Why is the multiplicative inverse of the additive identity nonexistent?

The real numbers are something called a field (this is from Abstract Algebra). By definition the inverse of the first function (addition with the reals) does not have an inverse of the second function (multiplication with the reals). This is simply a basic property of fields and without you having had at least a few months of an advanced (senior or graduate level) course in Abstract Algebra (Groups, Rings, and Fields) there is really no way that I can explain it.

I know that's a lousy answer, but that's the truth of the matter.

If you could get your hands on a book that covers Field Theory you'd see this in the book.

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Call e the additive identity and 1 the multiplicative identity. For all x in your field F

x = x*1 = x(1+e) = x + xe so that xe = e; Hence you can never have xe = 1 unless 1 = e. In which case x = e for all x in F, or F = {e}.