NetNews Usenet Archive 1993 #1/NN_1993_1.iso/spool/sci/math/17724

Internet Message Format | 1993-01-06 | 3.1 KB

Path: sparky!uunet!zaphod.mps.ohio-state.edu!usc!news.service.uci.edu!beckman.com!dn66!a_rubin
Newsgroups: sci.math
Subject: Re: Number of numbers (Was Re: PI)
Message-ID: <a_rubin.726276787@dn66>
From: [e-mail] (Arthur Rubin)
Date: 5 Jan 93 23:33:07 GMT
References: <[e-mail]> <[e-mail]>
 <[e-mail]> <[e-mail]>
Organization: Beckman Instruments, Inc.
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In <[e-mail]> mfriedma@uucp (Michael Friedman) writes:

>In article <[e-mail]> [e-mail] (G.J. McCaughan) writes:
>>In article <[e-mail]> [e-mail] (Mark Corscadden) writes:
>>>1)  ========== primitive recursive functions
>>>    The are the functions which you can compute in a typical general-
>>>    purpose computer language, except that before you enter any loop
>>>    you must first compute an upper bound on the maximum number of
>>>    times you will be allowed to pass through the loop, and must not
>>>    exceed this pre-computed upper bound.

>>  [description of "partial recursive", "total recursive"]

>>>Suppose you define a real number to be "primitive recursive" if there is
>>>a primitive recursive function which computes the Nth decimal digit as
>>>a function of N.  Same for a "total recursive" real number.

>>>Does anyone know whether PI is primitive recursive by the definition above?
>>>If not, this fact seems to capture a bit of Samuel's "gut feeling" that
>>>his example SUM(10^-(n!)) is "much simplier to calculate" than PI, since
>>>SUM(10^-(n!)) is clearly primitive recursive.

>>Sorry; pi is primitive recursive. Take any of the squillions of formulas
>>for pi which give error terms which actually give bounds on how far wrong
>>a given approximation is. (For instance, though I wouldn't dream of
>>actually computing pi with this, pi=4(1-1/3+1/5-1/7+1/9...) where the
>>error incurred by stopping at any given point is at most equal to the
>>size of the next term.) Then compute how many terms you need to take to
>>get the first N decimal places, and what sort of precision you need to
>>use; and run the algorithm that far. No unbounded loops in sight.

>Is this really true?  After all, we know that there are arbitrarily
>long strings of consecutive 0s and 9s in the decimal expansion of PI.
>What precision to I need to compute the Nth digit of PI if that Nth
>digit is at teh beginning of such a sequence?  After all, you can
>change the value of digit N by adding (for 9s) or subtracting (for 0s)
>and arbitrarily small value if the string of identical digits is long
>enough.

Well, actually, we don't know that there are arbitrarily long strings of
consecutive 0s and 9s in the decimal expansion of Pi.
--
Arthur L. Rubin: [e-mail] (work) Beckman Instruments/Brea
[e-mail] [e-mail] [e-mail] (personal)
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