# Chapter 18. lojbau mekso: Mathematical Expressions in Lojban ## 18.1. Introductory

lojbau mekso (Lojbanic mathematical-expression) is the part of the Lojban language that is tailored for expressing statements of a mathematical character, or for adding numerical information to non-mathematical statements. Its formal design goals include:

1. representing all the different forms of expression used by mathematicians in their normal modes of writing, so that a reader can unambiguously read off mathematical text as written with minimal effort and expect a listener to understand it;

2. providing a vocabulary of commonly used mathematical terms which can readily be expanded to include newly coined words using the full resources of Lojban;

3. permitting the formulation, both in writing and in speech, of unambiguous mathematical text;

4. encompassing all forms of quantified expression found in natural languages, as well as encouraging greater precision in ordinary language situations than natural languages allow.

Goal 1 requires that mekso not be constrained to a single notation such as Polish notation or reverse Polish notation, but make provision for all forms, with the most commonly used forms the most easily used.

Goal 2 requires the provision of several conversion mechanisms, so that the boundary between mekso and full Lojban can be crossed from either side at many points.

Goal 3 is the most subtle. Written mathematical expression is culturally unambiguous, in the sense that mathematicians in all parts of the world understand the same written texts to have the same meanings. However, international mathematical notation does not prescribe unique forms. For example, the expression

Example 18.1.

3 x + 2 y

contains omitted multiplication operators, but there are other possible interpretations for the strings 3x and 2y than as mathematical multiplication. Therefore, the Lojban verbal (spoken and written) form of Example 18.1 must not omit the multiplication operators.

The remainder of this chapter explains (in as much detail as is currently possible) the mekso system. This chapter is by intention complete as regards mekso components, but only suggestive about uses of those components – as of now, there has been no really comprehensive use made of mekso facilities, and many matters must await the test of usage to be fully clarified.

## 18.2. Lojban numbers

The following cmavo are discussed in this section:

 pa PA 1 re PA 2 ci PA 3 vo PA 4 mu PA 5
 xa PA 6 ze PA 7 bi PA 8 so PA 9 no PA 0

The simplest kind of mekso are numbers, which are cmavo or compound cmavo. There are cmavo for each of the 10 decimal digits, and numbers greater than 9 are made by stringing together the cmavo. Some examples:

Example 18.2.

 pa re ci one two three 123
 one hundred and twenty three

Example 18.3.

 pa no one zero 10
 ten

Example 18.4.

 pa re ci vo mu xa ze bi so no one two three four five six seven eight nine zero 1234567890
 one billion, two hundred and thirty-four million, five hundred and sixty-seven thousand, eight hundred and ninety.

Therefore, there are no separate cmavo for ten, hundred, etc.

There is a pattern to the digit cmavo (except for no, 0) which is worth explaining. The cmavo from 1 to 5 end in the vowels a, e, i, o, u respectively; and the cmavo from 6 to 9 likewise end in the vowels a, e, i, and o respectively. None of the digit cmavo begin with the same consonant, to make them easy to tell apart in noisy environments.

## 18.3. Signs and numerical punctuation

The following cmavo are discussed in this section:

 ma'u PA positive sign ni'u PA negative sign pi PA decimal point fi'u PA fraction slash ra'e PA repeating decimal ce'i PA percent sign ki'o PA comma between digits

A number can be given an explicit sign by the use of ma'u and ni'u, which are the positive and negative signs as distinct from the addition, subtraction, and negation operators. For example:

Example 18.5.

 ni'u pa negative-sign 1 -1

Grammatically, the signs are part of the number to which they are attached. It is also possible to use ma'u and ni'u by themselves as numbers; the meaning of these numbers is explained in Section 18.8.

Various numerical punctuation marks are likewise expressed by cmavo, as illustrated in the following examples:

Example 18.6.

 ci pi pa vo pa mu three point one four one five 3.1415

(In some cultures, a comma is used instead of a period in the symbolic version of Example 18.6; pi is still the Lojban representation for the decimal point.)

Example 18.7.

 re fi'u ze two fraction seven 2 7

Example 18.7 is the name of the number two-sevenths; it is not the same as the result of 2 divided by 7 in Lojban, although numerically these two are equal. If the denominator of the fraction is present but the numerator is not, the numerator is taken to be 1, thus expressing the reciprocal of the following number:

Example 18.8.

 fi'u ze fraction seven 1 7

Example 18.9.

 pi ci mu ra'e pa vo re bi mu ze point three five repeating one four two eight five seven .35142857142857...

Note that the ra'e marks unambiguously where the repeating portion 142857 begins.

Example 18.10.

 ci mu ce'i three five percent 35%

Example 18.11.

 pa ki'o re ci vo ki'o mu xa ze one comma two three four comma five six seven 1,234,567

(In some cultures, spaces are used in the symbolic representation of Example 18.11; ki'o is still the Lojban representation.)

It is also possible to have less than three digits between successive ki'o s, in which case zeros are assumed to have been elided:

Example 18.12.

 pa ki'o re ci ki'o vo one comma two three comma four 1,023,004

In the same way, ki'o can be used after pi to divide fractions into groups of three:

Example 18.13.

 pi ki'o re re point comma two two .022

Example 18.14.

 pi pa ki'o pa re ki'o pa point one comma one two comma one .001012001

## 18.4. Special numbers

The following cmavo are discussed in this section:

 ci'i PA infinity ka'o PA imaginary i, sqrt(-1) pai PA π, pi (approx 3.14159...) te'o PA exponential e (approx 2.71828...) fi'u PA golden ratio, Φ, phi, (1 + sqrt(5))/2 (approx. 1.61803...)

The last cmavo is the same as the fraction sign cmavo: a fraction sign with neither numerator nor denominator represents the golden ratio.

Numbers can have any of these digit, punctuation, and special-number cmavo of Sections 2, 3, and 4 in any combination:

Example 18.15.

 ma'u ci'i +∞

Example 18.16.

 ci ka'o re
 3i2 (a complex number equivalent to 3 + 2i)

Note that ka'o is both a special number (meaning i) and a number punctuation mark (separating the real and the imaginary parts of a complex number).

Example 18.17.

 ci'i no
 infinity zero ℵ0 (a transfinite cardinal)

The special numbers pai and te'o are mathematically important, which is why they are given their own cmavo:

Example 18.18.

 pai
 pi, π

Example 18.19.

 te'o
 e

However, many combinations are as yet undefined:

Example 18.20.

 pa pi re pi ci
 1.2.3

Example 18.21.

 pa ni'u re 1 negative-sign 2

Example 18.21 is not 1 minus 2, which is represented by a different cmavo sequence altogether. It is a single number which has not been assigned a meaning. There are many such numbers which have no well-defined meaning; they may be used for experimental purposes or for future expansion of the Lojban number system.

It is possible, of course, that some of these oddities do have a meaningful use in some restricted area of mathematics. A mathematician appropriating these structures for specialized use needs to consider whether some other branch of mathematics would use the structure differently.

More information on numbers may be found in Section 18.8 to Section 18.12.

## 18.5. Simple infix expressions and equations

The following cmavo are discussed in this section:

 du GOhA equals su'i VUhU plus vu'u VUhU minus pi'i VUhU times te'a VUhU raised to the power ny. BY letter “n” vei VEI left parenthesis ve'o VEhO right parenthesis

Let us begin at the beginning: one plus one equals two. In Lojban, that sentence translates to:

Example 18.22.

 li pa su'i pa du li re The-number one plus one equals the-number two. 1 + 1 = 2

Example 18.22, a mekso sentence, is a regular Lojban bridi that exploits mekso features. du is the predicate meaning x1 is mathematically equal to x2. It is a cmavo for conciseness, but it has the same grammatical uses as any brivla. Outside mathematical contexts, du means x1 is identical with x2 or x1 is the same object as x2.

The cmavo li is the number article. It is required whenever a sentence talks about numbers as numbers, as opposed to using numbers to quantify things. For example:

Example 18.23.

 le ci prenu
 the three persons

requires no li article, because the ci is being used to specify the number of prenu. However, the sentence

Example 18.24.

 levi sfani cu grake li ci This fly masses-in-grams the-number three.
 This fly has a mass of 3 grams.

requires li because ci is being used as a sumti. Note that this is the way in which measurements are stated in Lojban: all the predicates for units of length, mass, temperature, and so on have the measured object as the first place and a number as the second place. Using li for le in Example 18.23 would produce

Example 18.25.

 li ci prenu The-number 3 is-a-person.

which is grammatical but nonsensical: numbers are not persons.

The cmavo su'i belongs to selma'o VUhU, which is composed of mathematical operators, and means addition. As mentioned before, it is distinct from ma'u which means the positive sign as an indication of a positive number:

Example 18.26.

 li ma'u pa su'i The-number positive-sign one plus
 ni'u pa du li no negative-sign one equals the-number zero. +1 + -1 = 0

Of course, it is legal to have complex mekso on both sides of du:

Example 18.27.

 li mu su'i pa du li ci su'i ci The-number five plus one equals the-number three plus three. 5 + 1 = 3 + 3

Why don't we say li mu su'i li pa rather than just li mu su'i pa? The answer is that VUhU operators connect mekso operands (numbers, in Example 18.27), not general sumti. li is used to make the entire mekso into a sumti, which then plays the roles applicable to other sumti: in Example 18.27, filling the places of a bridi

By default, Lojban mathematics is like simple calculator mathematics: there is no notion of operator precedence. Consider the following example, where pi'i means times, the multiplication operator:

Example 18.28.

 li ci su'i vo pi'i mu du li reci The-number three plus four times five equals the-number two-three. 3 + 4 × 5 = 23

Is the Lojban version of Example 18.28 true? No! 3 + 4 × 5 is indeed 23, because the usual conventions of mathematics state that multiplication takes precedence over addition; that is, the multiplication 4 × 5 is done first, giving 20, and only then the addition 3 + 20. But VUhU operators by default are done left to right, like other Lojban grouping, and so a truthful bridi would be:

Example 18.29.

 li ci su'i vo pi'i mu du li cimu The-number three plus four times five equals the-number three-five. 3 + 4 × 5 = 35

Here we calculate 3 + 4 first, giving 7, and then calculate 7 × 5 second, leading to the result 35. While possessing the advantage of simplicity, this result violates the design goal of matching the standards of mathematics. What can be done?

There are three solutions, all of which will probably be used to some degree. The first solution is to ignore the problem. People will say li ci su'i vo pi'i mu and mean 23 by it, because the notion that multiplication takes precedence over addition is too deeply ingrained to be eradicated by Lojban parsing, which totally ignores semantics. This convention essentially allows semantics to dominate syntax in this one area.

(Why not hard-wire the precedences into the grammar, as is done in computer programming languages? Essentially because there are too many operators, known and unknown, with levels of precedence that vary according to usage. The programming language 'C' has 13 levels of precedence, and its list of operators is not even extensible. For Lojban this approach is just not practical. In addition, hard-wired precedence could not be overridden in mathematical systems such as spreadsheets where the conventions are different.)

The second solution is to use explicit means to specify the precedence of operators. This approach is fully general, but clumsy, and will be explained in Section 18.20.

The third solution is simple but not very general. When an operator is prefixed with the cmavo bi'e (of selma'o BIhE), it becomes automatically of higher precedence than other operators not so prefixed. Thus,

Example 18.30.

 li ci su'i vo bi'e pi'i mu du li reci The-number three plus four times five equals the-number two-three. 3 + 4 × 5 = 23

is a truthful Lojban bridi. If more than one operator has a bi'e prefix, grouping is from the right; multiple bi'e prefixes on a single operator are not allowed.

In addition, of course, Lojban has the mathematical parentheses vei and ve'o, which can be used just like their written equivalents ( and ) to group expressions in any way desired:

Example 18.31.

 li vei ny. su'i pa ve'o pi'i vei ny. su'i pa [ve'o] The-number ( n plus one ) times ( n plus one )
 du li ny. [bi'e] te'a re equals the-number n to-the-power two
 su'i re bi'e pi'i ny. su'i pa plus two times n plus 1. (n+1)(n+1) = n2 + 2n + 1

There are several new usages in Example 18.31: te'a means raised to the power, and we also see the use of the lerfu word ny, representing the letter n. In mekso, letters stand for just what they do in ordinary mathematics: variables. The parser will accept a string of lerfu words (called a lerfu string) as the equivalent of a single lerfu word, in agreement with computer-science conventions; abc is a single variable, not the equivalent of a × b × c. (Of course, a local convention could state that the value of a variable like abc, with a multi-lerfu name, was equal to the values of the variables a, b, and c multiplied together.)

The explicit operator pi'i is required in the Lojban verbal form whereas multiplication is implicit in the symbolic form. Note that ve'o (the right parenthesis) is an elidable terminator: the first use of it in Example 18.31 is required, but the second use (marked by square brackets) could be elided. Additionally, the first bi'e (also marked by square brackets) is not necessary to get the proper grouping, but it is included here for symmetry with the other one.

## 18.6. Forethought operators (Polish notation, functions)

The following cmavo are discussed in this section:

 boi BOI numeral/lerfu string terminator va'a VUhU negation/additive inverse pe'o PEhO forethought flag ku'e KUhE forethought terminator ma'o MAhO convert operand to operator py. BY letter “p” xy. BY letter “x” zy. BY letter “z” fy. BY letter “f”

The infix form explained so far is reasonable for many purposes, but it is limited and rigid. It works smoothly only where all operators have exactly two operands, and where precedences can either be assumed from context or are limited to just two levels, with some help from parentheses.

But there are many operators which do not have two operands, or which have a variable number of operands. The preferred form of expression in such cases is the use of forethought operators, also known as Polish notation. In this style of writing mathematics, the operator comes first and the operands afterwards:

Example 18.32.

 li su'i paboi reboi ci[boi] du li xa The-number the-sum-of one two three equals the-number six. sum(1,2,3) = 6

Note that the normally elidable number terminator boi is required after pa and re because otherwise the reading would be pareci= 123. It is not required after ci but is inserted here in brackets for the sake of symmetry. The only time boi is required is, as in Example 18.32, when there are two consecutive numbers or lerfu strings.

Forethought mekso can use any number of operands, in Example 18.32, three. How do we know how many operands there are in ambiguous circumstances? The usual Lojban solution is employed: an elidable terminator, namely ku'e. Here is an example:

Example 18.33.

 li py. su'i va'a ny. ku'e su'i zy du The-number “p” plus negative-of( “n” ) plus “z” equals
 li xy. the-number “x” . p + -n + z = x

where we know that va'a is a forethought operator because there is no operand preceding it.

va'a is the numerical negation operator, of selma'o VUhU. In contrast, vu'u is not used for numerical negation, but only for subtraction, as it always has two or more operands. Do not confuse va'a and vu'u, which are operators, with ni'u, which is part of a number.

In Example 18.33, the operator va'a and the terminator ku'e serve in effect as parentheses. (The regular parentheses vei and ve'o are NOT used for this purpose.) If the ku'e were omitted, the su'i zy would be swallowed up by the va'a forethought operator, which would then appear to have two operands, ny and su'i zy., where the latter is also a forethought expression.

Forethought mekso is also useful for matching standard functional notation. How do we represent z = f(x)? The answer is:

Example 18.34.

 li zy du li ma'o fy.boi xy. The-number z equals the-number the-operator f x. z = f(x)

Again, no parentheses are used. The construct ma'o fy.boi is the equivalent of an operator, and appears in forethought here (although it could also be used as a regular infix operator). In mathematics, letters sometimes mean functions and sometimes mean variables, with only the context to tell which. Lojban chooses to accept the variable interpretation as the default, and uses the special flag ma'o to mark a lerfu string as an operator. The cmavo xy. and zy. are variables, but fy. is an operator (a function) because ma'o marks it as such. The boi is required because otherwise the xy. would look like part of the operator name. (The use of ma'o can be generalized from lerfu strings to any mekso operand: see Section 18.21.)

When using forethought mekso, the optional marker pe'o may be placed in front of the operator. This usage can help avoid confusion by providing clearly marked pe'o and ku'e pairs to delimit the operand list. Example 18.32 to Example 18.34, respectively, with explicit pe'o and ku'e:

Example 18.35.

li pe'o su'i paboi reboi ciboi ku'e du li xa

Example 18.36.

li py. su'i pe'o va'a ny. ku'e su'i zy du li xy.

Example 18.37.

li zy du li pe'o ma'o fy.boi xy. ku'e

Note: When using forethought mekso, be sure that the operands really are operands: they cannot contain regular infix expressions unless parenthesized with vei and ve'o. An earlier version of the complex Example 18.119 came to grief because I forgot this rule.

## 18.7. Other useful selbri for mekso bridi

So far our examples have been isolated mekso (it is legal to have a bare mekso as a sentence in Lojban) and equation bridi involving du. What about inequalities such as x < 5? The answer is to use a bridi with an appropriate selbri, thus:

Example 18.38.

 li xy. mleca li mu The-number x is-less-than the-number 5.

Here is a partial list of selbri useful in mathematical bridi:

 du x1 is identical to x2, x3, x4, ... dunli x1 is equal/congruent to x2 in/on property/quality/dimension/quantity x3 mleca x1 is less than x2 zmadu x1 is greater than x2 dubjavme'a x1 is less than or equal to x2 [du ja mleca, equal or less] dubjavmau x1 is greater than or equal to x2 [du ja zmadu, equal or greater] tamdu'i x1 is similar to x2 [tarmi dunli, shape-equal] turdu'i x1 is isomorphic to x2 [stura dunli, structure-equal] cmima x1 is a member of set x2 gripau x1 is a subset of set x2 [girzu pagbu, set-part] na'ujbi x1 is approximately equal to x2 [namcu jibni, number-near] terci'e x1 is a component with function x2 of system x3

Note the difference between dunli and du; dunli has a third place that specifies the kind of equality that is meant. du refers to actual identity, and can have any number of places:

Example 18.39.

 py. du xy.boi zy. “p” is-identical-to “x” “z” p = x = z

Lojban bridi can have only one predicate, so the du is not repeated.

Any of these selbri may usefully be prefixed with na, the contradictory negation cmavo, to indicate that the relation is false:

Example 18.40.

 li re su'i re na du li mu the-number 2 + 2 is-not equal-to the-number 5. 2 + 2 ≠ 5

As usual in Lojban, negated bridi say what is false, and do not say anything about what might be true.

## 18.8. Indefinite numbers

The following cmavo are discussed in this section:

 ro PA all so'a PA almost all so'e PA most so'i PA many so'o PA several so'u PA a few no'o PA the typical number of da'a PA all but (one) of piro PA+PA the whole of/all of piso'a PA+PA almost the whole of piso'e PA+PA most of piso'i PA+PA much of piso'o PA+PA a small part of piso'u PA+PA a tiny part of pino'o PA+PA the typical portion of rau PA enough du'e PA too many mo'a PA too few pirau PA+PA enough of pidu'e PA+PA too much of pimo'a PA+PA too little of

Not all the cmavo of PA represent numbers in the usual mathematical sense. For example, the cmavo ro means all or each. This number does not have a definite value in the abstract: li ro is undefined. But when used to count or quantify something, the parallel between ro and pa is clearer:

Example 18.41.

 mi catlu pa prenu I look-at one person

Example 18.42.

 mi catlu ro prenu I look-at all persons

Example 18.41 might be true, whereas Example 18.42 is almost certainly false.

The cmavo so'a, so'e, so'i, so'o, and so'u represent a set of indefinite numbers less than ro. As you go down an alphabetical list, the magnitude decreases:

Example 18.43.

 mi catlu so'a prenu I look-at almost-all persons

Example 18.44.

 mi catlu so'e prenu I look-at most persons

Example 18.45.

 mi catlu so'i prenu I look-at many persons

Example 18.46.

 mi catlu so'o prenu I look-at several persons

Example 18.47.

 mi catlu so'u prenu I look-at a-few persons

The English equivalents are only rough: the cmavo provide space for up to five indefinite numbers between ro and no, with a built-in ordering. In particular, so'e does not mean most in the sense of a majority or more than half.

Each of these numbers, plus ro, may be prefixed with pi (the decimal point) in order to make a fractional form which represents part of a whole rather than some elements of a totality. piro therefore means the whole of:

Example 18.48.

 mi citka piro lei nanba I eat the-whole-of the-mass-of bread

Similarly, piso'a means almost the whole of; and so on down to piso'u, a tiny part of. These numbers are particularly appropriate with masses, which are usually measured rather than counted, as Example 18.48 shows.

In addition to these cmavo, there is no'o, meaning the typical value, and pino'o, meaning the typical portion: Sometimes no'o can be translated the average value, but the average in question is not, in general, a mathematical mean, median, or mode; these would be more appropriately represented by operators.

Example 18.49.

 mi catlu no'o prenu I look-at a-typical-number-of persons

Example 18.50.

 mi citka pino'o lei nanba I eat a-typical-amount-of the-mass-of bread.

da'a is a related cmavo meaning all but:

Example 18.51.

 mi catlu da'a re prenu I look-at all-but two persons

Example 18.52.

 mi catlu da'a so'u prenu I look-at all-but a-few persons

Example 18.52 is similar in meaning to Example 18.43.

If no number follows da'a, then pa is assumed; da'a by itself means all but one, or in ordinal contexts all but the last:

Example 18.53.

 ro ratcu ka'e citka da'a ratcu All rats can eat all-but-one rats.
 All rats can eat all other rats.

(The use of da'a means that Example 18.53 does not require that all rats can eat themselves, but does allow it. Each rat has one rat it cannot eat, but that one might be some rat other than itself. Context often dictates that itself is, indeed, the other rat.)

As mentioned in Section 18.3, ma'u and ni'u are also legal numbers, and they mean some positive number and some negative number respectively.

Example 18.54.

 li ci vu'u re du li ma'u the-number 3 − 2 = some-positive-number

Example 18.55.

 li ci vu'u vo du li ni'u the-number 3 − 4 = some-negative-number

Example 18.56.

 mi ponse ma'u rupnu I possess a-positive-number-of currency-units.

All of the numbers discussed so far are objective, even if indefinite. If there are exactly six superpowers (rairgugde, superlative-states) in the world, then ro rairgugde means the same as xa rairgugde. It is often useful, however, to express subjective indefinite values. The cmavo rau (enough), du'e (too many), and mo'a (too few) are then appropriate:

Example 18.57.

 mi ponse rau rupnu I possess enough currency-units.

Like the so'a-series, rau, du'e, and mo'a can be preceded by pi; for example, pirau means a sufficient part of.

Another possibility is that of combining definite and indefinite numbers into a single number. This usage implies that the two kinds of numbers have the same value in the given context:

Example 18.58.

 mi viska le rore gerku I saw the all-of/two dogs.
 I saw both dogs.

Example 18.59.

 mi speni so'ici prenu I am-married-to many/three persons.
 I am married to three persons (which is “many” in the circumstances).

Example 18.59 assumes a mostly monogamous culture by stating that three is many.

## 18.9. 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:

Example 18.60.

 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:

Example 18.61.

 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.

Example 18.62.

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

Example 18.63.

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

Example 18.64.

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

Example 18.62 through Example 18.64 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:

Example 18.65.

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

Example 18.66.

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

Example 18.67.

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

Example 18.68.

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

Each of these is a subtly different claim: Example 18.66 is true of two or any greater number, whereas Example 18.68 requires three persons or more. Likewise, Example 18.65 refers to zero, one, or two; Example 18.67 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,

Example 18.69.

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

is a meaningful claim.

Like the numbers in Section 18.8, 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 Section 6.7. Descriptions in general are outside the scope of this chapter.

## 18.10. Non-decimal and compound bases

The following cmavo are discussed in this section:

 ju'u VUhU to the base dau PA hex digit A = 10 fei PA hex digit B = 11 gai PA hex digit C = 12 jau PA hex digit D = 13 rei PA hex digit E = 14 vai PA hex digit F = 15 pi'e PA compound base point

In normal contexts, Lojban assumes that all numbers are expressed in the decimal (base 10) system. However, other bases are possible, and may be appropriate in particular circumstances.

To specify a number in a particular base, the VUhU operator ju'u is suitable:

Example 18.70.

 li panopano ju'u re du li pano The-number 1010 base 2 equals the-number 1 0.

Here, the final pa no is assumed to be base 10, as usual; so is the base specification. (The base may also be changed permanently by a metalinguistic specification; no standard way of doing so has as yet been worked out.)

Lojban has digits for representing bases up to 16, because 16 is a base often used in computer applications. In English, it is customary to use the letters A-F as the base 16 digits equivalent to the numbers ten through fifteen. In Lojban, this ambiguity is avoided:

Example 18.71.

 li daufeigai ju'u paxa du li rezevobi The-number ABC base 16 equals the-number 2748.

Example 18.72.

 li jaureivai ju'u paxa du li cimuxaze The-number DEF base 16 equals the-number 3567.

Note the pattern in the cmavo: the diphthongs au, ei, ai are used twice in the same order. The digits for A to D use consonants different from those used in the decimal digit cmavo; E and F unfortunately overlap 2 and 4 – there was simply not enough available cmavo space to make a full differentiation possible. The cmavo are also in alphabetical order.

The base point pi is used in non-decimal bases just as in base 10:

Example 18.73.

 li vai pi bi ju'u paxa du li pamu pi mu The-number F . 8 base 16 equals the-number 15 . 5.

Since ju'u is an operator of selma'o VUhU, it is grammatical to use any operand as the left argument. Semantically, however, it is undefined to use anything but a numeral string on the left. The reason for making ju'u an operator is to allow reference to a base which is not a constant.

There are some numerical values that require a base that varies from digit to digit. For example, times represented in hours, minutes, and seconds have, in effect, three digits: the first is base 24, the second and third are base 60. To express such numbers, the compound base separator pi'e is used:

Example 18.74.

 ci pi'e rere pi'e vono 3:22:40

Each digit sequence separated by instances of pi'e is expressed in decimal notation, but the number as a whole is not decimal and can only be added and subtracted by special rules:

Example 18.75.

 li ci pi'e rere pi'e vono su'i pi'e ci pi'e cici The-number 3 : 22 : 40 plus : 3 : 33
 du li ci pi'e rexa pi'e paci equals the-number 3 : 26 : 13. 3:22:40 + 0:3:33 = 3:26:13

Of course, only context tells you that the first part of the numbers in Example 18.74 and Example 18.75 is hours, the second minutes, and the third seconds.

The same mechanism using pi'e can be used to express numbers which have a base larger than 16. For example, base-20 Mayan mathematics might use digits from no to paso, each separated by pi'e:

Example 18.76.

 li pa pi'e re pi'e ci ju'u reno du li vovoci the-number 1 ; 2 ; 3 base 20 equals the-number 443

Carefully note the difference between:

Example 18.77.

 pano ju'u reno the-digit-10 base 20

which is equal to ten, and:

Example 18.78.

 pa pi'e no ju'u reno 1;0 base 20

which is equal to twenty.

Both pi and pi'e can be used to express large-base fractions:

Example 18.79.

 li pa pi'e vo pi ze ju'u reno The-number 1 ; 4 . 7 base 20
 du li revo pi cimu equals the-number 24 . 35

pi'e is also used where the base of each digit is vague, as in the numbering of the examples in this chapter:

Example 18.80.

 dei jufra panopi'epapamoi This-utterance is-a-sentence-type-of 10;11th-thing.
 This is Sentence 10.11.

## 18.11. Special mekso selbri

The following cmavo are discussed in this section:

 mei MOI cardinal selbri moi MOI ordinal selbri si'e MOI portion selbri cu'o MOI probability selbri va'e MOI scale selbri me ME make sumti into selbri me'u MEhU terminator for ME

Lojban possesses a special category of selbri which are based on mekso. The simplest kind of such selbri are made by suffixing a member of selma'o MOI to a number. There are five members of MOI, each of which serves to create number-based selbri with specific place structures.

The cmavo mei creates cardinal selbri. The basic place structure is:

x1 is a mass formed from the set x2 of n members, one or more of which is/are x3

A cardinal selbri interrelates a set with a given number of members, the mass formed from that set, and the individuals which make the set up. The mass argument is placed first as a matter of convenience, not logical necessity.

Some examples:

Example 18.81.

 lei mi ratcu cu cimei Those-I-describe-as-the-mass-of my rats are-a-threesome.
 My rats are three. I have three rats.

Here, the mass of my rats is said to have three components; that is, I have three rats.

Another example, with one element this time:

Example 18.82.

 mi poi pamei cu cusku dei I who am-an-individual express this-sentence.

In Example 18.82, mi refers to a mass, the mass consisting of me. Personal pronouns are vague between masses, sets, and individuals.

However, when the number expressed before -mei is an objective indefinite number of the kind explained in Section 18.8, a slightly different place structure is required:

x1 is a mass formed from a set x2 of n members, one or more of which is/are x3, measured relative to the set x4.

An example:

Example 18.83.

 lei ratcu poi zvati le panka The-mass-of rats which are-in the park
 cu so'umei lo'i ratcu are-a-fewsome-with-respect-to the-set-of rats.
 The rats in the park are a small number of all the rats there are.

In Example 18.83, the x2 and x3 places are vacant, and the x4 place is filled by lo'i ratcu, which (because no quantifiers are explicitly given) means the whole of the set of all those things which are rats, or simply the set of all rats.

Example 18.84.

 le'i ratcu poi zvati le panka cu se so'imei The-set-of rats which-are in the park is-a manysome.
 There are many rats in the park.

In Example 18.84, the conversion cmavo se swaps the x1 and the x2 places, so that the new x1 is the set. The x4 set is unspecified, so the implication is that the rats are many with respect to some unspecified comparison set.

More explanations about the interrelationship of sets, masses, and individuals can be found in Section 6.3.

The cmavo moi creates ordinal selbri. The place structure is:

x1 is the (n)th member of set x2 when ordered by rule x3

Some examples:

Example 18.85.

 ti pamoi le'i mi ratcu This-one is-the-first-of the associated-with-me rats.
 This is my first rat.

Example 18.86.

 ta romoi le'i mi ratcu That is-the-allth-of the associated-with-me rats.
 That is my last rat.

Example 18.87.

 mi raumoi le velskina porsi I am-enough-th-in the movie-audience sequence
 I am enough-th in the movie line.

Example 18.87 means, in the appropriate context, that my position in line is sufficiently far to the front that I will get a seat for the movie.

The cmavo si'e creates portion selbri. The place structure is:

x1 is an (n)th portion of mass x2

Some examples:

Example 18.88.

 levi sanmi cu fi'ucisi'e lei mi djedi cidja This-here meal is-a-slash-three-portion-of my day food.
 This meal is one-third of my daily food.

The cmavo cu'o creates probability selbri. The place structure is:

event x1 has probability (n) of occurring under conditions x2

The number must be between 0 and 1 inclusive. For example:

Example 18.89.

 le nu lo sicni cu sedja'o cu pimucu'o The event of-a coin being-a-head-displayer has-probability-.5.

The cmavo va'e creates a scale selbri. The place structure is:

x1 is at scale position (n) on the scale x2

If the scale is granular rather than continuous, a form like cifi'uxa (3/6) may be used; in this case, 3/6 is not the same as 1/2, because the third position on a scale of six positions is not the same as the first position on a scale of two positions. Here is an example:

Example 18.90.

 levi rozgu cu sofi'upanova'e xunre This-here rose is-8/10-scale red.
 This rose is 8 out of 10 on the scale of redness. This rose is very red.

When the quantifier preceding any MOI cmavo includes the subjective numbers rau, du'e, or mo'a (enough, too many, too few) then an additional place is added for by standard. For example:

Example 18.91.

 lei ratcu poi zvati le The-mass-of rats which-are in the
 panka cu du'emei fo mi park are-too-many by-standard me.
 There are too many rats in the park for me.

The extra place (which for -mei is the x4 place labeled by fo) is provided rather than using a BAI tag such as ma'i because a specification of the standard for judgment is essential to the meaning of subjective words like enough.

This place is not normally explicit when using one of the subjective numbers directly as a number. Therefore, du'e ratcu means too many rats without specifying any standard.

It is also grammatical to substitute a lerfu string for a number:

Example 18.92.

 ta ny.moi le'i mi ratcu That is-nth-of the-set-of associated-with-me rats.
 That is my nth rat.

More complex mekso cannot be placed directly in front of MOI, due to the resulting grammatical ambiguities. Instead, a somewhat artificial form of expression is required.

The cmavo me (of selma'o ME) has the function of making a sumti into a selbri. A whole me construction can have a member of MOI added to the end to create a complex mekso selbri:

Example 18.93.

 ta me li ny. su'i pa me'u moi That is the-number n plus one -th-of
 le'i mi ratcu the-set-of associated-with-me rats.
 That is my (n+1)-th rat.

Here the mekso ny. su'i pa is made into a sumti (with li) and then changed into a mekso selbri with me and me'u moi. The elidable terminator me'u is required here in order to keep the pa and the moi separate; otherwise, the parser will combine them into the compound pamoi and reject the sentence as ungrammatical.

It is perfectly possible to use non-numerical sumti after me and before a member of MOI, producing strange results indeed:

Example 18.94.

 le nu mi nolraitru cu me The event-of me being-a-nobly-superlative-ruler
 le'e snime bolci be vi la xel. cu'o has-the-stereotypical snow type-of-ball at Hell probability.
 I have a snowball's chance in Hell of being king.

Note: the elidable terminator boi is not used between a number and a member of MOI. As a result, the me'u in Example 18.93 could also be replaced by a boi, which would serve the same function of preventing the pa and moi from joining into a compound.

## 18.12. Number questions

The following cmavo is discussed in this section:

 xo PA number question

The cmavo xo, a member of selma'o PA, is used to ask questions whose answers are numbers. Like most Lojban question words, it fills the blank where the answer should go. (See Section 19.5 for more on Lojban questions.)

Example 18.95.

 li re su'i re du li xo The-number 2 plus 2 equals the-number what?
 What is 2 + 2?

Example 18.96.

 le xomoi prenu cu darxi do The what-number-th person hit you?
 Which person [as in a police lineup] hit you?

xo can also be combined with other digits to ask questions whose answers are already partly specified. This ability could be very useful in writing tests of elementary arithmetical knowledge:

Example 18.97.

 li remu pi'i xa du li paxono The-number 25 times 6 equals the-number 1?0

to which the correct reply would be mu, or 5. The ability to utter bare numbers as grammatical Lojban sentences is primarily intended for giving answers to xo questions. (Another use, obviously, is for counting off physical objects one by one.)

## 18.13. Subscripts

The following cmavo is discussed in this section:

 xi XI subscript

Subscripting is a general Lojban feature, not used only in mekso; there are many things that can logically be subscripted, and grammatically a subscript is a free modifier, usable almost anywhere. In particular, of course, mekso variables (lerfu strings) can be subscripted:

Example 18.98.

 li xy.boixici du li xy.boixipa su'i xy.boixire The-number x-sub-3 equals the-number x-sub-1 plus x-sub-2. x3 = x1 + x2

Subscripts always begin with the flag xi (of selma'o XI). xi may be followed by a number, a lerfu string, or a general mekso expression in parentheses:

Example 18.99.

 xy.boixino x0

Example 18.100.

 xy.boixiny. xn

Example 18.101.

 xy.boixi vei ny. su'i pa [ve'o] x(n+1)

Note that subscripts attached directly to lerfu words (variables) generally need a boi terminating the variable. Free modifiers, of which subscripts are one variety, generally require the explicit presence of an otherwise elidable terminator.

There is no standard way of handling superscripts (other than those used as exponents) or for subscripts or superscripts that come before the main expression. If necessary, further cmavo could be assigned to selma'o XI for these purposes.

The elidable terminator for a subscript is that for a general number or lerfu string, namely boi. By convention, a subscript following another subscript is taken to be a sub-subscript:

Example 18.102.

 xy.boi xi by.boi xi vo xb4

See Example 18.123 for the standard method of specifying multiple subscripts on a single object.

More information on the uses of subscripts may be found in Section 19.6.

## 18.14. Infix operators revisited

The following cmavo are discussed in this section:

 tu'o PA null operand ge'a VUhU null operator gei VUhU exponential notation

The infix operators presented so far have always had exactly two operands, and for more or fewer operands forethought notation has been required. However, it is possible to use an operator in infix style even though it has more or fewer than two operands, through the use of a pair of tricks: the null operand tu'o and the null operator ge'a. The first is suitable when there are too few operands, the second when there are too many. For example, suppose we wanted to express the numerical negation operator va'a in infix form. We would use:

Example 18.103.

 li tu'o va'a ny. du li no vu'u ny. The-number (null) additive-inverse n equals the-number zero minus n. -n = 0 − n

The tu'o fulfills the grammatical requirement for a left operand for the infix use of va'a, even though semantically none is needed or wanted.

Finding a suitable example of ge'a requires exhibiting a ternary operator, and ternary operators are not common. The operator gei, however, has both a binary and a ternary use. As a binary operator, it provides a terse representation of scientific (also called exponential) notation. The first operand of gei is the exponent, and the second operand is the mantissa or fraction:

Example 18.104.

 li cinonoki'oki'o du The-number three-zero-zero-comma-comma equals
 li bi gei ci the-number eight scientific three. 300,000,000 = 3 × 108

Why are the arguments to gei in reverse order from the conventional symbolic notation? So that gei can be used in forethought to allow easy specification of a large (or small) imprecise number:

Example 18.105.

 gei reno (scientific) two-zero 10 20

Note, however, that although 10 is far and away the most common exponent base, it is not the only possible one. The third operand of gei, therefore, is the base, with 10 as the default value. Most computers internally store so-called floating-point numbers using 2 as the exponent base. (This has nothing to do with the fact that computers also represent all integers in base 2; the IBM 360 series used an exponent base of 16 for floating point, although each component of the number was expressed in base 2.) Here is a computer floating-point number with a value of 40:

Example 18.106.

 papano bi'eju'u re gei (one-one-zero base 2) scientific
 pipanopano bi'eju'u re ge'a re (point-one-zero-one-zero base 2) with-base 2 .10102 x 21102

## 18.15. Vectors and matrices

The following cmavo are discussed in this section:

 jo'i JOhI start vector te'u TEhU end vector pi'a VUhU matrix row combiner sa'i VUhU matrix column combiner

A mathematical vector is a list of numbers, and a mathematical matrix is a table of numbers. Lojban considers matrices to be built up out of vectors, which are in turn built up out of operands.

jo'i, the only cmavo of selma'o JOhI, is the vector indicator: it has a syntax reminiscent of a forethought operator, but has very high precedence. The components must be simple operands rather than full expressions (unless parenthesized). A vector can have any number of components; te'u is the elidable terminator. An example:

Example 18.107.

 li jo'i paboi reboi te'u su'i jo'i ciboi voboi The-number array (one, two ) plus array (three, four)
 du li jo'i voboi xaboi equals the-number array (four, six). (1,2) + (3,4) = (4,6)

Vectors can be combined into matrices using either pi'a, the matrix row operator, or sa'i, the matrix column operator. The first combines vectors representing rows of the matrix, and the second combines vectors representing columns of the matrix. Both of them allow any number of arguments: additional arguments are tacked on with the null operator ge'a.

Therefore, the magic square matrix

 8 1 6 3 5 7 4 9 2

can be represented either as:

Example 18.108.

 jo'i biboi paboi xa pi'a jo'i ciboi muboi ze the-vector (8 1 6) matrix-row the-vector (3 5 7),
 ge'a jo'i voboi soboi re the-vector (4 9 2)

or as

Example 18.109.

 jo'i biboi ciboi vo sa'i jo'i paboi muboi so the-vector (8 3 4) matrix-column the-vector (1 5 9),
 ge'a jo'i xaboi zeboi re the-vector (6 7 2)

The regular mekso operators can be applied to vectors and to matrices, since grammatically both of these are expressions. It is usually necessary to parenthesize matrices when used with operators in order to avoid incorrect groupings. There are no VUhU operators for the matrix operators of inner or outer products, but appropriate operators can be created using a suitable symbolic lerfu word or string prefixed by ma'o.

Matrices of more than two dimensions can be built up using either pi'a or sa'i with an appropriate subscript numbering the dimension. When subscripted, there is no difference between pi'a and sa'i.

## 18.16. Reverse Polish notation

The following cmavo is discussed in this section:

 fu'a FUhA reverse Polish flag

So far, the Lojban notational conventions have mapped fairly familiar kinds of mathematical discourse. The use of forethought operators may have seemed odd when applied to +, but when applied to f they appear as the usual functional notation. Now comes a sharp break. Reverse Polish (RP) notation represents something completely different; even mathematicians don't use it much. (The only common uses of RP, in fact, are in some kinds of calculators and in the implementation of some programming languages.)

In RP notation, the operator follows the operands. (Polish notation, where the operator precedes its operands, is another name for forethought mekso of the kind explained in Section 18.6.) The number of operands per operator is always fixed. No parentheses are required or permitted. In Lojban, RP notation is always explicitly marked by a fu'a at the beginning of the expression; there is no terminator. Here is a simple example:

Example 18.110.

 li fu'a reboi ci su'i du li mu the-number (RP!) two, three, plus equals the-number five.

The operands are re and ci; the operator is su'i.

Here is a more complex example:

Example 18.111.

 li fu'a reboi ci pi'i voboi mu pi'i su'i the-number (RP!) (two, three, times), (four, five, times), plus
 du li rexa equals the-number two-six

Here the operands of the first pi'i are re and ci; the operands of the second pi'i are vo and mu (with boi inserted where needed), and the operands of the su'i are reboi ci pi'i, or 6, and voboi mu pi'i, or 20. As you can see, it is easy to get lost in the world of reverse Polish notation; on the other hand, it is especially easy for a mechanical listener (who has a deep mental stack and doesn't get lost) to comprehend.

The operands of an RP operator can be any legal mekso operand, including parenthesized mekso that can contain any valid syntax, whether more RP or something more conventional.

In Lojban, RP operators are always parsed with exactly two operands. What about operators which require only one operand, or more than two operands? The null operand tu'o and the null operator ge'a provide a simple solution. A one-operand operator like va'a always appears in a reverse Polish context as tu'o va'a. The tu'o provides the second operand, which is semantically ignored but grammatically necessary. Likewise, the three-operand version of gei appears in reverse Polish as ge'a gei, where the ge'a effectively merges the 2nd and 3rd operands into a single operand. Here are some examples:

Example 18.112.

 li fu'a ciboi muboi vu'u The-number (RP!) (three, five, minus)
 du li fu'a reboi tu'o va'a equals the-number (RP!) two, null, negative-of. 3 − 5 = -2

Example 18.113.

 li cinoki'oki'o du The-number 30-comma-comma equals
 li fu'a biboi ciboi panoboi ge'a gei the-number (RP!) 8, (3, 10, null-op), exponential-notation. 30,000,000 = 3 × 10 ^ 8

## 18.17. Logical and non-logical connectives within mekso

The following cmavo are discussed in this section:

 .abu BY letter “a” by BY letter “b” cy BY letter “c” fe'a VUhU nth root of (default square root) lo'o LOhO terminator for LI

As befits a logical language, Lojban has extensive provision for logical connectives within both operators and operands. Full details on logical and non-logical connectives are provided in Chapter 14. Operands are connected in afterthought with selma'o A and in forethought with selma'o GA, just like sumti. Operators are connected in afterthought with selma'o JA and in forethought with selma'o GUhA, just like tanru components. This parallelism is no accident.

In addition, A+BO and A+KE constructs are allowed for grouping logically connected operands, and keke'e is allowed for grouping logically connected operators, although there are no analogues of tanru among the operators.

Despite the large number of rules required to support this feature, it is of relatively minor importance in the mekso scheme of things. Example 18.114 exhibits afterthought logical connection between operands:

Example 18.114.

 vei ci .a vo ve'o prenu cu klama le zarci ( Three or four ) people go to-the market.

Example 18.115 is equivalent in meaning, but uses forethought connection:

Example 18.115.

 vei ga ci gi vo ve'o prenu cu klama le zarci ( Either 3 or 4 ) people go to-the market.

Note that the mekso here are being used as quantifiers. Lojban requires that any mekso other than a simple number be enclosed in parentheses when used as a quantifier. This rule prevents ambiguities that do not exist when using li.

By the way, li has an elidable terminator, lo'o, which is needed when a li sumti is followed by a logical connective that could seem to be within the mekso. For example:

Example 18.116.

 li re su'i re du The-number two plus two equals
 li vo lo'o .onai lo nalseldjuno namcu the-number four or-else a non-known number.

Omitting the lo'o would cause the parser to assume that another operand followed the .onai and reject lo as an invalid operand.

Simple examples of logical connection between operators are hard to come by. A contrived example is:

Example 18.117.

 li re su'i je pi'i re du li vo The-number two plus and times two equals the-number four. 2 + 2 = 4 and 2 × 2 = 4.

The forethought-connection form of Example 18.117 is:

Example 18.118.

 li re ge su'i gi pi'i re du li vo the-number two both plus and times two equals the-number four. Both 2 + 2 = 4 and 2 × 2 = 4.

Here is a classic example of operand logical connection:

Example 18.119.

 go li .abu bi'epi'i vei xy. te'a re ve'o su'i If-and-only-if the-number “a” times ( “x” power two ) plus
 by. bi'epi'i xy. su'i cy. du li no “b” times “x” plus “c” equals the-number zero
 gi li xy. du li vei va'a by. ku'e then the-number x equals the-number [ the-negation-of( b )
 su'i ja vu'u fe'a plus or minus the-root-of
 vei by. bi'ete'a re vu'u vo bi'epi'i .abu bi'epi'i cy. ( “b” power 2 minus four times “a” times “c”
 ve'o [ku'e] ve'o fe'i re bi'epi'i .abu ) ] divided-by two times “a” Iff a x 2 + b x + c = 0 , then x = - b ± b 2 - 4 a c 2 a

Note the mixture of styles in Example 18.119: the negation of b and the square root are represented by forethought and most of the operator precedence by prefixed bi'e, but explicit parentheses had to be added to group the numerator properly. In addition, the square root parentheses cannot be removed here in favor of simple fe'a and ku'e bracketing, because infix operators are present in the operand. Getting Example 18.119 to parse perfectly using the current parser took several tries: a more relaxed style would dispense with most of the bi'e cmavo and just let the standard precedence rules be understood.

Non-logical connection with JOI and BIhI is also permitted between operands and between operators. One use for this construct is to connect operands with bi'o to create intervals:

Example 18.120.

 li no ga'o bi'o ke'i pa the-number zero (inclusive) from-to (exclusive) one [0,1)
 the numbers from zero to one, including zero but not including one

Intervals defined by a midpoint and range rather than beginning and end points can be expressed by mi'i:

Example 18.121.

 li pimu ga'o mi'i ke'i pimu the-number 0.5 (inclusive) centered-with-range (exclusive) 0.5

which expresses the same interval as Example 18.120. Note that the ga'o and ke'i still refer to the endpoints, although these are now implied rather than expressed. Another way of expressing the same thing:

Example 18.122.

 li pimu su'i ni'upimu bi'o ke'i ma'upimu the-number 0.5 plus [-0.5 from-to (exclusive) +0.5]

Here we have the sum of a number and an interval, which produces another interval centered on the number. As Example 18.122 shows, non-logical (or logical) connection of operands has higher precedence than any mekso operator.

You can also combine two operands with ce'o, the sequence connective of selma'o JOI, to make a compound subscript:

Example 18.123.

 xy. xi vei by. ce'o dy. [ve'o] “x” sub ( “b” sequence “d” ) xb,d

## 18.18. Using Lojban resources within mekso

The following cmavo are discussed in this section:

 na'u NAhU selbri to operator ni'e NIhE selbri to operand mo'e MOhE sumti to operand te'u TEhU terminator for all three

One of the mekso design goals requires the ability to make use of Lojban's vocabulary resources within mekso to extend the built-in cmavo for operands and operators. There are three relevant constructs: all three share the elidable terminator te'u (which is also used to terminate vectors marked with jo'i)

The cmavo na'u makes a selbri into an operator. In general, the first place of the selbri specifies the result of the operator, and the other unfilled places specify the operands:

Example 18.124.

 li na'u tanjo te'u The-number the-operator tangent [end-operator]
 vei pai fe'i re [ve'o] du li ci'i ( π / 2 ) = the-number infinity. tan(π/2) = ∞

tanjo is the gismu for x1 is the tangent of x2, and the na'u here makes it into an operator which is then used in forethought

The cmavo ni'e makes a selbri into an operand. The x1 place of the selbri generally represents a number, and therefore is often a ni abstraction, since ni abstractions represent numbers. The ni'e makes that number available as a mekso operand. A common application is to make equations relating pure dimensions:

Example 18.125.

 li ni'e ni clani [te'u] The-number quantity-of length
 pi'i ni'e ni ganra [te'u] times quantity-of width
 pi'i ni'e ni condi te'u times quantity-of depth
 du li ni'e ni canlu equals the-number quantity-of volume. Length × Width × Depth = Volume

The cmavo mo'e operates similarly to ni'e, but makes a sumti (rather than a selbri) into an operand. This construction is useful in stating equations involving dimensioned numbers:

Example 18.126.

 li mo'e re ratcu su'i mo'e re ractu The-number two rats plus two rabbits
 du li mo'e vo danlu equals the-number four animals. 2 rats + 2 rabbits = 4 animals.

Another use is in constructing Lojbanic versions of so-called folk quantifiers, such as a pride of lions:

Example 18.127.

 mi viska vei mo'e lo'e lanzu ve'o cinfo I see ( the-typical family )-number-of lions.
 I see a pride of lions.

## 18.19. Other uses of mekso

The following cmavo are discussed in this section:

 me'o LI the mekso nu'a NUhA operator to selbri mai MAI utterance ordinal mo'o MAI higher order utterance ordinal roi ROI quantified tense

So far we have seen mekso used as sumti (with li), as quantifiers (often parenthesized), and in MOI and ME-MOI selbri. There are a few other minor uses of mekso within Lojban.

The cmavo me'o has the same grammatical use as li but slightly different semantics. li means the number which is the value of the mekso ..., whereas me'o just means the mekso ... So it is true that:

Example 18.128.

 li re su'i re du li vo The-number two plus two equals the-number four. 2 + 2 = 4

but false that:

Example 18.129.

 me'o re su'i re du me'o vo The-mekso two plus two equals the-mekso four.
 “2 + 2”=“4”

since the expressions 2 + 2 and 4 are not the same. The relationship between li and me'o is related to that between la djan., the person named John, and zo .djan., the name John

The cmavo nu'a is the inverse of na'u, and allows a mekso operator to be used as a normal selbri, with the place structure:

x1 is the result of applying (operator) to x2, x3, ...

for as many places as may be required. For example:

Example 18.130.

 li ni'umu cu nu'a va'a li ma'umu The-number -5 is-the-operator negation-of the-number +5.

uses nu'a to make the operator va'a into a two-place bridi

Used together, nu'a and na'u make it possible to ask questions about mekso operators, even though there is no specific cmavo for an operator question, nor is it grammatical to utter an operator in isolation. Consider Example 18.131, to which Example 18.132 is one correct answer:

Example 18.131.

 li re na'u The-number two applied-to-selbri
 mo re du li vo which-selbri? two equals the-number four. 2 ? 2 = 4

Example 18.132.

 nu'a su'i
 plus

In Example 18.131, na'u mo is an operator question, because mo is the selbri question cmavo and na'u makes the selbri into an operator. Example 18.132 makes the true answer su'i into a selbri (which is a legal utterance) with the inverse cmavo nu'a. Mechanically speaking, inserting Example 18.132 into Example 18.131 produces:

Example 18.133.

 li re na'u nu'a The-number two (the-operator the-selbri
 su'i re du li vo plus) two equals the-number four.

where the na'u nu'a cancels out, leaving a truthful bridi

Numerical free modifiers, corresponding to English firstly, secondly, and so on, can be created by suffixing a member of selma'o MAI to a digit string or a lerfu string. (Digit strings are compound cmavo beginning with a cmavo of selma'o PA, and containing only cmavo of PA or BY; lerfu strings begin with a cmavo of selma'o BY, and likewise contain only PA or BY cmavo.) Here are some examples:

Example 18.134.

 pamai
 firstly

Example 18.135.

 remai
 secondly

Example 18.136.

 romai all-ly
 lastly

Example 18.137.

 ny.mai
 nth-ly

Example 18.138.

 pasomo'o
 nineteenthly (higher order) Section 19

The difference between mai and mo'o is that mo'o enumerates larger subdivisions of a text. Each mo'o subdivision can then be divided into pieces and internally numbered with mai. If this chapter were translated into Lojban, each section would be numbered with mo'o. (See Section 19.7 for more on these words.)

A numerical tense can be created by suffixing a digit string with roi. This usage generates tenses corresponding to English once, twice, and so on. This topic belongs to a detailed discussion of Lojban tenses, and is explained further in Section 10.9.

Note: the elidable terminator boi is not used between a number and a member of MAI or ROI.

## 18.20. Explicit operator precedence

As mentioned earlier, Lojban does provide a way for the precedences of operators to be explicitly declared, although current parsers do not understand these declarations.

The declaration is made in the form of a metalinguistic comment using ti'o, a member of selma'o SEI. sei, the other member of SEI, is used to insert metalinguistic comments on a bridi which give information about the discourse which the bridi comprises. The format of a ti'o declaration has not been formally established, but presumably would take the form of mentioning a mekso operator and then giving it either an absolute numerical precedence on some pre-established scale, or else specifying relative precedences between new operators and existing operators.

In future, we hope to create an improved machine parser that can understand declarations of the precedences of simple operators belonging to selma'o VUhU. Originally, all operators would have the same precedence. Declarations would have the effect of raising the specified cmavo of VUhU to higher precedence levels. Complex operators formed with na'u, ni'e, or ma'o would remain at the standard low precedence; declarations with respect to them are for future implementation efforts. It is probable that such a parser would have a set of commonly assumed precedences built into it (selectable by a special ti'o declaration) that would match mathematical intuition: times higher than plus, and so on.

## 18.21. Miscellany

A few other points:

se can be used to convert an operator as if it were a selbri, so that its arguments are exchanged. For example:

Example 18.139.

 li ci se vu'u vo du li pa The-number three (inverse) minus four equals the-number one.
 3 subtracted from 4 equals 1.

The other converters of selma'o SE can also be used on operators with more than two operands, and they can be compounded to create (probably unintelligible) operators as needed.

Members of selma'o NAhE are also legal on an operator to produce a scalar negation of it. The implication is that some other operator would apply to make the bridi true:

Example 18.140.

 li ci na'e su'i vo du li pare The-number 3 non- plus 4 equals the-number 12.

Example 18.141.

 li ci to'e vu'u re du li mu The-number 3 opposite-of- minus 2 equals the-number 5.

The sense in which plus is the opposite of minus is not a mathematical but rather a linguistic one; negated operators are defined only loosely.

la'e and lu'e can be used on operands with the usual semantics to get the referent of or a symbol for an operand. Likewise, a member of selma'o NAhE followed by bo serves to scalar-negate an operand, implying that some other operand would make the bridi true:

Example 18.142.

 li re su'i re du li na'ebo mu The-number 2 plus 2 equals the-number non- 5.
 2 + 2 = something other than 5.

The digits 0-9 have rafsi, and therefore can be used in making lujvo. Additionally, all the rafsi have CVC form and can stand alone or together as names:

Example 18.143.

 la zel. poi gunta la tebes. pu nanmu Those-named “Seven” who attack that-named “Thebes” [past] are-men.
 The Seven Against Thebes were men.

Of course, there is no guarantee that the name zel. is connected with the number rafsi: an alternative which cannot be misconstrued is:

Example 18.144.

 la zemei poi gunta Those-named-the Sevensome who attack
 la tebes. pu nanmu that-named Thebes [past] are-men.

Certain other members of PA also have assigned rafsi: so'a, so'e, so'i, so'o, so'u, da'a, ro, su'e, su'o, pi, and ce'i. Furthermore, although the cmavo fi'u does not have a rafsi as such, it is closely related to the gismu frinu, meaning fraction; therefore, in a context of numeric rafsi, you can use any of the rafsi for frinu to indicate a fraction slash.

A similar convention is used for the cmavo cu'o of selma'o MOI, which is closely related to cunso (probability); use a rafsi for cunso in order to create lujvo based on cu'o. The cmavo mei and moi of MOI have their own rafsi, two each in fact: mem/ mei and mom/ moi respectively.

The grammar of mekso as described so far imposes a rigid distinction between operators and operands. Some flavors of mathematics (lambda calculus, algebra of functions) blur this distinction, and Lojban must have a method of doing the same. An operator can be changed into an operand with ni'enu'a, which transforms the operator into a matching selbri and then the selbri into an operand.

To change an operand into an operator, we use the cmavo ma'o, already introduced as a means of changing a lerfu string such as fy. into an operator. In fact, ma'o can be followed by any mekso operand, using the elidable terminator te'u if necessary.

There is a potential semantic ambiguity in ma'o fy. [te'u] if fy. is already in use as a variable: it comes to mean the function whose value is always f. However, mathematicians do not normally use the same lerfu words or strings as both functions and variables, so this case should not arise in practice.

## 18.22. Four score and seven: a mekso problem

Abraham Lincoln's Gettysburg Address begins with the words Four score and seven years ago. This section exhibits several different ways of saying the number four score and seven. (A score, for those not familiar with the term, is 20; it is analogous to a dozen for 12.) The trivial way:

Example 18.145.

 li bize eight seven 87

Example 18.145 is mathematically correct, but sacrifices the spirit of the English words, which are intended to be complex and formal.

Example 18.146.

 li vo pi'i reno su'i ze the-number four times twenty plus seven 4 × 20 + 7

Example 18.146 is also mathematically correct, but still misses something. Score is not a word for 20 in the same way that ten is a word for 10: it contains the implication of 20 objects. The original may be taken as short for Four score years and seven years ago. Thinking of a score as a twentysome rather than as 20 leads to:

Example 18.147.

 li mo'e voboi renomei the-number [sumti-to-mex] four twentysomes
 te'u su'i ze [end-sumti-to-mex] plus seven

In Example 18.147, voboi renomei is a sumti signifying four things each of which are groups of twenty; the mo'e and te'u then make this sumti into a number in order to allow it to be the operand of su'i.

Another approach is to think of score as setting a representation base. There are remnants of base-20 arithmetic in some languages, notably French, in which 87 is quatre-vingt-sept, literally four-twenties-seven. (This fact makes the Gettysburg Address hard to translate into French!) If score is the representation base, then we have:

Example 18.148.

 li vo pi'e ze ju'u reno the-number four ; seven base 20 4720

Overall, Example 18.147 probably captures the flavor of the English best. Example 18.145 and Example 18.146 are too simple, and Example 18.148 is too tricky. Nevertheless, all four examples are good Lojban. Pedagogically, these examples illustrate the richness of lojbau mekso: anything that can be said at all, can probably be said in more than one way.

## 18.23. mekso selma'o summary

Except as noted, each selma'o has only one cmavo.

 BOI elidable terminator for numerals and lerfu strings BY lerfu for variables and functions (see Section 17.11) FUhA reverse-Polish flag GOhA includes du (mathematical equality) and other non-mekso cmavo JOhI array flag KUhE elidable terminator for forethought mekso LI mekso articles (li and me'o) MAhO make operand into operator MOI creates mekso selbri (moi, mei, si'e, and cu'o, see Section 18.11) MOhE make sumti into operand NAhU make selbri into operator NIhE make selbri into operand NUhA make operator into selbri PA numbers (see Section 18.25) PEhO optional forethought mekso marker TEhU elidable terminator for NAhU, NIhE, MOhE, MAhO, and JOhI VEI left parenthesis VEhO right parenthesis VUhU operators (see Section 18.24) XI subscript flag

## 18.24. Complete table of VUhU cmavo, with operand structures

The operand structures specify what various operands (labeled a, b, c, ...) mean. The implied context is forethought, since only forethought operators can have a variable number of operands; however, the same rules apply to infix and RP uses of VUhU.

 su'i plus (((a + b) + c) + ...) pi'i times (((a × b) × c) × ...) vu'u minus (((a − b) − c) − ...) fe'i divided by (((a / b) / c) / ...) ju'u number base numeral string a interpreted in the base b pa'i ratio the ratio of a to b a:b fa'i reciprocal of/multiplicative inverse 1 / a gei scientific notation b × (c [default 10] to the a power) ge'a null operator (no operands) de'o logarithm log a to base b (default 10 or e as appropriate) te'a to the power/exponential a to the b power fe'a nth root of/inverse power bth root of a (default square root: b = 2) cu'a absolute value/norm | a | ne'o factorial a! pi'a matrix row vector combiner (all operands are row vectors) sa'i matrix column vector combiner (all operands are column vectors) ri'o integral integral of a with respect to b over range c sa'o derivative derivative of a with respect to b of degree c (default 1) fu'u non-specific operator (variable) si'i sigma (Σ) summation summation of a using variable b over range c va'a negation of/additive inverse -a re'a matrix transpose/dual a*

## 18.25. Complete table of PA cmavo: digits, punctuation, and other numbers.

• Table 18.1.  Decimal digits

 no non 0 pa pav 1 re rel 2 ci cib 3 vo von 4 mu mum 5 xa xav 6 ze zel 7 bi biv 8 so soz 9

• Table 18.2.  Hexadecimal digits

 dau A/10 fei B/11 gai C/12 jau D/13 rei E/14 vai F/15

• Table 18.3.  Special numbers

 pai π ka'o imaginary i te'o exponential e ci'i infinity (∞)

• Table 18.4.  Number punctuation

 pi piz decimal point ce'i cez percentage fi'u fi'u (from frinu; see Section 18.20) fraction (not division) pi'e mixed-base point ma'u plus sign (not addition) ni'u minus sign (not subtraction) ki'o thousands comma ra'e repeating-decimal indicator ji'i approximation sign ka'o complex number separator

• Table 18.5.  Indefinite numbers

 ro all rol so'a soj almost all so'e sop most so'i many sor so'i so'o sos several so'u sot few da'a daz all but

• Table 18.6. Subjective numbers

 rau enough du'e too few mo'a too many

• Table 18.7. Miscellaneous

 xo number question tu'o null operand

## 18.26. Table of MOI cmavo, with associated rafsi and place structures

 mei mem mei
 x1 is a mass formed from a set x2 of n members, one or more of which is/are x3, [measured relative to the set x4/by standard x4]
 moi mom moi
 x1 is the (n)th member of set x2 when ordered by rule x3 [by standard x4]
 si'e
 x1 is an (n)th portion of mass x2 [by standard x3]
 cu'o cu'o (borrowed from cunso; see Section 18.20)
 event x1 has probability (n) of occurring under conditions x2 [by standard x3]
 va'e
 x1 is at scale position (n) on the scale x2 [by standard x3]