## A Fuga Really Big Numbers

**By Alistair Cockburn**

Every parent knows that kids like to hurl really big numbers at each other. It starts with the five- and eight-year-olds.

"My space commander rules the whole world!"

"Yeah, well my space commander rules the whole star and all the planets."

"Yeah, well my commander rules two stars."

"Mine rules ten stars."

Now is the big moment for the five-year-old. Five-year-olds have to learn to count to 100 in kindergarten, so One Hundred is a really big number. It is so big and frightening to five-year-olds that they never name 101. Always 100.

"My space commander rules 100 stars!"

At this point the five-year-old will lose, because the eight-year-old can say, "Well mine rules 1,000 stars, so there." And the five-year-old can't say anything.

But the ten-year-old can, and jumps in with, "But my commander rules a million stars."

And now the five-year-old is back in the game. "Well my space commander rules a million million million million million million..." and keeps going until the other two walk away or mom or dad show up and say, "Quiet down in here and just play."

What kids need are some really big numbers. A jillion is good for a while, but it is not a real number, it's a fake. Quintillion is great if you can say it.

My kids were seriously impressed with googol, and even more impressed that it was named by a ten-year-old in search of a really big number. I can almost hear the conversation at the dinner table. The ten-year-old has recently learned about powers of ten:

"Dad, what's 10 to 5th power?"

"Ten thousand."

"Dad, what's 10 to the 10th power?"

"Ten billion." [Silence for a bit.]

"Dad, what's 10 to the 100th power?"

"It doesn't have a name." [Silence again.]

"I want to call it 'googol'."

"OK, that's fine."

"Dad, what's 10 to the googol?"

"Well, googol didn't have a name until a few seconds ago, so 10 to the googol doesn't have a name."

Power surges through the ten-year-old at the thought of having found a specific, real number that is so big it doesn't even have a name. The two decided on the name "googolplex," and they made the names public and popular. So that now, in our household, the conversation between the three kids runs on its course:

"Well, my space commander rules a googol stars!"

"Well, mine rules a googolplex stars!"

And they are stuck again. What we need are names for some seriously, really big numbers, bigger even than googolplex.

Let's first get straight that googolplex is a really big number. Googol is 10 to the 100th power, which is 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. Googolplex isn't just that number but has that many zeros in it. It simply has no other nameable name.

But it's no good having a top number. Kids need numbers beyond numbers, ways to name number larger than whatever numbers the other kids name. And they have to be actual numbers, not phonies like "jillion," or vague not-quite-numbers like "infinity." Part of the game is to keep using up more number names.

I have been obliged to find ways to make numbers grow faster and faster. My by-now six-year-old, Kieran, caught the other two off guard one day: "Mine rules gargoogolplex stars. Gargoogolplex is googolplex googolplexes." I liked the cleverness of Kieran's "gar-" prefix. It implies that there are as many of the number as the number itself. For example, "gar-four" is four fours. "Gar-million" is a million millions. And "gar-googolplex" is a googolplex googolplexes.

However, kids in junior high get around to noticing that four fours is really just four squared (4 x 4 = 4^{2}), and a million millions is a million squared (1,000,000 x 1,000,000 = (1,000,000)^{2}). Much more interesting would be to get 1,000,000^{1,000,000}, which doesn't have a name, as far as I know. But then, some of us have lived long enough to see what's coming. We won't just need a name for 1,000,000^{1,000,000}, but a name for: N^{N}, any number to the power of itself. Let's call it fz-whatever. Fz-four is 4^{4}, and fz-million is the 1,000,000^{1,000,000} we were looking for. We can already see where this is going. Fzgoogolplex is going to outdo gargoogolplex in just a moment, because gargoogolplex is only googolplex^{2}, and fzgoogolplex is googolplex^{googolplex}, and nobody is going to top that.

Not so. We made our own new math function, Fuga, to take us one step further in this game. We noticed that fzgoogolplex only has googolplex raised to the googolplex once. What about the ultimate five-year-old's answer? "Googolplex raised to the googolplex raised to the googolplex raised to the... (until the voice wears out)."

We came up with fuga (pronounced 'few-ga'). Fuga is a mixture of the musical word "fugue," and Kieran's "gar-" prefix. Fuga-number means "that number raised to that number that number of times." Fuga-2 is (2^{2}). Fuga-3 is ((3^{3})^{3}). Very soon, here, this is going to get hard to write. So we will leave fuga-4 and fuga-100 as exercises for the reader. Now we can name Fuga-gar-googolplex! (We could say fuga-googolplex, but why bother when we know that Kieran is just going to say Fuga-gar-googolplex right afterwards?)

There we have it. A number so big that it boggles the ears just in the speaking. Gar-googolplex raised to the gar-googolplex a gar-googolplex times! Bigger than the number of all the atoms in the universe! *[By a lot! Ed.]*