# mi'i'au PA …BOI experimental cmavo

digit/number: interval/range indicator for significant digits (determined by midpoint).

Given a well-formed digit string "xn x mi'i'au xm-1 x", where (a) "xi" is a member of selma'o PA (other than this word or similar words; including at most one instance of "pi") for all i, and (b) the string represents^×1 a finite number in base-b (taken to be ten by cultural convention in most human cases unless explicitly specified otherwise), the usage of this word in the aforementioned digit string yields an output of the interval (\sumi = 0^\infty(xn-i b^n-i) - b^m, \, \sum= 0^\infty(xn-i b^n-i) + b^m), after adjusting for "xk" in the original string being "pi" or "pi'e" vel sim. for at least one k (because that does not work in the summation notation without adjustment); notice the single minus sign in the expression which forms the lesser (left) endpoint; notice the placement of the comma in the interval, and the endpoint clusivity thereof (both endpoints are excluded), as well. Therefore, using/under the aforementioned notation and assumptions and specifications, usage of this word in "xn x mi'i'au xm-1 x" outputs an interval which is equivalent to the evaluation of/interval referenced by "eval("xn x") ga'o mi'i ke'i bm". Importantly, usage of this word generates an interval, not a specific number (even if such would be elliptical or vague) - meaning, among other things, that equality to such an expression would be set equality, and not numeric equality. Note that the interval which is generated excludes both endpoints but includes the center (which is the number specified by the string with this word ignored in/removed from it). As an example, where "M" represents this word: "2M000" yields (1000, 3000); meanwhile, "20M00" yields (1900, 2100); also, "19M69" yields (1869, 2069); likewise, "1.2M3" yields (1.13, 1.33). This word/function is useful for dates (example: "the mid-2000s" or "the period of two centuries which is centered on the year 1900"), ages (example: "they are in their mid-twenties", "they are older teens or young adults (fifteen to twenty-five years of age)", or "they are in their middle adult years (thirties or forties)"), or any estimate wherein the midpoint/expected value/average is known. The interval which is generated is a complete (math jargon) subset of the real numbers - there are no gaps and, in particular, the interval is not discrete (for example: it is not restricted to only the integers). Note that this word does not yield an interval of an arbitrary radius; use "mi'i" for that. Use a construct similar to "there exists a t in (the interval) re mu mi'i'au such that their age is measured to be (approximately) t in years" in order to express "they are in their mid-twenties (approximately 24 years old to 26 years old)"; the full English expression is wordy, but Lojban can make it concise in translation. See also: "mi'i", "bi'oi'au", "su'ai". (Footnote #1: this entire commentary section assumes that the method of interpretation is via a big-endian, traditional, unbalanced, positional, base-b numerical-representation system with b being an integer such that b > 1; however, the method of interpretation can be extended to other systems, such as p-adics or such as balanced integer or complex base-b systems, in natural and fairly self-evident ways, although no endeavor shall be made herein in order to do so and the assumptions about b and the method of interpretation should be as aforementioned, ignoring such possibilities for extension).

## In notes:

bi'oi'au (exp!)
digit/number: interval/range indicator for significant digits (determined by lesser endpoint).