[a_wild_dandan]

Look at it this way...

Standard definition of division function, d:

d(x, y) = x * y⁻, for all x and y EXCEPT 0

Author's modified, piecewise (https://en.wikipedia.org/wiki/Piecewise) definition:

d(x, y) = x * y⁻, for all x and y EXCEPT 0

d(x, y) = 0, for y = 0

He's just adding 0 to the domain of d(x, y) to extend the definition, and deliberately not using xy⁻ for that particular element of the domain. No inverse needed.

[sclv]

I know what he's doing. The problem is when you make it a different function (even by just extending it) then you change its equational properties. So equational properties that held over the whole domain of the function no longer hold over the extended domain. This is repaired by modifying the equational properties. But the modified equational properties mean that you now have a different system than before. So the whole thing is just playing around with words.

[mdpopescu]

> So the whole thing is just playing around with words.

Er... that's what mathematics is. It's a word game - we build systems from arbitrary rules and then explore the results.

Look through https://www.mathgoodies.com/articles/numbers for a bunch of uncommonly-defined numbers.