Ed Pegg writes: << I recently did a column on approximations. http://www.maa.org/editorial/mathgames/mathgames_02_14_05.html

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I urge other people to look at this incredibly interesting stuff. I admit that I don't understand very well how keenness is defined. Here's one item from that site that suggests some comments: << Richard Sabey's spectacular approximation to e, (1+9-47*6)3285, gets 18457734525360901453873570 decimal digits of accuracy due to high exponents and asymptotic effects.

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Yikes, I can barely count the number of digits in the number of digits! Since this is (1 + (very small))^(very large), and since e = lim_(t -> oo) (1 + 1/t)^t, and (1 + 1/t)^t is monotonic increasing for t > 1, this approximation suggests that9^(4^(7*6)) and 3^(2^85) must be fairly good approximations to one another. So, what would happen if we just used any incredibly large value of t with its own reciprocal? E.g., how close is (1 + 3^(-2^85))^(3^(2^85)) to e ? Or (1 + 9^(-4^(7*6)))^(9^(4^(7*6))) ? Now, I'm not sure if this is the same as keenness, but for me a nice measure of an approximation would be the "surprisingness" of it. This would be defined as a probabilty, roughly as follows: Suprisingness^(-1) = the probability that if one considers *all* the numbers with the same or smaller complexity to be randomly distributed within their range [smallest, largest] (or maybe this should be done logarithmically), that one of them gets as close as or closer than the approximation actually does. --Dan