Fraction (mathematics)#Equivalent fractions

Shorten a break means that you ( not 0) divides the numerator and denominator of the fraction by the same number. In the elementary fractions shortening is a method for simplifying fractions. In this case, the numerator and denominator of the given break by a common factor ( greater than 1) to be divided.

The value of the fraction remains the same when shortening: This gives a new representation of the same fraction. The number by which shortened one is called the reduction number.

The reversal of shortening is extending a fracture. However, while expand available for each fraction and with every natural number, requires the shortening that the numerator and denominator have a common divisor ( > 1). This is not the case, then the break is unkürzbar; then it is the basic representation of the respective fraction.

If you let other numbers than the common divisor as a reduction in numbers, the difference between extending and shortening disappears. Shortening by a number then nothing else than to expand with its reverse number.

Mathematical formulation

General: Are, and integers, and provided, is then considered

If you read this equation from left to right, then the fraction is reduced with, you read it from right to left, then the break with is extended.

To shorten it is helpful to divide the numerator and denominator of the fraction to its prime factors. In the same prime factors can then be easily pairs is emphasized to the numerator and denominator. For larger numbers, however, it is often easier to determine the greatest common divisor ( gcd ) with the Euclidean algorithm for the greatest common divisor is the largest number with which you can shorten a given fraction.

Examples

The examples show that the shortening of fractures usually is a very meaningful thing, as this substantial simplifications, which in particular facilitates any further calculations with the fractures significantly.

Generalization

Judging from the rational numbers away and considered other structures, then one recognizes that the ability to reduce fractions, a direct consequence of the way, as fractures are defined. One can thus simplify fractions in any quotient bodies eg. Localized to a ring R with a multiplicative subset S, then you can cut a break from RS, only elements of S and expand.

• Fractions