How Big is a Googolplex?

K.C. Cole is an award-winning science writer, whose 1998 book The Universe and the Teacup—The Mathematics of Truth and Beauty, I typically found at a used book sale. My wife chided me that it could now have only antiquarian interest. But I figured mathematics can’t have changed that much in 16 years. Two and two still make four, no?

The book broadly (and somewhat poetically) talks about the intersection between mathematics and life. It has some good stuff. One chapter discusses how goofy our risk perceptions can be. People worry about pesticide residues on fruit (annual U.S. death toll: zero) but not going for a drive (death toll: 30,000). Similarly, those terrified of child abduction drive kids to school – exposing them to the vastly greater auto accident risk. (All this echoed the “Freedom from Fear” chapter in my own very excellent book, The Case for Rational Optimism.)

However, not only did I also find some things I disagreed with, but some major bloopers.

Cole brings up one of my favorite paradoxes: “This sentence is false.” It contradicts itself. If the sentence is true, that means it’s false, so it can’t be true; but if it’s not, then it is true. However, Cole concludes this is no more paradoxical than the conflict between an American who thinks June is a summer month and an Australian who calls it winter. But that paradox is resolved with just a little more information. No additional information will resolve “this sentence is false.”
And how about this: “Those of us reared on Euclid swallowed without thinking all those axioms about the obviousness of such propositions as: two parallel lines never meet. Yet one only needs to look at the lines of longitude – which are parallel at the equator – to see that they do.” Hello? That’s non-Euclidean geometry! Euclid’s geometry applies only to flat surfaces, not curved ones (like the Earth’s).*

Then Cole says a googolplex is “a googol multiplied by itself a hundred times.” I’m no award-winning science writer, but even I knew this is wrong. (To confirm that, I googled it, of course.)

A googol is the number 10 to the hundredth power**; i.e., 10 multiplied by itself a hundred times; i.e., 1 followed by 100 zeroes. A googolplex (contrary to Cole) is the number 10 to the googol power; i.e., 1 followed by a googol zeroes.

These are very big numbers. Cole observes that we have trouble grasping how much bigger a billion is than a million, or a trillion than a billion. A billion is 1 followed by nine zeroes; a trillion by 12 zeroes; a quadrillion by 15 zeroes, and so on, for every three zeroes, through quintillion, sextillion, septillion, etc., each a thousand times bigger than the last. But we run out of those “illion” names long before reaching the end of all hundred zeroes in a googol.

Now, I asked myself, might Cole’s definition of a googolplex – a googol to the hundredth power – actually equal (the correct) 10 to the googol power? I didn’t think so, but how can one do this math? Not on a calculator! Too many zeroes. Indeed, there literally would not be enough space in the Universe for all the zeroes. But I thought there must be a simple, elegant solution. I finally puzzled it out.

First, 10 to the googol power is, again, 1 followed by a googol zeroes. So the number of zeroes in that number is 1 followed by 100 zeroes. Got that?

Next let’s look at Cole’s googol to the hundredth power. A googol has 100 zeroes. Multiply it by itself once and you get a number with 10,000 zeroes. Twice, and it’s 1,000,000 zeroes. Each multiplication increases the number of zeroes by 2. So doing it 100 times yields 200 zeroes – that is, a number whose number of zeroes is 1 followed by 200 zeroes.

Now that might sound like it’s double the result of 10 to the googol power – 100 versus 200 zeroes. However, remember that we’re comparing a number whose number of zeroes is 1 followed by a 100 zeroes with a number whose number of zeroes is 1 followed by 200 zeroes – far, far bigger. Cole would actually have been right had she said a googolplex equaled a googol to the fiftieth power. That would give you the 100 zeroes on each side. And this tells us just how far off she was: by a factor of exactly a googolplex! (a googol to the fiftieth power times a googol to the fiftieth power equals a googol to the hundredth power; i.e., Cole’s googol to the hundredth power equals a googolplex times a googolplex.)

Can any mathematicians out there tell me if I got this right?

By the way, yes, Google was named for googol, to evoke the vastness of the information accessible. But they inadvertently (?) got the spelling wrong!

When my daughter Elizabeth was eight, I explained googol to her. She was fascinated. Then she asked if the Universe would last a googol years.

Not a simple question. So I answered, “possibly.”

“Well,” she said, “if I’m eight now, then I’ll be a googol and eight.”

Now there’s an optimist for you.

* In fairness, Cole later does discuss non-Euclidean geometry.

** I’m not using superscripts because I don’t know how WordPress will display them on your screen.