On Monday, September 25, 2017 at 8:38:51 AM UTC-7, Dan Christensen wrote:
> > There is a thing called a wheel > > (https://en.wikipedia.org/wiki/Wheel_theory) > > In the Talk section, the author states, "In the ordinary sense of division, you cannot divide by zero." > > What he has done is invent an algebraic structure with a /-operator that is similar, but not identical to the multiplicative inverse.
Of course he has. Because (as has been shown numerous times here) "ordinary multiplicative inverse" in a field does not (and cannot) include 0 in its domain.
This is exactly what is done in the extension from the natural numbers to the integers, to the rationals, to the reals, to the complexes, ... that you love to rub our faces in with your "all mathematics can be developed from the Peano axioms" sloganeering.
What is important is that the restriction of the new, non-ordinary inverse agrees with the ordinary one on the natural embedding of the field into the wheel. Just like in the rational numbers, the image of 6 under the embedding divided by the image of 3 under the embedding yields the image of 2 under the embedding, so in the wheel, the image of 6 under the embedding divided by the image of 3 under the embedding yields the image of 2 under the embedding.