"but it fails to uphold the most important property division should have -- that it undoes multiplication"
I'm not sure I follow that as it's most important property. I'm not sure if division could even be defined as an operation that undoes multiplication.
Number theory, fields, and rings I believe make it clear while subtraction and addition can be viewed as the same function; multiplication and division cannot.
Apologize if that's not clear as to why that is; it's been a while since I read up on those being defined.
However I recommend One, Two, Three: Absolutely Elementary Mathematics by David Berlinski that gives in my opinion pretty good layman understanding of these nuances and number theory.
I'm not sure I follow that as it's most important property. I'm not sure if division could even be defined as an operation that undoes multiplication.
Number theory, fields, and rings I believe make it clear while subtraction and addition can be viewed as the same function; multiplication and division cannot.
Apologize if that's not clear as to why that is; it's been a while since I read up on those being defined.
However I recommend One, Two, Three: Absolutely Elementary Mathematics by David Berlinski that gives in my opinion pretty good layman understanding of these nuances and number theory.