Do Numbers Go On Forever?

Date: 04/08/2002 at 11:32:17
From: Christopher Retzlaff
Subject: Large numbers and infinity

Do numbers go on forever, or do they stop? What is the largest number 
that has a name?

P.S. from Mom: What's the best way to explain infinity to a 
kindergartener?!

Date: 04/08/2002 at 12:47:25
From: Doctor Peterson
Subject: Re: Large numbers and infinity

Hi, Christopher and Mom.

This is something very special about numbers, which is worth thinking 
about: numbers are just ideas, not anything you can see or touch, so 
there is nothing to stop them from going on forever. We have to use 
our imaginations and picture what it would be like if we found a 
"largest number." Suppose I came up to you and said, "The largest 
number is umpteen quillion." (I made that up!) You would just laugh at 
me and say, "No it's not! Umpteen quillion and one is bigger!" Do you 
see how you could make a fool of me if I tried to give you an answer 
like that?

Now, it's a little trickier to ask, "What is the largest number _that 
has a name_?" You could find the largest number name in a dictionary; 
that is probably "centillion." (Try looking it up.) But mathematicians 
are not satisfied with mere lists of words. Mathematics instead works 
with _rules_. For example, assume that centillion is the biggest 
number in your dictionary. This would not mean that "one centillion 
one" is not a named number; it's just not a separate word that needs a 
definition. We have rules for naming numbers that allow us to use a 
small list of names and put them together to make names for more 
numbers than a dictionary could hold. And using those rules, we can 
make numbers like "a centillion centillion," which is a whole lot 
larger.

Then, people can make up words that aren't in dictionaries at all. 
Back in the 1930's, a man named Edward Kasner, a mathematician, asked 
his eight-year-old nephew Milton Sirotta to think of a name to give to 
a large number. Mr. Kasner then said that a "googol" was the number 
you would write as a one followed by 100 zeroes. Sounds pretty big, 
doesn't it? But it's not even as large as the centillion I mentioned. 
However, they then used that new name to give a name to a number that 
is much, much larger, the "googolplex." This is written as a one 
followed by a googol of zeroes. Now, a googol is so large that there 
aren't that many particles in the whole universe; so even if you wrote 
a zero on every particle in the universe, you couldn't even _write 
out_ a googolplex, much less count it!

So probably "googolplex" is the answer to your second question (though 
other even larger numbers have been given special names like "Graham's 
Number"); it is in many dictionaries. But even kids have named larger 
numbers; what would you think a "googolplexplex" might be?

Now, if we can imagine and talk about numbers that are so big no one 
could ever even write them out, you can see why I said there is 
nothing that can ever stop numbers from getting larger. Our 
imaginations are bigger than the whole universe!

One last comment. The word "infinity" doesn't represent an actual 
number that is bigger than all the others; "infinite" just means 
"without end," and is a way of describing something that never comes 
to an end. There are infinitely many numbers, because there is no last 
number. And that's really all it means.

If you have not already seen it, our FAQ may be of interest:

   Large Numbers and Infinity
   http://mathforum.org/dr.math/faq/faq.large.numbers.html 

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/