Definition talk:Value of Continued Fraction

Definedness of value
Evidently, there are some restrictions on a (finite, for now) continued fraction to have a value: the length 1 fcf $[1,0]=1+\frac10$ does not have one, for example. It's not hard to come up with a definition of evaluability, but in fact there are two nonequivalent notions (the terminology here is new; nobody has formalized it afaik): Calculatable implies evaluable, but the example above shows that the converse does not hold.
 * calculatable: intuitively, if you can can calculate the epression starting from the innermost fraction. Rigorously, defined by induction at the same time as the value.
 * evaluable: intuitively, if you simplify the fcf to a fraction of the form $p/q$ with $q\neq0$, with rules like $\frac1{4+\frac 5 0} = \frac0{0\cdot 4+5}=\frac 05$. Rigorously, if the last denominator is nonzero; equivalently, using the matrix product here.

What do you think about defining these two notions? Any other suggestions for terminology? --barto (talk) (contribs) 03:48, 18 January 2018 (EST)
 * Another idea is to let calculatable=evaluable or any near-synonym, and define the weaker one to be weakly evaluable. --barto (talk) (contribs) 03:53, 18 January 2018 (EST)


 * We can of course cut this Gordian knot by imposing the condition that the coefficients of a CF are non-zero. Which we have already done. --prime mover (talk) 04:21, 18 January 2018 (EST)
 * Even with that restriction there are less apparent pathologies like $1+\cfrac1{2+\cfrac1{-3+\cfrac52}}$. (And in a general field, there is no notion of positivity.) --barto (talk) (contribs) 08:39, 18 January 2018 (EST)


 * Do you have a source to cite for this? --prime mover (talk) 08:51, 18 January 2018 (EST)
 * Nope, as I stated in the beginning. --barto (talk) (contribs) 09:57, 18 January 2018 (EST)


 * Original research then? --prime mover (talk) 10:00, 18 January 2018 (EST)
 * In the sense of technicalities all authors have figured out while they wrote those chapters, but Proofwiki may be the first to write down, yes. --barto (talk) (contribs) 10:45, 18 January 2018 (EST)