Sunday, January 27, 2008

Really big numbers. And I mean, REALLY big.

onsider a favorite song of mine, Built To Spill’s “Randy Describe Eternity”:

Built to SpillRandy Described Eternity

The song asks you to imagine a very large number by means of an analogy. The full lyrics go like this:

every thousand years

this metal sphere

ten times the size of Jupiter

floats just a few yards past the earth

you climb on your roof

and take a swipe at it

with a single feather

hit it once every thousand years

`til you’ve worn it down

to the size of a pea

yeah I’d say that’s a long time

but it’s only half a blink

in the place you’re gonna be

where you gonna be

where will you spend eternity

I’m gonna be perfect from now on

I’m gonna be perfect starting now

stop making that sound

stop making that sound

I will say I forgot

but it was only yesterday

and it’s all you had to say

Despite the technical difficulties of having a metal sphere that large pass that close to the Earth, it’s a cool metaphor. So how long of a time are we talking about here? Let’s come up with an estimate.

First consider how much a swipe of a feather would remove from the sphere. Let’s take a guess and say that 100 swipes would remove a square millimeter. (That’s being conservative, I think; could you really complete dissolve a grain of sand by brushing it a hundred times with a feather?)

Now consider the size of the sphere, “ten times the size of Jupiter”. Let’s assume they mean ten times the volume, which would be 10 × 1.43128 × 1015 km³, or 1.43128×1016 km³, which in cubic millimeters is 1.43128 × 1034 mm³. The “size of a pea” amount left over is incidental.

So if we just multiple that figure by 100 (feather swipes) times 1000 years, and we get 1.43128 × 1039 years. Let’s call this Randy’s number. Is it older than the current age of the universe? Yes, by a long shot. The universe is only about 1.37 × 1010 years; Randy’s number is 100 000 000 000 000 000 000 000 000 000 times larger. Yeah, I’d say that’s a long time.

Clifford Pickover’s Mazes For The Mind mentions a few other large (estimated) numbers:

  • The talking number — the total number of words spoken by humans since the dawn of history: 1016

  • The Coney Island number — the total number of grains of sand on the beach at Coney Island: 1020

  • The ice age number — the total number of ice crystals necessary to form the ice age: 1030

Douglas Hoftstadter in Metamagical Themas mentions a few others:

  • The Hemoglobin number — the number of hemoglobin molecules in the human body: 6 x 1021

  • Rubik’s constant — the number of possible configurations of a Rubik’s cube: 4.3 x 1019

Huge numbers, to be sure. Larger still is the infamous googol, which is 10100. Now consider the googolplex, which is 10googol. It is said that such a number could never be written down, as it would require more space than is available in the known universe.

But even these numbers are peanuts compared to some other numbers which have been envisioned. Consider tetration, Knuth’s up-arrow notation, Steinhaus-Moser notation, or Conway’s chained arrow notation.

Finally, just to fry your brain a little more, consider a number proposed in the excellent web comic XKCD, in his strip What XKCD Means”:

The Ackermann function with Graham’s number as its arguments is a number orders of magnitude larger than we can imagine. Actually, you can hardly say it’s “orders of magnitude” larger since that only implies exponentially larger; this number goes way, way, way beyond that!

But it’s still finite. And if you divide it by infinity, you still get zero. That kills me.


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