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., Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \frac{5}{5} = 1 } ). We can represent all rational numbers with fractions.

Fractions are a special case of ratios. For instance, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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., Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 : $\scriptstyle {\frac {3}{4}}$).

Forms

A vulgar fraction (or common fraction) simply refers to a numerator divided by a denominator (e.g., Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \frac{5}{11}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 natural number greater than 1 can be reprented by an an improper fraction, since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle 7 \div 1 = \frac{7}{1}} .

A mixed number is the sum of an integer and a proper fraction (e.g., Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle 2 \frac{3}{4}} ). An improper fraction is equivalent to a mixed number. To convert from one form to another, see Mixed number to improper fraction and Improper fraction to mixed number

Arithmetic 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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a, b, c, d \in \mathbb{Z} } and $b\neq 0\,$, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \tfrac{4}{20} } is not in lowest terms because both 4 and 20 can be exactly divided by 4, giving Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle - \left( \frac{a}{b} \right) = \frac{-a}{b} = \frac{a}{-b} }

The multiplicative inverse of a fraction is :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\frac{a}{b}\right)^{-1} = \frac{b}{a} \mbox{ if } a \neq 0} .

Addition

Formally, apply this algorithm to add two fractions :

1. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{a}{b} + \frac{c}{d} = \frac{ad+bc}{bd}}
2. $e=gcd(ad+bc,bd)\,$
3. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \frac{a}{b} } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \frac{13}{12} }

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

Substraction

Formally, apply this algorithm to substract two fractions :

1. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{a}{b} - \frac{c}{d} = \frac{ad-bc}{bd}}
2. $e=gcd(ad-bc,bd)\,$
3. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \frac{a}{b} } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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.

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. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e = gcd(ac, bd) \,}
3. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \frac{3}{5} \times \frac{1}{3} }  ?

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \frac{3}{15} }

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{3 \div 3}{15 \div 3} = \frac{1}{5} } .

Division

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

Formally, apply this algorithm to divide two fractions :

1. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \frac{3}{5} \div \frac{1}{4} }  ?

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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.

See also

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