Changes

Proposal: Digit Strings which Represent Continued Fractions

, 1 year ago
m
- building alternative description
For the purposes of this article: All expressions are big-endian and microdigits are in traditional decimal. PEMDAS is obeyed.

== 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 ). 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.
[[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.