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In conventional mathematics, ∞ is not a number, and such things as "1/∞" is only a qualitative description of some limit process. So any "algebra" involving ∞ must be taken in this perspective.

However, there are non-standard areas of mathematics where such things as "infinities" are well-defined, but they are rarely taught in an undergraduate level, as they usually don't bring much more than the usual "standard" mathematics.

For example, there is Abraham Robinson's non-standard Analysis, which can lead to non-standard Calculus. Here, numbers of "infinite" values are introduced to the usual real numbers, as well as "infinitesimal" values, adding the appropriate axioms to the usual system. A positive infinitesimal ε would be greater than 0, but smaller than any *real* number. 1-ε would be a number smaller than 1 by an infinitesimal, but would not be a real number; one calls these hyperreal numbers (and don't expect a digital representation!). 1/ε would be an "infinite", greater than any real number.

I've seen such a non-standard Calculus book long ago; some proofs appear quicker, but a lot is similar to regular Calculus, and there's no new results. It was sometimes presented as an easier alternative to the epsilon-delta approach to calculus.

It was interesting, but not much more (imho).

At least, it made rigorous the notion of "infinitesimal", which always has been fidgy in conventional maths, until stronger notions of limits appeared (epsilon-delta, etc) to replace them.

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//Answer to Additional Details:

Again, you must make a distinction between the quantitative and the qualitative "infinities".

If I say "infinity" is not a number, it does not mean 1/3 is not a real number. As I said, when you see the ∞ symbol in conventional mathematics, it implies somewhere some limiting process, or some qualitative assertion.

For example, the sum 3/10 + 3/10^2 + 3/10^3 + ... + 3/10^n will be as close as we like to 1/3, with an appropriate positive integer n. This is what is *meant* when we say 0.3333333.... (ad infinitum) is the same as (or the decimal representation of) 1/3. An "infinite string of decimals" implies a limit process, with an ever increasing (hopefully determined) precision. The decimal representation depends of such a limiting process, it is *defined* by it, it does not exist without it: whenever you use the decimal representation (especially with an ever expanding string of decimals), it *uses* a limiting process, that is how it is actually made and defined...

In the same vein, writing 0.9999... always adding 9's, only make sense when we *define* what the expression "always adding 9's" actually means; and in the decimal notation, it does mean that the sum 9/10 + 9/10^2 + 9/10^3 + ... + 9/10^n gets as close to 1 (in this case) as we want.

To summarize: in standard math, "infinity" usually arises in a qualitative meaning most of the times, to quickly state what an actual limiting process does (in a finite but -hopefully- determined way).

Without this distinction, it is easy to philosophize into conclusions that goes off what is originally meant. It is important that the terms we use must be as precise as possible (even when they can't in the absolute), otherwise, we enter into two monologues that could hardly converge into a discussion.

I'd like to add, though, that this important distinction in mathematics shouldn't necessarily imply that the same should be done in other fields of knowledge and philosophy; but when one does talk about mathematical infinity, one should be careful that the diverse meanings be identified and distinguished so that a coherent discussion ensue.

Or whatever, it's your choice.

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You seem to be treating ∞ as if it represents a number in the set of real numbers or possibly an integer. This is a mistake. If you look up the definitions of the real numbers or the integers you will find they contain no such element so statements such as 1/∞ = 0 aren't valid.

The best way to understand the kind of questions you are asking is to study the theory of limits.

I hope this helps.

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