If identities exist for a certain binary operation then we can talk about
inverses (should they exist). If e is an identity then a is an
inverse of b if 
. Again, we could
define a left or right inverse (how should that be done?).
Also we need to point out that an element might have more than one inverse or
for that matter, no inverse. Certainly if identities do not exist then it does
not make sense to talk about inverses. Even if there is an identity then
elements may not have an inverse. For multiplication on the integers
there is an identity, namely 1, but 2 does not have an inverse.
In our familiar real numbers the concept of inverse allows us to solve equations such as x+2=5 or 3x = 11. However, we also need one additional property that we'll see in the next section.