On Wed, 27 Jun 2012 19:55:50 +0000 (UTC), Herman Rubin <email@example.com> 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?