Expand

Extend a fracture means that the counter and the denominator (but not 0) of the fraction with the same number is multiplied . The value of the fraction remains the same: You get a new representation of the same fraction . The number that expands with which one is called coverage factor simply or as extension number called. Any number (except 0) can be the expansion factor. In elementary fractions, natural numbers that are greater than 1 are used as expansion numbers .

The reverse of widening is shortening a fraction, which in turn is nothing other than widening with the reciprocal .

Examples

Elementary fractions

The fraction can be expanded by 2 by multiplying the numerator (above) and denominator (below) by a factor of 2: ${\displaystyle {\tfrac {2}{3}}}$

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

${\displaystyle {\frac {2}{3}}}$and are representations for the same fraction; therefore there are equal signs between them. ${\displaystyle {\frac {4}{6}}}$

Likewise, expanding with 3, 4, 5 and so on delivers

${\displaystyle {\frac {2}{3}}={\frac {2\cdot 3}{3\cdot 3}}={\frac {6}{9}},\quad {\frac {2}{3}}={\frac {2\cdot 4}{3\cdot 4}}={\frac {8}{12}},\quad {\frac {2}{3}}={\frac {2\cdot 5}{3\cdot 5}}={\frac {10}{15}}}$ and so on - all representations of the same fraction.

Negative sign

Expanding with (–1) becomes

${\displaystyle {\frac {-2}{-3}}={\frac {-2\cdot (-1)}{-3\cdot (-1)}}={\frac {2}{3}}.}$

In accordance with the rules for division, two negative signs can be omitted.

Make denominators rational

See the separate article on the rationalization process .
When irrational numbers appear, it is sometimes difficult to tell whether two fractions represent the same fraction. Therefore, the convention applies to look for a representation in which the denominator is a rational number.

${\displaystyle {\tfrac {1}{\sqrt {2}}}}$should therefore be expanded with: ${\displaystyle {\sqrt {2}}}$

${\displaystyle {\frac {1}{\sqrt {2}}}={\frac {1\cdot {\sqrt {2}}}{{\sqrt {2}}\cdot {\sqrt {2}}}}={\frac {\sqrt {2}}{2}}.}$

algebra

When converting terms , the result is often a representation of the term that is clear and uses as few characters as possible. In the following example, the number of characters can be reduced from 20 to 12 by expanding with ( a  -  b ):

${\displaystyle {\frac {\left(a-b\right)^{3}}{a^{3}+a^{2}b+ab^{2}+b^{3}}}={\frac {\left(a-b\right)^{3}\left(a-b\right)}{\left(a^{3}+a^{2}b+ab^{2}+b^{3}\right)\left(a-b\right)}}={\frac {\left(a-b\right)^{4}}{a^{4}-b^{4}}}.}$

However, this transformation is only correct if applies (because then one does not expand with 0). In this case , the first expression is 0, while the second and third expression are undefined (there the 0 is in both the numerator and the denominator). ${\displaystyle a\neq b}$ ${\displaystyle a=b\neq 0}$

Applications

Addition and subtraction

Expanding is particularly needed when adding and subtracting fractions. The fractions involved are given the same name so that their denominators match.

Example: Find the sum of the fractions and . ${\displaystyle {\tfrac {3}{4}}}$ ${\displaystyle {\tfrac {5}{6}}}$

The two denominators are 4 and 6. The common denominator must be a multiple of both 4 and 6: a common multiple. Of course, the product of the denominator is always a common multiple: 6 · 4 is 6 times 4 and 4 times 6. However, the product is often not the smallest possible number and leads to unnecessary computational effort. In our example it is easy to see that 12 is also a common multiple of 4 and 6.

How the smallest suitable number can be found even in more difficult cases is explained under Smallest common multiple . This is also called the lowest common denominator or main denominator of the given fractions. In the example, 12 is the main denominator.

In order to bring both fractions to the denominator 12, we have to expand the first summand with 3, the second with 2:

${\displaystyle {\frac {3}{4}}+{\frac {5}{6}}={\frac {3\cdot 3}{4\cdot 3}}}$ + ${\displaystyle {\frac {5\cdot 2}{6\cdot 2}}={\frac {9}{12}}+{\frac {10}{12}}.}$

As is well known, fractions with a common denominator are added by adding their numerators and keeping the denominator ( distributive law ):

${\displaystyle {\frac {9}{12}}}$+ = ${\displaystyle {\frac {10}{12}}}$ ${\displaystyle {\frac {19}{12}}.}$

Sometimes the result of an addition or subtraction can be shortened . This is not the case with, but this can still be written as a mixed number : ${\displaystyle {\tfrac {19}{12}}}$ ${\displaystyle {\tfrac {19}{12}}={1}{\tfrac {7}{12}}.}$

to compare

Expanding can also be useful to determine which of the two fractions is the larger. In any case, the goal is to give the fractions the same name - as with adding - and then to check which has the larger numerator in this representation.

But there are often easier ways: To determine whether is greater or less than , it is sufficient to expand the first fraction by 3: ${\displaystyle {\tfrac {3}{4}}}$ ${\displaystyle {\tfrac {9}{11}}}$

${\displaystyle {\frac {3}{4}}={\frac {3\cdot 3}{3\cdot 4}}={\frac {9}{12}}<{\frac {9}{11}},}$

because a twelfth is a smaller fraction than an eleventh. Here, instead of the denominators of the fractions, their numerators have been made equal - sometimes a practical method when comparing fractions, but which is not suitable for addition / subtraction.

1. ↑ This convention had its special justification before calculating machines were widely used. In written arithmetic , √2: 2 = 1.4142…: 2 is a simple problem that is easy to calculate for every reasonable number of √2 digits, while 1: √2 = 1: 1.4142… even with a few digits of √2 requires an enormous amount of computing power.