Shorten

Shortening of a fracture means that the counter and the denominator (not 0) of the fraction by the same number divided . In elementary fractions, the abbreviation is a method to simplify fractions. Here, the numerator and denominator are the given fracture by a common divisor dividing (greater than 1).

The value of the fraction remains the same when you shorten it: you get a new representation of the same fraction . The number by which one cuts is called the cut number .

The reverse of shortening is expanding a fraction. However, while expanding is possible with every fraction and with every natural number, shortening requires that the numerator and denominator have a common factor (> 1). If this is not the case, the fraction cannot be shortened ; it is then the basic representation of the relevant fraction.

If other numbers than the common divisors are allowed as reduction numbers, the difference between expanding and reducing disappears. Shortening by a number is nothing other than expanding with its inverse number.

Mathematical formulation

General: Are , and integers, where and are assumed, then applies ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle c}$ ${\displaystyle b\neq 0}$ ${\displaystyle c\neq 0}$

${\displaystyle {\frac {a\cdot c}{b\cdot c}}\;=\;{\frac {a}{b}}}$

If you read this equation from left to right, the fraction is shortened with , if you read it from right to left, the fraction is expanded with . ${\displaystyle (ac)/(bc)}$ ${\displaystyle c}$ ${\displaystyle a/b}$ ${\displaystyle c}$

In order to reduce it, it is helpful to decompose the numerator and denominator of the fraction into their prime factors . The same prime factors can then simply be crossed out in pairs in the numerator and denominator. With larger numbers, however, it is often easier to find the greatest common divisor ( GCF ) using the Euclidean algorithm , because the GCF is the largest number with which a given fraction can be reduced.

Examples

${\displaystyle {\frac {6}{8}}\;=\;{\frac {3\cdot 2}{4\cdot 2}}\;=\;{\frac {3}{4}}}$

${\displaystyle {\frac {200}{8800}}\;=\;{\frac {1\cdot 200}{44\cdot 200}}\;=\;{\frac {1}{44}}}$

The examples show that shortening fractions is usually a very useful thing, because it results in considerable simplifications, which in particular makes it much easier to calculate further with the fractions.

generalization

If you move away from the rational numbers and look at other structures, you see that the ability to abbreviate fractions is a direct consequence of the way fractions are defined. You can thus z. B. shorten fractions in any quotient field . If one localizes a ring R with a multiplicative subset S , then a fraction from R _S can only be shortened and expanded with elements of S.

See also

• Can be shortened