Chapter 18: lojbau mekso: Mathematical Expressions in Lojban

9. Approximation and inexact numbers

The following cmavo are discussed in this section:

ji'i    PA  approximately
su'e    PA  at most
su'o    PA  at least
me'i    PA  less than
za'u    PA  more than

The cmavo “ji'i” (of selma'o PA) is used in several ways to indicate approximate or rounded numbers. If it appears at the beginning of a number, the whole number is approximate:

✥9.1  ji'i vo no
approximation four zero
approximately 40

If “ji'i” appears in the middle of a number, all the digits following it are approximate:

✥9.2  vo no ji'i mu no
four zero approximation five zero
roughly 4050
    (where the “four thousand” is exact,
    but the “fifty” is approximate)

If “ji'i” appears at the end of a number, it indicates that the number has been rounded. In addition, it can then be followed by a sign cmavo (“ma'u” or “ni'u”), which indicate truncation towards positive or negative infinity respectively.

✥9.3  re pi ze re ji'i
two point seven two approximation
2.72 (rounded)

✥9.4  re pi ze re ji'i ma'u
two point seven two approximation positive-sign
2.72 (rounded up)

✥9.5  re pi ze pa ji'i ni'u
two point seven one approximation negative-sign
2.71 (rounded down)

Examples 9.3 through 9.5 are all approximations to “te'o” (exponential e). “ji'i” can also appear by itself, in which case it means “approximately the typical value in this context”.

The four cmavo “su'e”, “su'o”, “me'i”, and “za'u”, also of selma'o PA, express inexact numbers with upper or lower bounds:

✥9.6  mi catlu su'e re prenu
I look-at at-most two persons

✥9.7  mi catlu su'o re prenu
I look-at at-least two persons

✥9.8  mi catlu me'i re prenu
I look-at less-than two persons

✥9.9  mi catlu za'u re prenu
I look-at more-than two persons

Each of these is a subtly different claim: ✥9.7 is true of two or any greater number, whereas ✥9.9 requires three persons or more. Likewise, ✥9.6 refers to zero, one, or two; ✥9.8 to zero or one. (Of course, when the context allows numbers other than non-negative integers, “me'i re” can be any number less than 2, and likewise with the other cases.) The exact quantifier, “exactly 2, neither more nor less” is just “re”. Note that “su'ore” is the exact Lojban equivalent of English plurals.

If no number follows one of these cmavo, “pa” is understood: therefore,

✥9.10    mi catlu su'o prenu
I look-at at-least [one] person

is a meaningful claim.

Like the numbers in c18-§8, all of these cmavo may be preceded by “pi” to make the corresponding quantifiers for part of a whole. For example, “pisu'o” means “at least some part of”. The quantifiers “ro”, “su'o”, “piro”, and “pisu'o” are particularly important in Lojban, as they are implicitly used in the descriptions introduced by the cmavo of selma'o LA and LE, as explained in Chapter 6. Descriptions in general are outside the scope of this chapter.