Is there any possibility to replace zero (null) and his inverse (quantity of zero raised to the power of negative one is indefinite) with an infinitesimal for calculations to avoid "zero ring" and other problems related to zero and his multiplicative inverse?
For example, if we will do that, can we convert problem, like:
"Assume there was some number ⋆ that we add to the real numbers R with the property 0⋅⋆=⋆⋅0=1. From this it follows that for any number x∈R we have x = x ⋅ 1= x ⋅ 0 ⋅ ⋆ = 0 ⋅ ⋆ = 1.
This is a contradiction of course, since there are real numbers that are not equal to 1. Hence, you can't add a multiplicative inverse of 0 to the reals. If you attempt to, you end up with a structure where all elements are equal to 1, the zero ring."
... into such problem:
"Assume there were some numbers ω="infinities" and ε="infinitesimals" that we add with the property ε ⋅ ω = ω ⋅ ε = 1. From this, it follows that for any number x ∈ R we have:
- x = x ⋅ 1 = x ⋅ ε ⋅ ω;
- x ÷ ε = ω * x;"
P.S. Sorry for the poor explanation - I'm not a Mathematician ;D