Field/Rules for fractions/Exercise

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

1. ${}{\frac {x}{1}}=x\,,$

2. ${}{\frac {1}{z}}=z^{-1}\,,$

3. ${}{\frac {1}{-1}}=-1\,,$

4. ${}{\frac {0}{z}}=0\,,$

5. ${}{\frac {z}{z}}=1\,,$

6. ${}{\frac {x}{z}}={\frac {xw}{zw}}\,$

7. ${}{\frac {x}{z}}\cdot {\frac {y}{w}}={\frac {xy}{zw}}\,,$

8. ${}{\frac {x}{z}}+{\frac {y}{w}}={\frac {xw+yz}{zw}}\,.$

Does there exist an analogue of formula (8) that arises when one exchanges addition with multiplication (and division with subtraction), that is

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

Show that the popular formula

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

does not hold.

Create a solution