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I have seen from past posts on the topic of infinity that there is some ambiguity with the concept infinity and whether it is a number etc. From what I can gather the terms number and infinity are imprecise, yet can be useful in both academic and colloquial usage. There are many excellent answers, beautifully written to other questions on this topic. Many of the answers I do not understand, so it is likely that this question has been answered, if so my apologies.

Still I will ask this question: Does the "number" infinity have any digits in it?

Does the "number" infinity have an infinite number of each digit?

Intuitively to me it seems that infinity does have some digits in it, but we cannot say which ones or how many.

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If you've seen past posts on the topic of infinity, one thing you should glean from them at some point is that there is no "the" infinity - you can append min and max values to the order topology on $\Bbb R$ for instance, or put a complex infinity on $\Bbb C$ to make the Riemann sphere, or talk about infinitessimals and infinite numbers in the hyperreals (in which there are an infinite number of infinite numbers), or infinite cardinal numbers (of which there are again infinite), or the field of surreal numbers (a proper class), etc. To be precise you must specify the infinity you speak of. – blue Mar 12 '14 at 17:05
By "number", people usually mean either real number, natural number, or integer. $\infty$ is none of those, and there is no imprecision about this. There are extremely precise ways of adding different concepts (all called $\infty$, but different in fact) to different number systems. There is again no imprecision, though. If there's one thing I'd like to emphasize, it's that there is no wishy-washyness here. – mixedmath Mar 12 '14 at 17:06
@seaturtles, thank you for your comment. I did glean that there were different infinities and contexts, but did not understand them enough to actually specify which one I meant. I know this would be an undertaking but if there are some contexts where the answer does make more sense, I'd love to see that addressed. – Paul Mar 12 '14 at 17:27
(FWIW I did not vote.) The closest thing to "infinity" having a digital expansion I can think of is that there are infinite series (comprised of terms in $\Bbb Q$) which diverge to $\infty$ in $\Bbb R$ but converge to a value in the so-called p-adic numbers, and $p$-adic numbers are essentially numbers written in base $p$ which have only finitely many digits to the right of the "decimal" point but potentially infinitely many to the left (so, a power series in $p$). – blue Mar 12 '14 at 17:58
No, you misunderstand me. I appreciated your first comment very much and your second comment explaining when my question might make sense - though I don't understand it :), I can grasp it a little. Fascinating stuff. – Paul Mar 12 '14 at 18:01
up vote 7 down vote accepted

You are confusing numbers with numerals. Numerals are symbols that represent numbers. Numbers do not have any intrinsic representation as sequences of digits or anything else. Instead, we devise different schemes for representing numbers with numerals. For example, in one scheme, we use sequences of digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 to represent certain numbers; the numeral 119 represents a certain number. But there is nothing privileged or special about this numeral; in a different, similar system, the same number is represented with the numeral 1110111; in a different, less similar system the same number is represented with the numeral 百十九, in another system it is represented with the numeral CXIX, in a different system it is represented with the numeral one hundred and nineteen, and in a different system again it is represented with a certain pattern of electron flow in a chunk of silicon.

So the question of whether a certain number "has digits in it" is a category error. Numbers never have digits. Some systems of numeration use digits, and numerals in those systems have digits in them. But the number of digits will depend on which system you are using. 119 is a three digit numeral, and 1110111 is a seven-digit numeral, but they both represent the same number.

The question that does make sense to ask is whether a certain system of numerals can represent a certain number. For example, some systems are able to represent the number one-half. One might write it in one system as $\frac12$, and in another system as 0.5. Some systems simply have no representation for one-half.

So we can ask if the standard decimal system, the one which uses digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, has a representation of the number infinity, and if so how many digits are used to represent it. And the answer is no, as usually understood, this system has no representation for the number infinity. (Or, more precisely, for any of the several numbers called "infinity".)

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Beautiful exposition, resolving a very common misconception. – Sammy Black Mar 12 '14 at 17:02
Thank you for coming down to my level! Often I don't realize all the hidden assumptions I am making when addressing a problem. You have unmasked several of them in a few minutes. Much appreciated. – Paul Mar 12 '14 at 17:24
@Paul Confusing numbers and numerals is an extremely common error, and I am glad to be able to help you clear it up. – MJD Mar 12 '14 at 17:27

There is no natural way to express any of the various objects called "infinity" using decimal digits. There are artificial ways, but none are standardized. More common is to use aleph, a Hebrew letter, instead.

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It looks like this: $\aleph$ – TonyK Mar 12 '14 at 17:19
Short and to the point. I like how this math SE, has support for all these symbols. It reminds me of latex. – Paul Mar 12 '14 at 17:28

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