Let
be elements in a field and suppose that
and
are not zero. Prove the following fraction rules.
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![{\displaystyle {}{\frac {x}{1}}=x\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7726c5274bfca0a0675f777a2b267f128bdf6773)
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![{\displaystyle {}{\frac {1}{z}}=z^{-1}\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0cb6f560ae3533855eff66b2e3634f213d4f5bf)
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![{\displaystyle {}{\frac {1}{-1}}=-1\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d33e4048c7b56006e72c27e909d4fb99aa6848a)
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![{\displaystyle {}{\frac {0}{z}}=0\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95992602052e4ab8d9593a556d1658d1715de0c4)
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![{\displaystyle {}{\frac {z}{z}}=1\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1214c5baf39a9c3fafe6369643e69a6f1ff1c8d7)
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![{\displaystyle {}{\frac {x}{z}}={\frac {xw}{zw}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2266090d42e38af933bd7f534ba1d8e4a0b9f7c)
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![{\displaystyle {}{\frac {x}{z}}\cdot {\frac {y}{w}}={\frac {xy}{zw}}\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/717e70395683c6d78cbfaa16e8d2ce34d686ad9e)
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![{\displaystyle {}{\frac {x}{z}}+{\frac {y}{w}}={\frac {xw+yz}{zw}}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0b5658c5d20bf6d91447a7437518877886f6991)
Does there exist an analogue of formula (8), which arises when one replaces addition by multiplication (and subtraction by division), that is
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![{\displaystyle {}(x-z)\cdot (y-w)=(x+w)\cdot (y+z)-(z+w)\,?}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0fc3611de915a95f229d55f61ed528ff123ae3e7)
Show that the popular formula
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![{\displaystyle {}{\frac {x}{z}}+{\frac {y}{w}}={\frac {x+y}{z+w}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fdd89b9acb2e2fc5ebaab34bf4d750711cee8435)
does not hold.