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A rhetorical question, obviously. But check out this gentleman, who can calculate the 13th root of a 200-digit random number in his head, in just a little over a minute: http://www.smh.com.au/news/world/mathlete-works-out-200digit-number-in-head/2007/12/12/1197135500016.html.
Amazing! Not really all that useful, perhaps, but still amazing. . .
Best,
--- Les
[http://www.lesbell.com.au]
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For this alternative to work, when one would need a calculator, apparently one would have to have access to an array of such focused, specialized, and well-trained people -- roughly one person ("button") for each major function available on the assembled crowd-powered calculator.
(I imagine a high probability of error -- not due to faults in any one computation, necessarily, but in the communication of intermediate results among "function processors" during chain calculations.)
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And now you're describing the first computers. ;)