Talk:Infinite Numbers

There is some logic to requiring an encyclopaedia only to contain only known and accepted knowledge, but this is such a lovely, straightforward and understandable extension to real numbers and infinite sequences that it would be a shame not to allow it. Please do not delete this if you are not a mathematician yourself and if you invite a mathematician to comment on this page, please have her/him read the article as it stands with all contents contained in it. — Preceding unsigned comment added by BenHeideveld (talk • contribs) 14:59, 11 August 2011

One of Wikipedia's fundamental content policies is Wikipedia:No original research, in which original research is defined as "material—such as facts, allegations, ideas, and stories—for which no reliable published source exists". If, as is stated in the article, there is "no literature on this subject to refer to", then the article is in conflict with this policy and must be deleted. --88.104.47.107 (talk) 19:01, 11 August 2011 (UTC)[reply]

Can any mathematician be asked to give her/his opinion, remember when someone said he murdered someone (my contention that this is new and therefore there are no references, which by Wikipedia policy requires deletion) that is insufficient evidence to hang the guy, his guilt still has to be proven beyond a reasonable doubt. So, let's not delete just yet but let's try to get a mathematician to look at this, and perhaps it turns out it's not new and research-like, but it's in fact old hat in disguise, knowledge in stead of research so to say. — Preceding unsigned comment added by BenHeideveld (talk • contribs) 01:55, 13 August 2011 (UTC)[reply]

Please review WP:OR. None of the content of the article was permissible under Wikipedia rules. However the search term is plausible for the transfinite number article, where I have therefore redirected it. --Trovatore (talk) 05:40, 13 August 2011 (UTC)[reply]

OK, I looked a little more closely at BenHeideveld's request, and I'll respond. The "infinite integers" proposed are something like the p-adic numbers, with the difference that p is usually taken to be a prime number, and 10 is not prime. You can still define the 10-adic numbers, but they're not really used for much as far as I'm aware, whereas the p-adics do come up in a fair number of contexts. You can further extend the p-adic integers by allowing a finite string to the right of the radix point; I forget what those are called, maybe "p-adic rationals" or some such.

However, I'm not aware of anyone who's managed to come up with an interesting number-like structure by allowing the digits to extend infinitely far in both directions from the radix point. The problem is that it's not clear how you define multiplication (or perhaps even addition, not sure about that), because digits can change infinitely many times when you're adding up partial sums.

In any case, p-adic numbers are not "infinite" in any usual sense of the word, and more to the point are not called infinite, so it does not make sense to have the search term Infinite Numbers point to a discussion of the p-adics. --Trovatore (talk) 05:56, 13 August 2011 (UTC)[reply]

The closest analog to what is defined in this article are the n-adic rationals for any integral n > 2. You write numbers in base n, but you're allowed an infinite number of numbers before the decimal point and a finite number of numbers after the decimal point. The n-adic integers are what you get by allowing yourself no numbers after the decimal point. If n is a product kl, then the n-adics are the product of the k-adics and the l-adics. The 10-adic integers have been discovered and named the "infinite integers" before, by Archimedes Plutonium. They are the product of the 2-adics and the 5-adics.

You could allow yourself a finite number of numbers before the decimal point and an infinite number of numbers after the decimal point. This is almost the base n expansion of the real numbers. You can define addition and multiplication by the usual rules, but you no longer have 0.999... = 1. However, if you put the obvious topology on this ring and insist that the ring be Hausdorff in that topology, then you get the real numbers.

If you try to allow an infinite number of numbers both before and after the decimal point, then you run into problems with multiplication: Consider what should go in the ones digit; it's the sum of the tens digit times the tenths digit, the hundreds digit times the hundredths digit, etc. There's an infinite sum in there, and that sum may not converge. One thing that is sort of similar is to work to doubly infinite numbers is the space of L¹ functions on the integers (i.e., all absolutely convergent doubly infinite sequences) under convolution. That's not really the same, though (since there's no carries). Ozob (talk) 13:44, 13 August 2011 (UTC)[reply]

I thank Trovatore and Ozob, clearly mathematicians, for taking the time to reveiw the contents of my page. I like what Ozob had to say, although the Archimedes Plutonium link left me unsettled... bien etonne de se trouver ensemble, I guess. I have one more thing to say. I think the redirect to transfinite numbers is negligent. Both ordinal numbers and cardinal numbers have nothing to do with decimal representations. Yes you may assign ordinals and cardinals to any sequence, and infinite reals and infinite integers are two-sided and one-sided infinite sequences, but I think you get what I'm saying. I propose that Ozob comes up with a sensible redirect that at least preserves what I (and Archimedes) meant when we (re)introduces infinite integers. Thanks again, and take care! — Preceding unsigned comment added by BenHeideveld (talk • contribs) 21:14, 14 August 2011 (UTC)[reply]

The problem is that if someone types "Infinite Numbers" into the search bar, they are more likely to want to see Transfinite number or Infinite set or Cardinal number than p-adic number. CRGreathouse (t | c) 19:23, 15 August 2011 (UTC)[reply]