5}}$

The result is an irreducible fraction.

Additive inverse

The additive inverse of a fraction is :

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

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. Set its numerator to the addition of the numerators.
   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. Set its numerator to the substraction of the numerators.
   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.

Mixed number to improper fraction

A mixed number can be converted to an improper fraction with this algorithm :

1. Multiply the integer by the denominator of the fractional part.
2. Add the numerator of the fractional part to that product.
3. Add both fractions.

Here is an example.

$2{\frac {3}{4}}=2+{\frac {3}{4}}$

$={\frac {2}{1}}+{\frac {3}{4}}$

$={\frac {2\times 4}{1\times 4}}+{\frac {3}{4}}$

$={\frac {8}{4}}+{\frac {3}{4}}$

$={\frac {11}{4}}$

Mixed number addition and substraction

Convert the mixed numbers to improper fraction before adding or substracting. Not doing so may give you a wrong result. Moreover, it may happen that you need to add a mixed number to a fraction.

For instance, what is the sum of $\scriptstyle {\frac {1}{3}}$ and $\scriptstyle -2{\frac {3}{4}}$ ? The minus sign applies equally to the fraction $\scriptstyle {\frac {3}{4}}$ : $\scriptstyle -2{\frac {3}{4}}=-\left(2{\frac {3}{4}}\right)=-\left(2+{\frac {3}{4}}\right)=-2-{\frac {3}{4}}$. The answer is $\scriptstyle -{\frac {29}{12}}$ or $\scriptstyle -2{\frac {5}{12}}$ .

Improper fraction to mixed number

An improper fraction can be converted to a mixed number with this algorithm :

1. Divide the numerator by the denominator.
2. The quotient (without remainder) becomes the whole part and the remainder becomes the numerator of the fractional part.
3. The new denominator is the same as that of the original improper fraction.

Here is an example.

${\frac {11}{4}}={\frac {8}{4}}+{\frac {3}{4}}$

(since $11=8+3=2\times 4+3\,$)

$=2+{\frac {3}{4}}$

$=2{\frac {3}{4}}$

See also

• Continued fraction
• Rational function
• Partial fraction
• Field of fractions