There is a debate amongst my friends. I say that if numbers are truly INFINITE, than every single combination of letters must describe a number. Therefore, a billion, a trillion, a jigazillion, and a fapillion are all valid numbers at some point. In fact, even
StraightDopeMessageBoardillion is a valid number if there are infinite combinations.
My friend maintains that since numbers use a naming methodology, the names will only increase in length, but will never get “awkward names.” In his case, he says you’ll sooner see “duodecahexaseptaoctillion” than made-up names like “bazillion” or “kajillion.”
Even though we completely disagree, assuming we’re discussing a number set that is INFINITE, how can either one of us be wrong?
I was about to say that we may never have to face the problem as, after we get to a certain point, the number we are describing is larger than the number of fundamental particles in the universe and may not be terribly useful, but that may not be true; for example the number of possible combinations in a cipher key could exceed the number of particles and we’d still have to talk about it.
I’m still not entirely sure what you’re asking, but it seems that the string of letters that precedes the ~illion must inevitably get longer as we run out of (actually, meaningful) combinations of shorter sequences, but these things are not subject to a fundamental law of the universe, but rather to human reasoning within the academic community - for this kind of reason we have ‘Googol’ - which doesn’t follow the regular ~illion pattern.
Given the nature of infinity any possibility will eventually happen (I realize that is an improvable statement), another question is what number would be large enough to be StraightDopeMessageBoardillion and if it was to large would we be accused of trying to overcompensate for something?
There is no particular reason at all why a given number must have a name, other than convenience. And it’s unlikely that any new number names will be invented.
Sethadam, you are wrong. Now go and apologize, I’m sure your friend feels bad enough.
Just because there are an infinity of numbers, and an infinity of combinations, that doesn’t mean that all the combinations have to be names of numbers. What if we decided (we haven’t yet, but what if) that all number names do have to end in -illion, like your examples? There would still be an infinite number of names (i.e., more than enough), but not all names would be numbers–for instance, “Sethadam” would not, because it doesn’t end in -illion.
Yes and no. It’s possible to come up with a naming scheme where all combinations of letters are used and all possible numbers are described. It’s also possible to come up with a naming scheme that describes all numbers without using all combinations of letters, or one that uses all combinations of letters and fails to hit all the numbers.
A naming scheme is a correspondence between a set of names and a set of objects which satisfies two properties:[ol][li]Every object has a name.No two objects have the same name.[/ol]The defining property of an infinite set is that it has a proper subset which has exactly the same number of elements as the set itself. The set of all letter combinations has the same number of elements as the set of all numbers because you can list all the letter combinations in dictionary order, breaking them up by length (this is called a lexicographical ordering, if you want to search for a more precise description).[/li]
If you replace the set of all letter combinations with the set of all letter combinations that end in ‘e’, you still have the same number of letter combinations to work with. Likewise, if you replace the set of all numbers by the set of all even numbers, you still have the same number of elements to work with. So you can use either of those subsets in place of the original sets, and it’ll work.
How do I know this? You can tell if two sets have the same number of elements if you can set up a correspondence between them that has the same properties as a naming scheme. That type of correspondence is called a bijection, and two sets are defined to have the same size (or cardinality) if there is a bijection between them. The bijection from the set of all letter combinations to the set of all letter combinations that end in ‘e’ just consists of adding an ‘e’ to the end of each word. The bijection between the set of all whole numbers and the even whole numbers consists of doubling each whole number.
Note also that the properties of a bijection guarantee that each element of one set corresponds to a unique element of the second set. So whenever you have a bijection going one way, you can construct one going the other way, called the inverse. I’ll leave it to the reader to construct the inverses of the above bijections.
Well, that’s long-winded, isn’t it? Does it help, or am I just flapping my (virtual) lips to hear the pretty noises?
I definitely appreciate some scholarly perspective, but I remain skeptical. Given that all nominals are subjective in English numerals, and many do not follow a convention, I find flaw. Infinity to me presumes non-exclusion (right?), so how can any letter combination not be a valid number? Infinity should account for every case imaginable that follows the given subset, and in this case, the only defined rules seem to be that it be both pronounceable and unique. Had we developed a number system that evaluated validity based on existing criteria, like numbers with 00s ending with -illion, perhaps this debate would be moot, but in this case, how can the number “CecilAdams” be any less valid than “two?”
Perhaps some numbers might be unfathomably large, but how many electrons exist in the universe? How many quarks?
It seems to me that this is more a flaw in human interpretation of the concept of infinity than comprehension of the number system. The human mind isn’t equipped with the logic to decipher “infinite.”
Wrong. There are an infinite number of prime numbers. Does that mean that at some point, 6 has to be prime? Of course not. The primes is an infinite set that excludes 6. Similarly, the set of all number names is an infinite set. It could definitely exclude “kajillion”.
Why no just use scientific notation at a certain point? 1 X 10^20 (or just 10^20) rather than StraightDopeMessageBoardillion. Its easier and more reliable than just saying a made up name.
Why no just use scientific notation at a certain point? 1 X 10^20 (or just 10^20) rather than StraightDopeMessageBoardillion. Its easier and more reliable than just saying a made up name.
According to this site we’re at least covered for numbers up to 10[sup]303[/sup], which, in the U.S. is a “centillion.” (in Europe a centillion is 10[sup]600[/sup].)
Of course, if 10[sup]303[/sup] is “one centillion”, then we’re actually covered up to 10[sup]306[/sup] - 1, which is “nine hundred ninety-nine centillion, nine duotrigintillion, nine hundred ninety-nine untrigintillion, …”
Wrong.
Look at it like this. Say some guys got together and for lack of a better thing to do decided to go and name as many numbers as they can. They could create a naming pattern but sooner or later their limitations would force them to abandon it. They have limited sounds which with to say the name and limited time to say it. Also to keep track they would need to invent symbols to keep from spending years writing the number. Also, at some point they would have to begin repeating names, maybe adding a word or sound to show it a repetition.
So ‘B’ could be the written form of a number and StraightDopeMessageBoardillion could be a number. Also the second six could be the name for a prime number so when it comes to infinity anything can go.
I declare The number 447wterm5 to be pronunced Cek (seek)!
Wrong.
I’ll invent a naming pattern for you right here:
I’ll call 1 “la”.
I’ll call 2 “lala”.
3 is “lalala”.
.
.
.
And any positive integer n would be named “lalala…la”, (n "la"s).
You might think this would be an objection to my above naming system; however, you would be wrong. Any naming system will eventually have to start giving arbitrarily large names (you can only name finitely many numbers if the name has to be less than n letters).
I suppose this, in theory, could be done. For example, I could write the number 1 as a vertical line. The number 2 as a vertical line with a horizontal line across it; 3 would be a vertical line with two horizontal lines across it, and so on, introducing another horizontal line to the symbol for each new integer. This system would suck for readability, however. It wouldn’t save time writing the number, either. In short, there’s no practical way of inventing new symbols with which to name the numbers that would be of any benefit. And no matter how many symbols you invent, if you have only finitely many of them, the names of the numbers would still get arbitrarily large.
This would certainly be a no-no; if I have two different numbers named “Cek”, how would I ever know which number was being referred to by the name “Cek”?
You could do this; in fact, that’s what I did above.
