[BlackFingolfin]

I am a professor for algebra at a research university. I make a point out of teaching my students that `a/b` is NOT the same as multiplying `a` by the multiplicative inverse of `b`.

The standard example is that we have a well-defined and useful notion of division in the ring Z/nZ for n any positive integer even in cases were we "divide" by an element that has no multiplicative inverse. Easy example: take n=8 then you can "divide" 4+nZ by 2+nZ just fine (and in fact turn Z/nZ into a Euclidean ring), even though 2+nZ is not a unit, i.e. admits no multiplicative inverse.