Monday, October 26, 2009

On Googols and Google, Googolplex and Infinity: The Truth About Big Numbers

When I was in high school, I read a book called Infinity: Beyond the Beyond the Beyond. I don’t remember much about it, but I’ll never forget the title. The concept of infinity in its … well, infiniteness… can keep my mind occupied for a long time. And the idea of going “beyond the beyond” — and then beyond that! — provided more delicious food for thought. I sometimes think about that title and the mind-candy of endlessness when I’m speaking at a school, as I was last week in West Chester, PA, and someone asked, “What’s the biggest number?” It’s a question I often hear. The conversation usually goes something like this:

Child: What’s the biggest number in the whole wide world?

David: Do you think there is such a thing as the biggest number?

Audience: half “Yes,” half “No”

David: Will someone please tell me what you think the biggest number is.

Children, variously: billion, trillion, quadrillion, quintillion, googol, googolplex, etc.

David: Hang on. Let’s suppose you think “quintillion” is the biggest number. Then what about “quintillion-and-one”? Isn’t that bigger? And if that's the biggest, what about “quintillion-and-two” — even bigger, right?

This usually leads to a triumphant retort about an enormous number familiar to many children (much less familiar to their parents and teachers):

Child: Googol has to be the biggest!

David: What’s a googol?

Many children know that “googol” is the name for a very large number — a one followed by a hundred zeros. This is an exciting concept. In my book G is for Googol: A Math Alphabet Book, I tell the story of how “googol” got its name from a nine-year old boy. Surely it is tempting to call googol “the biggest number,” but that’s a non-starter.

Me: If you think googol is the biggest number, then what about googol-and-one? Or two googol? Or a googol googol?

Almost inevitably, at this point someone proffers an even bigger number, “googolplex.” It is true that the word “googolplex” was coined to mean a one followed by a googol zeros. It’s way bigger than a measly googol! Googolplex may well designate the largest number named with a single word, but of course that doesn’t make it the biggest number. In a last-ditch effort to hold onto the hope that there is indeed such a thing as the largest number…

Child: Infinity! Nothing is larger than infinity!

True enough, but there is nothing as large as infinity either: infinity is not a number. It denotes endlessness. A number designates a specific amount.

So, finally we get to a consensus: There is no such thing as the largest number. Yet numbers as large as googol or googolplex continue to tantalize, and well they should. To me the most fascinating thing about googol is how incredibly enormous it actually is. Writing those hundred zeros, while tedious, would take only a minute or two, yet the amount represented is, as I stated in G is for Googol, “more than the number of hairs on the head of everyone in the world, more than the number of blades of grass on all the lawns of the world, more than the number of grains of sand on all the beaches of the world — even more than the number of atoms in the universe.”

The estimated number of atoms is a one followed by 72 zeros (ten to the 72nd power, but I can’t do exponents in this blog). Let’s suppose the astrophysicists who estimated the number of atoms are way off. For the moment, let's imagine that the actual (though unknowable) number of atoms is a hundred times as what they claim. So it would be a one followed by 74 zeros —still way, way, way less than a googol.

The number “googol” is, in fact, useless — except as food for a hungry mathematical mind. And it is an especially nourishing numerical treat for young hungry minds. In fact, a child possessing just such a hungry young mind corrected me when I once said, “There isn't a googol of anything, anywhere.” The boy countered, “There are more than a googol numbers. The number of numbers is infinite.” Right he was! Now I modify the statement: “There isn't a googol of any physical object.”

I am less enthusiastic about the point that was made by sixth graders in a class that sent me a stack of letters. All had the same basic theme, reflected by this one:

Dear Mr. Schwartz,

How do you know how many hairs are on the head of every person in the world? You probably haven’t met every person in the world. Even if you have, babies are being born every minute. People are losing hair every day!

No argument there but, unfortunately, this class didn’t seem to have a good understanding of the importance (and legitimacy) of estimation.

Now, with the ascendancy of a certain multi-billion dollar online enterprise, it is necessary for me to include in any discussion of googol the following important inequality, lest there be confusion:

It is interesting to note that the item on the right was the result of a spelling error. When Larry Page and his friends were choosing a name for a start-up company, he attempted to name it after the huge number “googol.” Instead, he committed what is probably the most famous (and lucrative) misspelling in history. Regardless, there is absolutely no doubt that both “googol” and “google” are mighty big.

1 comment:

Vicki Cobb said...

Great post, David. My head hurts from imagining. Here's what Einstein said: “Imagination is more important than knowledge. For knowledge is limited to all we now know and understand, while imagination embraces the entire world, and all there ever will be to know and understand.”
In other words, imagination is infinite.