Since you're doing this work on your own without any formal training, it might be worthwhile to explain some concepts that might be unfamiliar to you.
First, you have designed a system of numbers, which is great. And you've made a bit of an attempt to describe operations on those numbers, which you wish to consider identical to multiplication and addition. Rather than trying to explain the mechanics of these operations in gritty detail in limits to infinity, it might be more worthwhile to take the operations for granted, and then show that your set of numbers, together with your versions of multiplication and addition (call them M and A), are related in some manner to some known structure.
In other words, you have a set, we'll call them T. Then, you've got two operations, M and A. What you want is to let them be similar to some set of numbers (reals, complex, rationals, whatever) with the known analogues of multiplication and addition. So, you want to draw a relationship between (T,A,M) and (S,+,*).
In algebra, a set with two operations, that satisfies a few properties I won't go into (because they're not complicated; you can look them up) are called "rings". You can pile on some additional properties to rings to get domains, fields, etc. However, rings are more general and it makes sense to talk about them at first.
When we talk about two rings being "equivalent", we're talking about isomorphism. Mathematically, two rings are isomorphic if there is a bijective homomorphism between the respective sets. What that means is that if you put your two sets side by side, you can define some function that maps something in your set to something in the Reals (complex, rationals, integers, etc) and vice versa. In addition, this function has to have some properties, namely that multiplication and addition are respected. So if you had f : T -> R, then you need to have f(x A y) = f(x) + f(y) and f(x M y) = f(x)*f(y). In other words, adding two numbers in your set, then taking the map over to the reals is the same thing as taking the map to the reals for both numbers independently, then adding them together.
Isomorphism is very powerful, because if you can show that two rings are isomorphic, then any property that holds in one set automatically holds in the other.
This way, if you can show that your number system is isomorphic to some known number system, then you know that your number system is consistent up to the details of your operations M and A. Once you've shown that your systems are isomorphic, then you can mop up the details between the difference in the operations. This makes it a lot easier to develop the constructions you want to do, particularly when dealing with things that go out to infinity. It also eliminates the need to try to bandy about awkward arguments about notation and places, etc.