On Thu, May 16, 2019 at 5:42 AM Bill Gosper <billgosper@gmail.com> wrote:
Defn: "Regular CF" just means unit numerators. True or false?: Two consecutive 0s in a regular CF cancel each other out. True or false?: Tacking a 0 onto the end of a finite regular CF erases the (formerly) final term. Therefore, tacking on two 0s erases the last two terms! Or does the 2nd 0 erase the first before the first erases the previous? —rwg
It drastically matters whether you append them one-at-a-time or both at once: In[325]:= ContinuedFraction[FromContinuedFraction[{1, 2, 2, 2, 0}]] Out[325]= {1, 2, 2} In[326]:= FromContinuedFraction[{1, 2, 2, 0}] Out[326]= 3/2 In[327]:= FromContinuedFraction[{1, 2, 2, 2, 0, 0}] Out[327]= 17/12 Note added in obfuscation: As we saw yesterday, three consecutive i
<https://i.ytimg.com/vi/BGnoZ9ndYJc/maxresdefault.jpg> (=√-1) terms cancel each other out: In[151]:= FromContinuedFraction[{1, 2, 2, 2}]
Out[151]= 17/12
%^2/2
289/288
In[152]:= FromContinuedFraction[{1, 2, 2, 2, I, I, I}]
Out[152]= 17/12
In[153]:= FromContinuedFraction[{1, 2, 2, I, I, I, 2}]
Out[153]= 17/12
Another paradox: Twelve consecutive Out[405]= x -> (I (1 + Sqrt[3]))/Sqrt[2] terms provide zero information: In[409]:= FromContinuedFraction[{1, 2, 2, 2, x, x, x, x, x, x, x, x, x, x, x, x} /. %405] // Simplify Out[409]= 17/12 But six of them maybe say something. ? In[411]:= FromContinuedFraction[{1, 2, 2, 2, x, x, x, x, x, x} /. %405] // FullSimplify Out[411]= (31124 - 11 I Sqrt[2] + 60 Sqrt[3] + 61 I Sqrt[6])/22084 An x contains how much information? —rwg