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I was thinking of how numbers are mentally approximated, and the fact
that (at least to me) it seems there's a sense in which 1 and 1000 are
more "mentally distinct" than a billion and a trillion, even though both
differ by a factor of a thousand and the latter are far more different
in absolute terms. A cute and illustrative little math puzzle arose
from this musing.
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
In a psychological sense these numbers seem pretty similar to me, but
just how different are they mathematically? Consider the ratio A/B
measuring how many times bigger A is than B. It's interesting to
consider how accurately you can estimate this ratio without calculating
it explicitly. For instance, is it true that A/B is greater than a
trillion (than is, A is more than a trillion times bigger than B)?
More explicitly, 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.
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On 3/6/2012 11:13 AM, Kevin Wampler wrote:
>
> A = 10^10^10^11
> B = 10^10^10^10
> A/B = 10^10^10^x
Note: My newsreader tries to be too smart and utterly botches the
formatting of these equations. They should read equivalently to:
A = 10^(10^(10^11))
B = 10^(10^(10^10))
A/B = 10^(10^(10^x))
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On 06/03/2012 19:13, Kevin Wampler wrote:
> I was thinking of how numbers are mentally approximated, and the fact
> that (at least to me) it seems there's a sense in which 1 and 1000 are
> more "mentally distinct" than a billion and a trillion, even though both
> differ by a factor of a thousand and the latter are far more different
> in absolute terms.
That's funny, I could have sworn I said the same thing yesterday... ;-)
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On 3/6/2012 1:11 PM, Orchid Win7 v1 wrote:
>
> That's funny, I could have sworn I said the same thing yesterday... ;-)
>
I either missed the post or was unable to correctly parse your intent
from what you wrote then. I was thinking it was Warp who was
(implicitly) pointing out this while you were saying that humans judge
numbers logarithmically (plus some sort of "size-numbness" thing which I
couldn't connect to your preceding reasoning). In either case the point
itself is pretty obvious, but it's still interesting to see really clear
cases of it in action.
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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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