# Körper/Bruchrechenregeln/Aufgabe/en

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Let ${\displaystyle {}x,y,z,w}$ be elements in a field and suppose that ${\displaystyle {}z}$ and ${\displaystyle {}w}$ are not zero. Prove the following fraction rules.

1. ${\displaystyle {\frac {x}{1}}=x,}$
2. ${\displaystyle {\frac {1}{-1}}=-1,}$
3. ${\displaystyle {\frac {0}{z}}=0,}$
4. ${\displaystyle {\frac {z}{z}}=1,}$
5. ${\displaystyle {\frac {x}{z}}={\frac {xw}{zw}},}$
6. ${\displaystyle {\frac {x}{z}}\cdot {\frac {y}{w}}={\frac {xy}{zw}},}$
7. ${\displaystyle {\frac {x}{z}}+{\frac {y}{w}}={\frac {xw+yz}{zw}}.}$

Does there exist an analogue of formula (7), which arises when one replaces addition by multiplication (and subtraction by division), that is

${\displaystyle {}(x-z)\cdot (y-w)=(x+w)\cdot (y+z)-(z+w)\,?}$

Show that the “popular formula”

${\displaystyle {}{\frac {x}{z}}+{\frac {y}{w}}={\frac {x+y}{z+w}}\,}$

does not hold.