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Topic: computing k-th digit of a real number independently of the previous ones
Replies: 10   Last Post: Jul 1, 2012 4:12 PM

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AMX

Posts: 35
Registered: 8/22/09
Re: computing k-th digit of a real number independently of the
previous ones

Posted: Jun 30, 2012 5:49 PM
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On Wed, 27 Jun 2012 19:55:50 +0000 (UTC),
Herman Rubin <hrubin@skew.stat.purdue.edu> wrote:

>>>>> Q: Is it possible to prove the existence of a real number with
>>>>> the property that in order to compute its k-th digit, we must
>>>>> compute all the previous ones?

>
>>>> This is not the property of the number, this is property of way
>>>> of computation (algorithm) and also representation of the number.

>
>>> For many cases, one need not compute the preceding digits,
>>> but the subsequent ones.

>> [..]
>>> Then using number theory, one could get the contribution
>>> of each term to the desired digit and the subsequent ones,
>>> and hence compute the k-th digit without computing any
>>> previosu ones.

>
>> How general is this approach? Can I compute 10th and subsequent
>> decimal digit for Euler number
>> e = sum 1/n!

>
> I will illustrate how to compute the 11th digit.
>
> The terms for n < 3 do not contribute. for n = 3, 1/6 is 1
> in the first digit, and 6 thereafter. We keep track of
> this for digits ten and onward.
>
> For n=4, we have1/24. It likewise is 6 in all digits from
> 11 on. We can add these to get 12 in all digits from 11 on,
> or carrying out the additions, 3 in those digits.
>


In fact, I expected some magic theorem =:-)

So, let us make this problem a bit more complex. Suppose, we have
calculator with only 10 digits. Is it still possible to compute
11th and subsequent digits?

AMX





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