Invisible <voi### [at] devnull> wrote:
> Occasionally you'll read about something, and the author remarks that 
> "this has more than 10^100 possible combinations, which is more than the 
> number of atoms in the universe", or something similar.

> Of course, 10^80 and 10^90 don't *sound* all that much different. They 
> *are* in fact extremely different (specifically, one is a thousand times 
> bigger!), but they don't look all that different. Something like 10^496 
> is rather difficult to grasp mentally.

  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.

  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.)

  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?

  Most people who have never heard of the concept of exponential growth
will usually give a guess which is at least some thousands. However, the
real answer is a surprisingly small number.

  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.

-- 
                                                          - Warp

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