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Would this be an appropriate time to discuss dividing by zero in modular arithmetic? Dividing by zero definitely plays an interesting role there. I'm not saying any more until either Ricky or Mathisfun gives me the go ahead, because I'm afraid that going into detail might be confusing and hence a bad idea.
The integers modulo n always form a ring, and in such an algebraic structure, my proof that a*0 = 0 holds. Thus, division by zero again does not make sense. Now you can change definitions around to make it work (for example, having 1/a no longer meaning "the multiplicative inverse of a" would do it), but that isn't exactly modular arithmetic anymore.
Ricky, can you be a bit more simplistic? And if it's not "exactly modular arithmetic" than what is it?
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
If you drop the condition that division means "multiplication by multiplicative inverse", then you can end up turning modular arithmetic into a Wheel. If, on the other hand, you drop the distributive property (which I believe is required to prove that 0*a = 0, but not certain), then you'd probably be studying something that hasn't been looked at all too much before. I know of no name for a ring without the distributive property.
Since distributivity is the only ring axiom to mention both addition and multiplication, a ring without distributivity is nothing more than an abelian group and a semigroup that happen to be defined on the same set.
Right, it's the relation between multiplication and addition that gives a ring its interesting properties.
Wow, thanks for giving this so much attention. I see now why division by zero is still a "no-no" in modular arithmetic. I admit (begrudgingly) that the "wheel" stuff is way over my head, but I'm very appreciative that attention was given to my question. Thanks.
There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.