On 3/6/2012 11:13 AM, Kevin Wampler wrote:
>
> Assume exponentiation is right-associative, so a^b^c = a^(b^c). Now
> consider the two similar-appearing numbers A and B where:
>
> A = 10^(10^(10^11))
> B = 10^(10^(10^10))
>
> let's write:
>
> A/B = 10^(10^(10^x))
>
> Obviously x is less than 11, and it turns out it's greater than -11.
> It's an interesting experiment to make an intuitive "gut instinct" guess
> as to the correct value of x without calculating anything, and then
> actually calculate it out in full and see how accurate your intuition is.
>

It seems people aren't too interested in this, but I thought I'd provide 
the answer in case someone out there was quietly curious about it.  If 
that's you, make a "gut instinct" guess now since I'll be giving the 
answer next.

Assuming my calculations are correct, it turns out that the correct 
value of x is *very* close to 11.  So close in fact, that the decimal 
representation of 11-x begins to differ from zero only in the 
ninty-billion-and-twelfth decimal place.  Said another way, the decimal 
representation of x is "10.<ninty-billion-and-eleven 9s><other stuff>".

Thus, in some meaningful psychological sense, the value of A/B is very 
very close to the value of A, despite the fact that any absolute or 
logarithmic comparison would indicate that A is fantastically larger 
than A/B.  It's a simple and kind of neat concrete illustration of both 
just how bad our brains are are reasoning about large quantities 
directly, and how flexible our brains are are at nevertheless 
representing these quantities in ways that can be reasoned about indirectly.

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