Warp wrote:

>   In general, the human mind tends to think linearly and cannot easily
> grasp the concept of exponential growth, no matter how much it's explained.
> Heck, even people who have studied technical subjects filled with math
> often have hard time grasping the concept of exponential growth.

Yeah, pretty much.

Also, many people don't seem to comprehend that "five billion" and "five 
thousand billion" are quite different numbers - people seem to go "oh 
yeah, something with 'billion' in it. I guess it's really big then?"

In a way, I think this is where metric measurements come in handy. Most 
people are able to "get" that KB is small change, MB is moderately big, 
and GB is very large (and TB is utterly huge). On the other hand, if you 
said "oh yeah, I have three thousand million bytes of RAM", people would 
just kinda blank that, I suspect.

>   This is sometimes used to present thinking problems with surprising
> results. The most classic one is the problem of the chessboard and the
> grains of wheat (or in some versions rice), as supposedly some man
> presented to some king (although this is almost certainly just an invented
> story).
> 
>   In other words: The man wanted as reward one grain on the first square
> of the chessboard, and for each successive square double the previous
> (ie. 2 on the second square, 4 on the third, 8 on the fourth and so on).
> This sounded reasonable to the king, so he accepted. Only when his men
> started actually counting how many grains of wheat that would make, did
> they realize the impossibility of the request. (Just the grains at square
> 45 or such are more numerous than the yearly production of wheat of the
> entire world.)

I heard that the total would be 2^64-1 grains which is "more than the 
toal number of grains that has ever existed on Earth" - a figure far 
exceeding merely the yearly production of wheat.

I have absolutely *no idea* whether this description is actually 
accurate or not - which is why I'm after a table of big numbers! ;-)

>   Another funny example, which you can use on someone: Assume you have
> a really, really large piece of cardboard which is 1 mm thick. Also assume
> that you can fold it in half as many times as you want (thus doubling its
> thickness each time you fold it). How many times do you have to fold it
> before the thickness reaches the Moon?

Weirdly, you can only fold a piece of paper about 7 times. I have 
literally no idea why. Brainiac tried it with a huge roll of industrial 
tissue paper (so it's both very large and extremely thin). To make the 
7th fold, they actually drove a van over the thing, but still it didn't 
"really" fold convincingly. I guess it's due to the large turn radius or 
something...

>   As for number series which grow very fast, I like the following one,
> because it's easy to understand and state, and grows incredibly fast:
> 
> - Let's denote a series of exponentials with (!^n).
>   n(!^1) just means n!
>   n(!^2) means (n!)! (the exponential of the exponential of n)
>   n(!^3) means ((n!)!)!
>   etc.
> 
> - The number series is: f(n) = n(!^n)
> 
>   This series grows *very* fast. The result of f(5) (represented as a regular
> decimal number) is probably larger than the combined hard drive space of all
> the hard drives in the entire world could store. It's probably so large that
> even if each atom in our planet could be used to represent a bit of storage,
> it wouldn't be enough to store the entire number.

Heck, apparently n! is [VERY approximately] proportional to n^n, which 
gives you some idea just how damned fast it grows. (Faster than any 
normal exponential function.)

A degree N polynomial grows faster than any possible degree N-1 polynomial.

An exponential function grows faster than any polynomial of finite degree.

A factorial function grows faster than any possible exponential 
function. [And we've already established just how ****ing fast that grows!]

Your n(!^2) is guaranteed to grow even faster still.

Finally, n(!^n) grows faster than any other function yet mentioned.

Still, take heart. The busy beaver function grows so fast it's not even 
a computable function... (!!)

-- 
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*

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