Friday, February 17, 2006

The Googolplex

Time to laugh a little.

The googolplex is a famous number so large that it cannot be written out in a conventional manner. The number of digits in a googolplex exceeds the space available to put them in, even if you printed each digit at the size of an atom and used all the space in the known universe.

A googolplex is unimaginably larger than a googol, which can be written out as a 1 followed by a hundred zeroes, or 10 to the power of 100. Let me write that out (with some help from cutting and pasting):

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.

Another way to write it out is as 10 to the power of 100, which is 10 multiplied by itself in a string that includes a hundred instances of "10".

10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10.

That's a large enough number. The estimated total number of electrons, protons and neutrons in the known universe is 10 to the power of 80, which is just a tiny fraction of a googol. If you took away the last 20 zeroes in the number above, you would have the total number of elementary particles in the universe. If you imagine that number as P, you get a googol by multiplying P in the following manner:

P x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10.

or,

P times 100,000,000,000,000,000,000.

How would you say this equation? "P times a hundred ..." What? What's the term that comes after millions, billions, trillions, and zillions? Well, if you type "millions, billions, trillions, zillions" into our old friend Google, you find that Word Reference lists a certain "jillion" coming after zillions. Unfortunately it is merely defined as a term for any indeterminately large number. Other sites list a slang term, "kazillion". A more formal table of terms is at Zillions and Zillions!, where I found out that my way of looking at it is American; the European scale is different.

In the formal terms, zillions don't even exist. In the American system, trillions are followed instead by quadrillions, and then by quintillions. So an American would say the above equation as, "P times 100 quintillion." In England you'd say, "P times 100 trillion." Sticking to the American terminology, you would say that the total number of elementary particles in the universe is estimated to be a hundred-quintillionth part of a googol.

A googol, incidentally, would be listed in the U.S. as 10 million trigintillions. I have no idea, even with the table in front of me, how to say that in Europe.

So that's a googol -- 100 quintillion times the number of elementary particles in the known universe.

Here is where it gets funny -- the googolplex. This is a number with a sense of humor. You arrive at it by multiplying 10 by itself in a string such as the one above, where "10" appears one-hundred times -- except that you have to write it out a googol times. That's right. The number typed out above, with a hundred zeroes, is the number of times that the "10" has to appear in such an equation. And each time it appears, of course, the total number grows ten times larger -- but that is by no means the hard part of trying to imagine a googolplex. What's beyond our normal reasoning is the number of times that "10" appears in the equation. If you take on the process of stuffing a "10" into each elementary particle in the universe, and do that entire process 100 quintillion times, you get to write out the equation. Never mind the result.

And that is still just so much talk.

Take your average person weighing around 150 lbs., or 70 kilograms. That person is comprised of about 7 octillion atoms, written out with 27 zeroes. The body's total number of electrons, protons and neutrons is 64 octillion, or 64,000,000,000,000, 000,000,000,000,000. That's the number we're starting with.

Multiply that by 1 quintillion (that's 18 zeros). Take 85,000 of those and you've got the mass of the earth. Take 300 earths and you have Jupiter. Take 1,100 Jupiters, and you've got our sun. According to Sten Odenwald's Back to the Astronomy Cafe, estimates for the total mass of our galaxy, the Milky Way, tend to fall around 700 billion times our sun's mass -- contained in around 1 trillion stars. The total number of stars in the universe is estimated -- give or take a factor of 10, of course -- at 80,000,000,000,000,000,000,000. That's 80 sextillion stars. (A sextillion comes after quintillions). If you want to say it in terms of the Milky Way's trillion stars, you'd say 80 billion trillion stars. 80 billion Milky Way masses.

Now take the result and multiply it, as we worked out it above, by 100 quintillion. That's a googol.

And that's the number of times you will write out the number "10" on the black board, just to write out the equation for a googolplex.

But I did promise a little laughter.

A very funny site is Googolplex. That will spin your brain. It's one of the oldest pages on the web; back in 1995 it was awarded a distinction as one of the top 5% internet sites.

The page shows you how to print out a googolplex, if you're so inclined. Maybe that's why it's also listed as one of the most useless sites on the Web.

I say that laughter is vastly useful, however. From that site, and with a little exploration, I found myself laughing harder than I have in years -- especially at a site called Gizoogle. Translate any text, or even a web page, into jive.

Now a Googolplex has a 1 followed by 10 ta tha powa of 100 zeroes.

Gizoogle is listed at a page called Pointless Sites. Another distinguished member on that list: The Infinite Cat Project.

All this madness started a week ago by watching Carl Sagan's Cosmos, Episode 9, "The Lives of the Stars." That's the one that has Carl explaining the Googolplex, trying to write it out on a roll of paper, explaining mathematical infinity (a googolplex is no closer to infinity than is the number 1), exploring the Table of Elements, describing the fates of stars with enormous masses, flattening Alice's tea party under excessive G's, and threatening to cut a slice of apple pie down to atoms (it takes 90 successive cuts). Best of the "Cosmos" episodes.

I highly recommend meditating on the googolplex. A bit of madness will take hold of you for a while, but it will pass.

5 Comments:

Anonymous Anonymous said...

It's P times one hundred quintillion if I counted correctly.

July 16, 2007 12:50 PM  
Blogger James Justin Harrell said...

Protons and neutrons are not elementary particles.

http://en.wikipedia.org/wiki/Elementary_particle

You also did not sight a source for you claim that the estimated total number of protons, neutrons, and electrons is 10^80.

Pretty poor writing.

May 22, 2008 3:48 PM  
Blogger Kevin Rosero said...

It was definitely an oversight, not to cite Carl Sagan's Cosmos as the source of the estimate. I do note in the post that his discussion of the googolplex in the Cosmos film is what got me started.

"By comparison [to a googolplex], the total number of atoms in your body is about 10^28, and the total number of elementary particles -- protons, neutrons and electrons -- in the observable universe is 10^80." (Cosmos, p. 181).

On the next page Sagan goes on to say that protons have been bombarded at high energy with other protons, and that we've begun to observe still smaller units, more "fundamental units" in his words.

When he wrote Cosmos, other elementary particles, besides the three famous ones, had already been discovered.

http://en.wikipedia.org/wiki/Timeline_of_particle_discoveries

But in his book he referred to elementary particles as neutrons, protons and electrons. Why, exactly, I don't know. Perhaps because he was writing for a lay readership he was keeping to the best-known definition (the one we all got in grade school) of elementary particles as neutrons, protons and electrons.

Certainly it's that popular basic definition that I'm using; but I should have given the citation to make that clear -- not least because there may be other and newer estimates out there.

May 22, 2008 7:45 PM  
Blogger James Justin Harrell said...

Thank you for taking the time to respond.

May 25, 2008 9:44 PM  
Anonymous Anonymous said...

James....
You are an idiot.
It is a BLOG! Cope with the fact that people who blog are usualy to envolved with themselves to care and they also are too pre-occupied with their own self image of imporntance to write proper;y!

October 19, 2009 10:16 AM  

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