# Changes

## Proposal: Digit Strings which Represent Continued Fractions

, 1 year ago
m
no edit summary
For the purposes of this article: All expressions are big-endian and microdigits are in traditional decimal. PEMDAS is obeyed.

== Mode Activation ==
== Original Explanation ==
Let $z = a_0 + b_0/(a_1 + b_1/(a_2 + b_2/(\dots))) = a_0 + \underset{i=0}{\overset{\infty}{\mathrm K}} \big(\frac{b_i}{a_{i+1}}\big)$, where $a_i$ and $b_i$ are integers for all $i$ (see [https://oeis.org/wiki/Continued_fractions#Gauss.27_Kettenbruch_notation Kettenbruch notation here]). In fact, for all i, we will canonically restrict $a_{(i+1)}$ and $b_i$ to nonnegative integers such that if $b_j = 0$, then $a_k = 1$ and $b_k = 0$ for all $k \geq j$; this is a perfectly natural and standard set of restrictions to make and does not actually diminish the set of numbers which are expressible in this format, but the restriction is not technically necessary for Lojban. Then we will denote $z$ by the continued fraction representation $z = (a_0 : b_0, a_1 : b_1, a_2 : b_2, \dots)$; the whole rhs representation is called a string. Notice that the integer part is included. In this format, for each $i$, "$a_i : b_i$" forms a single unit called a macrodigit; for each $i$, "$a_i$" and "$b_i$" each are microdigits; the colon ("$:$") separates microdigits and the comma ("$,$") separates macrodigits. Microdigits can be expressed in any base or other representation and macrodigits could be reversed or slightly rearranged (such as being of form "$b_i : a_{(i+1)}$"; however, for our purposes here microdigits will be expressed in big-endian traditional decimal and macrodigits will be formed and ordered as shown; the specification herein proposed will obligate the user to express the macrodigits in the form which is shown (id est: of form "$a_i : b_i$"; within any given macrodigit, the first microdigit expressed represents $a_i$ and the second (and final) microdigit expressed represents $b_i$, only) but the other features aforementioned are not guaranteed, although they may normally be assumed as a contextless default. In order to be clear: in this representation, each macrodigit will consist of exactly two microdigits - namely, $a_i$ and $b_i$ in that order, for all $i$ - and these microdigits will be separated explicitly by "pi'e"; meanwhile, macrodigits will be separated explicitly by "pi". In this representation, I will denote a not-explicitly-specified microdigit by a pair of consecutive underscores ("$\_\_$"). In the 'big-endian' arrangement of the macrodigits (as herein depicted), the first microdigit ($a_0$) represents the 'integer part' of the expression.
In this system, let "pi'e" represent ":" and let "pi" represent ",", each bijectively. Then the basic method of expressing a continued fraction is to just read $(a_0 : b_0, a_1 : b_1, a_2 : b_2, \dots)$ where each microdigit is expressed in some base which represents integers, the parenthesis are not mentioned, the separators being named/pronounced as before, "ra'e" being used in order to create cyclic patterns or to extend the string indefinitely, and the string being terminated as any numeral string could or would be. The interpretation of the whole string according to these rules for continued fractions would be specified via JUhAU.
The string will terminate and be interpreted as a number formed from the specified continued fraction as all other digits strings do (see my other work).
-- [[User:Krtisfranks|Krtisfranks]] ([[User talk:Krtisfranks|talk]]) 08:10, 9 March 2018 (UTC)
== Alternative Explanation ==
The description in this section is intended to provide the same results as those in the "Original Explanation" section.

Zeroth, the mode must be activated (as described previously).

=== Step 1: Simplest (and Finite) Case ===

Let $n \in \mathbb{N} \cup \{0\}$. Consider the continued fraction $a_0 + \underset{i=0}{\overset{n}{\mathrm K}} \big(\frac{b_i}{a_{i+1}}\big)$, where $\forall i, a_i$ & $b_i$ will be explicitly defined, particularly as they arise. We can enforce conditions on the sequences $(a_i)_i$ & $(b_i)_i$, but we will ignore such details, because we just need formal continued fractions.