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: 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.

Calculatable implies evaluable, but the example above shows that the converse does not hold.

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)

Bear in mind that the study of continued fractions is one of the oldest fields of number theory, and may in fact date back to the times before mathematicians were comfortable with negative numbers, let alone zero, so by default all such coefficients were strictly positive. I'm guessing. --prime mover (talk) 11:40, 18 January 2018 (EST)