TITLE: What digits is the "number" $\infty$ composed of?
QUESTION [6 upvotes]: 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.

REPLY [3 votes]: 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.