Show that 0 cannot have a multiplicative inverse in any of our number systems.

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- Oct 26th 2009, 08:45 AMtigergirlMultiplicative inverse 71.7
Show that 0 cannot have a multiplicative inverse in any of our number systems.

- Oct 26th 2009, 09:34 AMTheEmptySet
We will need the fact that $\displaystyle 0\cdot x=0$ for all x.

This can be proved if needed.

Now supose that $\displaystyle 0$ does have an inverse. Call it $\displaystyle e$

Then by defintion of multiplicive inverse

$\displaystyle e \cdot 0= 0 \cdot e=1$

But by our theorem above $\displaystyle 0=e\cdot 0 =1$ this is a contradiction. So the assumption that zero has an inverse must be false.