# Kids and Infinity

I was web surfing and came across a very impressive observation by a three-year-old on GoogolPower:

One evening, my three-year-old, in his usual stalling-tactic mode, asked me one more question: “Mom how many words are there in the English language?” I answered, “Well, I guess there are about 600,000 and a few more are added each year.” Then we talked about other languages. He pondered for a while and then asked, “Isn’t there a name for each number?”

I mostly wanted to post that just because it is such a smart question by the little guy! But it did get me to thinking, *is* there really a name for *every *number? Certainly there are names for far more than 600,000 numbers. But I suspect that we do run out of analogs for million, billion, trillion, etc. at some point (although there is a clear naming pattern…). Edward Zaccaro’s awesome book Challenge Math lists the names for large numbers up to vigintillion, and I guess we could keep using latin numbers to keep going, but would we really have a name for *every *number?

How do you define

name? Of course we can construct terms for any number, but just as for chemical compounds, there is no guarantee the names will be concise or not awkward.For instance, Avogadro’s number is 6.02214179 × 1023 (per mole)

This is low enough to use standard terms:

Six hundred two sextillion, two hundred fourteen quintillion, and one hundred seventy-nine quadrillion.

However, most people (I, included) would name it using scientific notation:

Six point zero two two one four one seven nine times ten to the twenty-third (power).

But I know that’s not quite what you had in mind. You may be interested in Wikipedia’s article which has a lot more, including extendable schemes.

Six hundred two sextillion, two hundred fourteen quintillion, and one hundred seventy-nine quadrillion.However, most people (I, included) would name it using scientific notation:Six point zero two two one four one seven nine times ten to the twenty-third (power).

And clearly each of these is more than one “word”. I thought about chemical names too. Another place where long names are constructed. In that case the pieces are more likely to be combined into a single “word”.

The table in the wikipedia article you referenced is very close to the Challenge Math version, though the latter does not show the differences between American and British usage.

I assume that those people who decide how many “words” there are in the English language leave out specialized terminology like that.

The simple answer is, “No.” Unless you want to get into theology: Has God given a name to every number? Because we finite humans can’t even begin to count just the numbers between one and zero…

Just because we can’t count all the numbers doesn’t mean they don’t all have names. Although we can’t count them all, any given rational number can be named (using multiple words). So I would argue that there are an infinity of number names. Most non-repeating infinite decimals do not have finite names, except those we have explicitly named, such as e and pi.

But just looking at the whole numbers…. each number can be exactly named in scientific notation using a combination of words, even though there are too many for any finite being to count.

My question was, can each number be named exactly in “standard” terms. We have words for up to twenty triples (or sextuples, if you’re British) of zeros, and it’s fairly obvious to generate the words for many more. But probably not an infinity of them. Of course you could just name them by a combination of digits, but that doesn’t introduce any new words to the language, so it’s not as interesting.

I must agree with Mathmom; that we can’t count something is no reason that it can’t have a name.

If it were useful, I think it would be relatively easy to come up with an extendable number scheme, Mathmom. Perhaps using something like Archimedes’ powers of 10^8…though numbers with many significant digits would certainly get cumbersome…

I suppose it depends on what you call a “name.” While I agree that pi and e and the square root of 2 have names, what about a random irrational number? A random transcendental number? Or for that matter, even a large and random integer? If it takes you an hour or longer to rattle off the digits in order, can that really be considered a name? Would we call a long-winded description like “the redheaded boy—the older one, not that little brat—who lives in the brick house on the corner of X street and Y boulevard—you know, the one who threw a paper airplane into the principal’s office” a name?

Denise, I agree that random transcendental numbers would be hard to name. Maybe so for random irrationals too. I’m mainly thinking about whole numbers. Is “sixteen” a name? Is “Six billion, four-hundred eighty-five million, two-hundred sixty-five thousand, five-hundred thirty-two” a name? If you think of the latter as a name, my question is, can we construct a “name” like that for any given large whole number? We have names that occur in the dictionary for the numbers 10^3, 10^6, 10^9… up to 10^60. (Plus a few special ones like googol & googolplex) There are obvious candidates for the names of the next “several” terms in that sequence, based on the latin names for numbers. But at some point, we’re going to be talking about a number so large that we’d need to use these names just to describe the number sets of zeros the new number has. And then if we don’t have an obvious latin construct for something that large, I think we’re finally out of luck. I think there’s only so far we can go even with the generative naming schemes.

Refer to the last sentence in the wikipedia article cited above:

“John Horton Conway and Richard Guy have proposed[12] a consistent set of conventions which permit the system to provide “English names”, in principle, for any integer whatever.”

It’s not an official naming scheme, but it would do what you are looking for.

(Take a look at the cited book for details.)

Thanks for the pointer. I don’t think that reference was there last time I looked at that page, but I don’t know to check the history of the page to know for sure. Looks like a fun book in any case.