Multiplicative inverse 71.7

Quote:

Originally Posted by tigergirl

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

We will need the fact that $0\cdot x=0$ for all x.

This can be proved if needed.

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

Then by defintion of multiplicive inverse

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

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