Monthly Archives: December 2010

The Googolplex

googol A recent question was posed to me on Facebook by a family member and because it’s related to one of my kids’ go-to big-numbers I thought I’d share the exchange.  Here was the question:

How do you correctly notate a googleplex [sic] in scientific notation. I think it would be 10^10^100 but some of my co-workers disagree.

For those of you that might be unfamiliar, a googolplex is, in fact, one of the largest numbers that can be easily described to the non-mathematically oriented.  You’ll obviously be aware that Google is the company that introduced a technology that changed the way we use the internet.  They’ve become so ubiquitous that the very word is now a verb synonymous with internet search.

Not everyone is aware that the name “Google” is actually a corruption of googol which is the number 1 followed by 100 zeros, i.e.,

10,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000

It also has the name ten duotrigintillion, but googol has a nice ring to it.  After a bit of research, using Google in fact, I learned a couple of interesting facts from “Mathematics and the Imagination” by Edward Kasner and James Newman.  The name “googol” was first introduced by Dr. Kasner:

The name "googol" was invented by a child (Dr. Kasner’s nine-year-old nephew) who was asked to think up a name for a very big number, namely 1 with one hundred zeroes after it. He was very certain that this number was not infinite, and therefore equally certain that it had to have a name.

At the same time, Dr. Kasner suggested an even larger number which he dubbed the googolplex which would be formed much like the googol number except instead of 1 followed by 100 zeros, it would be 1 followed by a googol of zeros.  Take a minute to let that soak in.  The very writing down of the number in standard decimal form would not be possible, according to Kasner, even “if you went to the farther star, touring all the nebulae and putting down zeros every inch of the way.”

The question above proposes an alternative to writing it down the “old-fashioned” way.  A googol could easily be written with exponentiation,

[tex]10^{100}[/tex]. 

So, we might also be able to write a googolplex as

[tex]10^{10^{100}}[/tex],

which is 10 raised to a googol.  Going back to the original question,

How do you correctly notate a googleplex [sic] in scientific notation. I think it would be 10^10^100 but some of my co-workers disagree.

My Response:

Yes, kind of. Technically, scientific notation is of the form, A x 10^N, where A is a number between 1 and 10 and N is any integer. I would say yours is the mathematical notation for a googolplex.

Also, there is not a general agreement as to which order you should do the exponentiation. For example,

[tex]10^{left(10^{100}right)}[/tex]

is a googolplex (1 followed by a googol of zeros).  However,

[tex]left(10^{10}right)^{100}[/tex]

is not a googolplex. It turns out to be a 1 followed by 1000 zeros, much smaller than a googolplex.

Fortunately that answer was clear enough for him and to quote my family member, “The things tech geeks talk about at work… lol”.