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Old 2004-12-12, 20:38   #58
jinydu
 
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Dec 2003
Hopefully Near M48

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Of course we can make such predictions. For instance, I can prove with 100% certainty that every digit in the decimal expansion of 1/3 is 3. Although such precise predictions about the decimal expansion of pi cannot be made (at least not yet), this certainly doesn't mean that pi is random.

If you think about it, a decimal expansion is a rather arbitrary way of representing a number. In the case of a real number between 1 and 10, the representation is:

d0 + d1*10^-1 + d2*10^-2 + d3*10^-3 + d4*10^-4 + ...

where d0, d1, d2, d3, d4 ... are integers from 0 to 9. But what is so special about this form of representation? Its just one way to express a (potentially infinite) series. The observation that the digits of pi are unpredictable just shows that a decimal expansion is not a very good way to represent pi. But this doesn't mean that all representations will be unsuccessful. In fact, some other infinite series representations are not only successful, but stunningly beautiful. Once again, I cite the equation:

1 + 1/4 + 1/9 + 1/16 + 1/25 + 1/36 + ... = (pi^2)/6
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