# What digits is the “number” $\infty$ composed of?

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.

• 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. – davidlowryduda 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
• All the digits are eight, they are rotated 90°, and there is only one of them. – Elements in Space Apr 28 '16 at 15:25

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".)

• 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
• @Paul: Infinity is not even a number, and this post is misleading. Talking about infinity as if it were a number is no better than talking about Unicorn as if it were an animal. – user21820 Feb 6 at 17:13
• The question does not mention real numbers. Nor does it mention decimal expansions. – MJD Feb 7 at 15:27

There is no ambiguity regarding infinity, particularly with regard to whether or not it is a number. There is, perhaps, some ambiguity to the novice reader regarding infinity, as the term infinity pops up in a lot of different places in mathematics, and can refer to a number of distinct (though related) concepts. For example:

• In analysis, we are often interested in describing the limiting behaviour of sequences. Some sequences oscillate, such as the sequence $$1, -1, 1, -1, 1, -1, \dotsc = \{ (-1)^n \}_{n\in\mathbb{N}};$$ some sequences "approach a limit", such as $$1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \dotsc = \left\{ \frac{1}{n} \right\}_{n\in\mathbb{N}}$$ which "approaches" zero; and some sequences "blow up" or are "unbounded", such as $$1, 2, 3, 4, 5, \dotsc = \{n\}_{n\in\mathbb{N}}.$$ In the case of an unbounded sequence, we use the shorthand $$\lim_{n\to\infty} a_n = \infty.$$ If we want to be very formal about this, what it means is that if we pick any number $$M$$ (no matter how large), we can always find some natural number $$N$$ so that $$a_n$$ is bigger than $$M$$ whenever $$n$$ is bigger than $$N$$. In other words, this notation means that the terms $$a_n$$ grow without bound. To avoid having to write down a bunch of technical notation every time we see something like this, we give a single definition of what it means for "a limit to be equal to infinity," then use the shorthand notation shown above. The symbol $$-\infty$$ ("negative infinity") is understood similarly.

The overall point is that $$-\infty$$ and $$+\infty$$ are not a real numbers in this setting. They can be seen as useful symbols which give a shorthand for a more technical definition, or they may be regarded as points in a topological space which extends the real numbers (which would make $$\pm\infty$$ affinely-extended real numbers). They don't behave like real numbers, as, for example, $$0\cdot\infty$$ and $$\infty-\infty$$ cannot be sensibly defined.

• In set theory, it is often desirable to describe the cardinality of a set, where cardinality is a more rigorous way of describing "size". In a way that can be made formal, there is no "largest" natural number, so the set of natural numbers must be of infinite size (i.e. it has infinite cardinality). There are infinitely many natural numbers—here, the use of "infinity" is somewhat distinct from the application to analysis. "Infinity" means something slightly different here. However, there is no ambiguity—the context of the usage makes it clear what is going on. That being said, there is also notation for this kind of infinity: the cardinality of the natural numbers is often denoted by $$\operatorname{card}(\mathbb{N}) = \aleph_0,$$ where that last symbol is the Hebrew letter "aleph". Something fun to know is that in this context, there are different of infinities. For example, $$\operatorname{card}(\mathbb{R}) = \mathfrak{c},$$ where, in a way that can be made formal, $$\aleph_0 < \mathfrak{c}$$.

Note that $$\aleph_0$$, $$\mathfrak{c}$$, and their compatriots (there are other similar symbols with related meanings) are not numbers, though each one can reasonably called "infinity." In the language of mathematics, these guys are "infinite cardinals."

• There is also a notion of an "infinite ordinal." For example, the symbol $$\omega$$ (a lowercase Greek omega) is is a number-like object which has the property that $$x < \omega$$ for all real numbers $$x$$. In some sense, $$\omega$$ is probably the closest thing to a "number" infinity that I am going to discuss. We can do arithmetic with $$\omega$$, for example (though that arithmetic is a little more complicated than the usual—for example $$1+\omega \ne \omega + 1$$), but $$\omega$$ is still not really a number. Alternatively, if you want to call $$\omega$$ a number, you have to be very careful about defining what you mean by "number" before you go any further. In particular, $$\omega$$ doesn't fit in well with the real numbers. Instead, $$\omega$$ is useful in providing a generalization of the natural numbers.

In particular, the ordinals extend the idea of indexing. Given a set, we may want to label every element of that set so that the elements can be put into order. This kind of thing is useful when arguing by induction, for example. The "natural" index set is $$\mathbb{N}$$, but what if we need indices beyond those contained in $$\mathbb{N}$$? The symbol $$\omega$$ is simply if first index after $$\mathbb{N}$$, and $$\omega+1$$ is the next index after that. Infinite ordinals extend the idea of addressing objects by a well-order (i.e. of saying what comes first, second, next, &c.).

Note that in each of these contexts, the notion of "infinity" is unambiguous, at least from a mathematical point of view. The notion of "number" is also unambiguous here, though, again, this notion might depend on context. For example, we might desire limits to exist in our space, so we might define $$\pm\infty$$ to be numbers. This gives us the set of extended real numbers, i.e. $$\overline{\mathbb{R}} := \mathbb{R} \cup \{-\infty,+\infty\}.$$ Here, positive and negative infinity are numbers, but they are not real numbers—they are extended real numbers. Something similar can probably be done in other contexts, as well.

To address your second question: when you describe the "digits" of a number, you are talking about a decimal expansion or representation of a real number (where we include the natural numbers, integers, and rationals as subsets of the reals). If an object is not a real number, then it does not make any sense to discuss its digits. Infinity is not a real number. Conclude from this what you will.

• To add to this excellent answer, one must be clear about what precisely each mathematical notion refers to and what structures are related to it. For instance, the cardinals capture the notion of size of unstructured sets in a world where every pair of sets has an injection from one to the other. In contrast, the ordinals capture the notion of indices of a well-ordering. So the ordinal ω is simply the first index after natural number indices, and of course is distinct from the ordinal ω+1, which is the first index after ω. But ω and ω+1 have the same size. – user21820 Mar 1 at 6:58
• (See this post for more on how cardinals and ordinals extend the notions of sizes and indices of natural numbers.) And both ordinals and cardinals are completely different from the notion of infinite points in a 'continuous space'. The affinely-extended reals have two points at infinity, which arise naturally from the compactification of the reals, and has nicer topological properties but $∞-∞$ and $0·∞$ and $∞/∞$ remain undefined. As you can see, the points at infinity are still points, not numbers in any ordinary sense. – user21820 Mar 1 at 7:03
• @user21820 But what is a number then, and what is the difference between numbers and points? Are p-adic numbers numbers? Are hyperreal numbers numbers? I've even heard "transfinite numbers" used to refer to cardinals and ordinals. I don't think there is a concrete definition of "number" in mathematical practice such that we can say that $\infty$ is definitely not a number (as opposed to other examples, e.g. the Dirac delta function is definitely not a function from reals to reals). – pregunton Mar 1 at 7:54
• @pregunton Again, as I pointed out near the end of my answer, what we mean by "number" is unambiguous, but dependent upon context. So, for example, $\pm\infty$ are not real numbers, but they are extended real numbers. The perceived ambiguity may be a problem of communication between mathematician and non-mathematician, but when mathematician refers to something as a "number", it is nigh certain that they have a specific notion in mind. – Xander Henderson Mar 1 at 13:30
• @pregunton: You certainly know that (exactly as Xander said), the context is crucial in mathematics, but mathematically untrained people/students do not know that, so for them "number" is something with digits in it... That's why my comments are what they are; I think it is a pedagogical mistake to call anything beyond complex numbers as numbers, even though (as we both know) a lot of mathematical objects have been and will still be called 'numbers', including cardinal numbers, p-adic numbers, even surreal numbers! By the way, no need to apologize (for no offense). =) – user21820 Mar 2 at 5:47

To adress the "digits" part of the question, while sticking to decimal representation, let's begin by defining what it really means for a number to have certain digits.

For exmple, "one hundred and twenty three point four" has the decimal expression $$123.4$$ because $$123.4 = 1\cdot \textbf{100} + 2 \cdot \textbf{10} + 3\cdot \textbf{1} + 4\cdot \textbf{0.1}$$.

Similarly 1/3 can be written as $$3 \cdot \textbf{0.1} + 3 \cdot \textbf{0.01} + 3 \cdot \textbf{0.001} + 3 \cdot \textbf{0.0001} + ... = \sum_{k=1}^{\infty} 3 \cdot 10^{-k}$$

For simplicity, let's forget the decimals parts and focus on integers. A number $$n$$ will have a decimal expression $$a_{k} a_{k-1}... a_1 a_0$$ if, and only if, it can be expressed as:

$$\sum_{i=0}^{k} a_i \cdot 10^i$$

Note that this only makes sense for number with a finite ($$k$$) number of digits, so does not really apply to the case of "infinity". But still, similarly to how we can get decimal expressions for numbers like $$\frac{1}{3}$$ using infinitely many numbers after the decimal point, one could ask himself whether it is also posible to work with infinitely many numbers before the decimal point. Writing something like:

$$\sum_{i=0}^{\infty} a_i \cdot 10^i$$

The problem is that the above sum will diverge to infinity unless there is a number $$k$$ such that $$a_i=0$$ for every $$i>k$$. In other words, with these "decimal expressions", you can get to infinity in as many ways as you want. In short, if we accepted that convention $$33333333....=\infty$$, but also $$44444444....=\infty$$, and $$123456789123456789....=\infty$$

In conclusion, there is no "standard" way of writing numbers that assigns a particular set of digits to "infinity". If we want to expand our "common-use" decimal numbers to have such a feature, we will have to accept that "infinity" can be written pretty much anyway we like, which does not have a lot of practical use.