This blog is maintained outside office hours. The views stated in this blog do not necessarily reflect the views of our employers.

Monday, May 30, 2005

Number naming

Having seen an excel sheet with all sorts of numbers for the last week i just remembered on a web site i (or someone else, anyway a site i came to know about) that tackles the naming of numbers for example... 3 zeros - thousand.. why the hell thousand i ask ?

anyway here's what resulted in a well invested 15 mins by Alistair Cockburn

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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."

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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.

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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.

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My by-now six-year-old, Kieran, caught the other two off guard one day:

- "Mine rules gargoogolplex stars. Gargoogolplex is googolplex googolplexes."

He was heading off the

- "Mine rule two googolplex stars" argument before it even got started. Winner for the day.

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I have been obliged to find ways to make numbers grow faster and faster. We 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. A good number.
Well, pretty good, but 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: NN, 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.

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Not so. We made our own new math function, Fuga, to take us one step farther in this gamec. 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.) Fuga-four is (((4^4)^4)^4). Fuga-one-hundred is (...((100^100)^100) ... ^100) with a hundred hundreds listed in there. 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!

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Whew. Done for the day. They'll play with that for quite some time before one of them notices that we only said fuga- once ("Well my space commander rules fuga-fuga-fuga-fuga-fuga-fuga-fuga-fuga- ..."

Half of a half

mmmm, is this cheese i'm smelling ??

hey i'm currently the external correspondent with the biggest number after the # sign

since this is my first post in here i would like to post a quote from my favourite cartoon character... Peter Griffin, to a man who lives on a wheel chair.

"And Joe, I've had new neighbors before, but none of them were half the man you are. Since you're half a man already, that splits them into some kind of fraction I can't even measure."

Wednesday, May 25, 2005

Of Sheep and the Philosophy of Numbers

Good grief, it's a month since anyone bothered to update the blog. We've got a good excuse... mainly due to work, heat (of the climatic variety, not the other sort. Dirty, dirty people.), the total and abject sense of apathy brought on by excessive doses of both of the above, and good old fashioned bone laziness. Oh, and the fact that we totally forgot it existed. Oops.

Anyway, here's a quick filler until we find something mildly interesting to post about (provided we remember to do so.)

While #2 and I were having a cigarette and discussing the vicissitudes of the Visual Basic concepts of null, empty and nothing (heartfelt thanks to Big Bill for C#. All hail C#.) I wondered how the hell one explains the concept of null.

I vaguely remember something about apples and pears, way back in my childhood years, but for some bizarre reason which #2 attributed to the sun, we started discussing them in terms of sheep. Now one, two or even 6, 000, 639 sheep can be explained fairly easily. Fractions can be explained with equal ease, although it's a bit messy if we start cutting sheep up. (Also, #3 sustains that if we slice up a sheep it no longer counts as a sheep since it's dead.)

We're a bit on shaky ground with zero. Basically, there are zero sheep running loose on my desk as I type this. There are zero sheep running about the room, and #3 is currently doing things I wouldn't care to repeat in polite company to zero sheep at the moment. There are zero sheep wherever there are no sheep.

The zero sheep is everywhere. It is all around us. Even now in this very room. You can see it when you look out your window. Or when you turn on your television. You can feel it when you go to work. When you go to Church. When you pay your taxes. It is the world that has been pulled over your eyes to blind you from the truth.

(paraphrase from The Matrix)

We couldn't bring our heads around the idea of a negative sheep.

Things got even freakier as we discussed null. In decent programming languages null values just don't exist. It's different from a zero value - a zero implies that there are zero units of a given type, while a null implies that there isn't anything, not even a zero.

In the end we concluded that there can be null sheep. You can have a sign pointing to zero or more sheep. That is a value. Then you can have a sign that is not pointing, which is the null value.

I think it's time for coffee and red pills now.