Is the estimation of number's name's length and comma-grouping feasible? I am thinking in a mathematical problem that probably is already formulated and even solved. It is about big integers and someting else.
Let n be an integer positive number. 


*

*For n := 1,000 we have written words to name it: “thousand” in English, “mil” in Spanish, “mille” in French, “Tausend” in German, and “sen” or “ban” in Japanese (written in latin alphabet).

*For n := 1,000,000 we have “million”.

*For n := 1E9 (“E” is read “times ten raised to the power of” and the number, as everybody knows) many people will call it “billion”, but in most european countries people will say “thousand million”, and call 1E12 “billion”, while is somehow more common to hear “trillion”. 


Suppose now that we have a very big number, for instance: n := 3.1416E9999, which is 31416 followed by 9995 zeroes, if I'm not wrong. Questions:


*

*If we were to actually write this rather unfathomable number, and separate in “groups” of three digits as is the custom (counting from the “right” hand side of the number, or the “end” if you prefer), and marking this groups with a comma, then:


*

*How many commas would there be?

*Where would go the number's first comma? 

*

*3,141,600...

*31,416,000...

*314,160,000...



*What function could give us the length of the word used to pronounce the whole number? And what is the length of word used to say the leftmost part, the largest “unit” or “group”?
I think that the first question is a straight induction problem, but it's too tricky for me. And about the second one, I am no sure if I can solve it. First of all, it may be impossible to compose a single word with so many Greek or Latin prefixes.
I've been looking up the CPAN for a perl module that could answer my question, but my research was unsuccessful. I want to know what is the length of that prefix (maybe even in syllables rather than in letters) in order to calculate how much time would it take to pronounce the number. This should be an estimate because humans usually stop to breathe when talking, and there's also the need to drink water and sleep, and some words take much time to pronunce them than others. And why do I want to know this? Because I want to know exactly when a number is pronounceable and unpronounceable.
I don't know if these results can be useful or simply trivial, but if the mathematical science cannot answer this, I think we are doomed.
Please, spare me for my grammatical mistakes, since English is not my first language.
 A: Say that our number is  $a \mathrm E n$, or in other words $a\times 10^n$ where $1<a<10$.  Also suppose $a$ is a rational number with less than $n$ decimal digits. (So that $a\times 10^n$ is an integer)  For example, in the above case, $a=3.1416$ and $n=9999$, and $a$ certainly has less than $n$ digits.  
1:  The number of commas will be:  $$\left\lfloor \frac{n-1}{3}\right\rfloor.$$  
2: To find where the first comma goes, look at the remainder of $n$ after division by 3, call this $r$.  If the remainder is one, the first comma goes one to the right of the first number.  If the remainder is two, the first comma goes two to the right of the first number.  Lastly if the remainder is 0 (that is, $n$ is divisible by 3) the first comma goes three to the right of the first number.
So in our case, since $n=9999$ is divisible by 3, we see that the first comma goes three to the right, and hence we would write:  314, 160, 000, ...
3: It is not clear how long the word could be, but we can put a makeshift minimum number of letters.  Since there are 26 characters in the alphabet, and we need a word to represent every 3 zeros, the "word" representing $\left(10^3\right)^n$ would require at least $$\left\lceil \frac{\log n}{\log 26}\right\rceil$$ letters.  
