lojbau mekso: Mathematical Expressions in Lojban

mkhtml: 1.1

- 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.

```
1.1) $3x\; +\; 2y$
```

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

2.1) pa re ci one two three 123 one hundred and twenty three 2.2) pa no one zero 10 ten 2.3) 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.

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

3.1) ni'u pa negative-sign 1 -1

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

3.3) re fi'u ze two fraction seven 2/7

3.4) fi'u ze fraction seven 1/7

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

3.6) ci mu ce'i three five percent 35%

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

3.8) 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:

3.9) pi ki'o re re point comma two two .022 3.9) pi pa ki'o pa re ki'o pa point one comma one two comma one .101012001

The following cmavo are discussed in this section:

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

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

```
4.1) ma'u ci'i
$+$¥
```

4.2) ci ka'o re $3i2$ (a complex number equivalent to ``$3\; +\; 2i$'')

4.3) ci'i no infinity zero $$À_{0}(a transfinite cardinal)

4.4) pai pi 4.5) te'o e

4.6) pa pi re pi ci 1.2.3 4.5) pa ni'u re 1 negative-sign 2

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 Sections 8 to 12.

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

```
5.1) li pa su'i pa du li re
the-number one plus one equals the-number two.
$1\; +\; 1\; =\; 2$
```

5.2) le ci prenu the three personsrequires no ``li'' article, because the ``ci'' is being used to specify the number of ``prenu''. However, the sentence

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

5.4) li ci prenu the-number 3 is-a-personwhich is grammatical but nonsensical: numbers are not persons.

```
5.5) li ma'u pa su'i ni'u pa
du li no
the-number positive-sign one plus negative-sign one
equals the-number zero
$+1\; +\; -1\; =\; 0$
```

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

```
5.6) li mu su'i pa
du li ci su'i ci
the-number five plus one
equals the-number three plus three
$5\; +\; 1\; =\; 3\; +\; 3$
```

```
5.7) li ci su'i vo pi'i mu
du li reci
the-number three plus four times five
equals the-number two-three
$3\; +\; 4\; \times \; 5\; =\; 23$
```

Is the Lojban version of Example 5.7 true? No! ``$3\; +\; 4\; \times \; 5$'' is indeed 23, because the usual conventions of mathematics state that multiplication takes precedence over addition; that is, the multiplication ``$4\; \times \; 5$'' 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:

```
5.8) li ci su'i vo pi'i mu
du li cimu
the-number three plus four times five
equals the-number three-five
$3\; +\; 4\; \times \; 5\; =\; 35$
```

```
5.9) 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
$3\; +\; 4\; \times \; 5\; =\; 23$
```

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

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

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 py. BY letter ``p'' xy. BY letter ``x'' zy. BY letter ``z'' ma'o MAhO convert operand to operator fy. BY letter ``f''

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

```
6.2) li py. su'i va'a ny. ku'e su'i zy du li xy.
the-number ``p'' plus negative-of( ``n'' ) plus ``z''
equals the-number ``x''
$p\; +\; -n\; +\; z\; =\; x$
```

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

```
6.3) li zy du li ma'o fy.boi xy.
the-number z equals the-number the-operator f x
$z\; =\; f(x)$
```

6.4) li pe'o su'i paboi reboi ciboi ku'e du li xa 6.5) li py. su'i pe'o va'a ny. ku'e su'i zy du li xy. 6.6) li zy du li pe'o ma'o fy.boi xy. ku'e

7.1) 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 [namcu jibni, number-near] terci'e x1 is a component with function x2 of system x3

```
7.2) 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.

```
7.3) li re su'i re na du li mu
the-number 2 + 2 is-not equal-to the-number 5.
$2\; +\; 2$½ 5
```

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

8.1) mi catlu pa prenu I look-at one person 8.2) mi catlu ro prenu I look-at all personsExample 8.1 might be true, whereas Example 8.2 is almost certainly false.

8.3) mi catlu so'a prenu I look-at almost-all persons 8.4) mi catlu so'e prenu I look-at most persons 8.5) mi catlu so'i prenu I look-at many persons 8.6) mi catlu so'o prenu I look-at several persons 8.7) mi catlu so'u prenu I look-at a-few persons

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

8.9) mi catlu no'o prenu I look-at a-typical-number-of persons 8.10) mi citka pino'o lei nanba I eat a-typical-amount-of the-mass-of bread.

8.11) mi catlu da'a re prenu I look-at all-but two persons 8.12) mi catlu da'a so'u prenu I look-at all-but a-few personsExample 8.12 is similar in meaning to Example 8.3.

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

8.14) li ci vu'u re du li ma'u the-number 3 - 2 = some-positive-number 8.15) li ci vu'u vo du li ni'u the-number 3 - 4 = some-negative-number 8.16) mi ponse ma'u rupnu I possess a-positive-number-of currency-units.

8.17) mi ponse rau rupnu I possess enough currency-units.

8.18) mi viska le rore gerku I saw the all-of/two dogs. I saw both dogs. 8.19) 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 8.19 assumes a mostly monogamous culture by stating that three is ``many''.

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

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

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)

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)

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

9.10) mi catlu su'o prenu I look-at at-least [one] personis a meaningful claim.

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

10.1) li pa no pa no ju'u re du li pa no the-number 1010 base 2 equals the-number 10

10.2) li daufeigai ju'u paxa du li rezevobi the-number ABC base 16 equals the-number 2748 10.3) li jaureivai ju'u paxa du li cimuxaze the-number DEF base 16 equals the-number 3567

10.4) li vai pi bi ju'u paxa du li pamu pi mu the-number F.8 base 16 equals the-number 15.5

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

```
10.6) li ci pi'e rere pi'e vono su'i pi'e ci pi'e cici
du li ci pi'e rexa pi'e paci
the-number 3:22:40 plus :3:33 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 10.5 and Example 10.6 is hours, the second minutes, and the third seconds.

10.7) 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

10.8) pano ju'u reno the-digit-10 base 20which is equal to ten, and:

10.9) pa pi'e no ju'u reno 1;0 base 20which is equal to twenty.

10.10) li pa pi'e vo pi ze ju'u reno du li re vo pi ci mu the-number 1;4.7 base 20 equals the-number 24.35

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

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

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

Some examples:

11.1) 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:

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

In Example 11.2, ``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 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:

11.3) lei ratcu poi zvati le panka cu so'umei fo lo'i ratcu the-mass-of rats which are-in the park 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.

11.4) 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 11.4, 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 Chapter 6.

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

Some examples:

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

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

11.7) mi raumoi le velskina porsi I am-enough-th-in the movie-audience sequence I am enough-th in the movie line.Example 11.7 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.

- x1 is an (n)th portion of mass x2

Some examples:

11.8) 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.

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

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

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

11.10) le vi rozgu cu sofi'upanova'e xunre This rose is 8/10-scale red This rose is 8 out of 10 on the scale of redness. This rose is very red.

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

11.12) ta ny.moi le'i mi ratcu that is-nth-of the-set-of my rats That is my nth rat.

11.13) ti me la nu,IORK. [me'u] this-here pertains-to what-I-call ``New York''. This is New York (or is New York-related).

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

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

The following cmavo is discussed in this section:

xo PA number question

```
12.1) li re su'i re du li xo
the-number 2 plus 2 equals the-number what?
What is $2\; +\; 2$?
```

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

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

The following cmavo is discussed in this section:

xi XI subscript

13.1) 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 $x$_{3}= x_{1}+ x_{2}

13.2) xy.boixino $x$_{0}13.3) xy.boixiny. $x$_{n}13.4) xy.boixi vei ny. su'i pa [ve'o] $x$_{n+1} }

13.5) xy.boi xi by.boi xi vo $x$_{b4}

See Example 17.10 for the standard method of specifying multiple subscripts on a single object.

More information on the uses of subscripts may be found in Chapter 19.

The following cmavo are discussed in this section:

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

```
14.1) li tu'o va'a ny. du
li no vu'u ny.
the-number (null) additive-inverse n equals
the-number zero minus n
$-n\; =\; 0\; -\; n$
```

```
14.2) li cinonoki'oki'o
du li bi gei ci
the-number three-zero-zero-comma-comma
equals the-number eight scientific three.
$300,000,000\; =\; 3\; \times \; 108$
```

```
14.3) gei reno
(scientific) two-zero
$1020$
```

14.4) papano bi'eju'u re gei pipanopano bi'eju'u re ge'a re (one-one-zero base 2) scientific (point-one-zero-one-zero base 2) with-base 2 $.1010$_{2}× 2^{1102}

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

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

- 8 1 6 3 5 7 4 9 2

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

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

The following cmavo is discussed in this section:

fu'a FUhA reverse Polish flag

16.1) 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:

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

16.3) li fu'a ciboi muboi vu'u du li fu'a reboi tu'o va'a The-number (RP!) (three, five, minus) equals the-number (RP!) two, null, negative-of. $3\; -\; 5\; =\; -2$ 16.4) li cinoki'oki'o du li fu'a biboi ciboi panoboi ge'a gei the-number 30-comma-comma equals the-number (RP!) 8, (3, 10, null-op), exponential-notation $30,000,000\; =\; 3\; \times \; 108$

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

Despite the large number of rules required to support this feature, it is of relatively minor importance in the mekso scheme of things. Example 17.1 exhibits afterthought logical connection between operands:

17.1) vei ci .a vo ve'o prenu cu klama le zarci ( three or four ) people go-to the market.Example 17.2 is equivalent in meaning, but uses forethought connection:

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

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

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

17.4) 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 $2\; \times \; 2\; =\; 4$.

The forethought-connection form of Example 17.4 is:

17.5) 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 $2\; \times \; 2\; =\; 4$.

Here is a classic example of operand logical connection:

17.6) go li .abu bi'epi'i vei xy. te'a re ve'o su'i by. bi'epi'i xy. su'i cy. du li no gi li xy. du li vei va'a by. ku'e su'i ja vu'u fe'a vei by. bi'ete'a re vu'u vo bi'epi'i .abu bi'epi'i cy. ve'o [ku'e] ve'o fe'i re bi'epi'i .abu if-and-only-if the-number ``a''-times-( ``x'' power two ) plus ``b''-times-``x'' plus ``c'' equals the-number zero then the-number x equals the-number [ the-negation-of( b ) plus or minus the-root-of ( ``b''-power-2 minus four-times-``a''-times-``c'' ) ] divided-by two-times-``a''. Iff $ax2+\; bx\; +\; c\; =\; 0$, then $x\; =-b\; \pm $Ö(b^{2}- 4ac) ----------------------- $2a$

```
17.7) 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
```

17.8) li pimu ga'o mi'i ke'i pimu the-number 0.5 plus-or-minus 0.5which expresses the same interval as Example 17.7. 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:

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

17.10) xy. xi vei by. ce'o dy. [ve'o] ``x'' sub ( ``b'' sequence ``d'' ) $x$_{b,d} }

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

```
18.1) li na'u tanjo te'u
vei pai fe'i re [ve'o] du li ci'i
the-number the-operator tangent
( pi / 2 ) = the-number infinity
$tan(pi/2)\; =$¥
```

``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.

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

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

18.4) 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.

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.

```
19.1) li re su'i re du li vo
the-number two plus two equals the-number four
$2\; +\; 2\; =\; 4$
```

but false that:

19.2) me'o re su'i re du me'o vo the-mekso two plus two equals the-mekso four ``2 + 2'' = ``4''

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

19.3) li ni'umu cu nu'a va'a li ma'umu the-number $-5$ is-the-negation-of the-number $+5$uses ``nu'a'' to make the operator ``va'a'' into a two-place bridi.

```
19.4) li re na'u mo re du li vo
the-number two what-operator? two equals the-number four
$2\; ?\; 2\; =\; 4$
19.5) nu'a su'i
plus
```

In Example 19.4, ``na'u mo'' is an operator question, because ``mo'' is the selbri question cmavo and ``na'u'' makes the selbri into an operator. Example 19.5 makes the true answer ``su'i'' into a selbri (which is a legal utterance) with the inverse cmavo ``nu'a''. Mechanically speaking, inserting Example 19.5 into Example 19.4 produces:

19.6) li re na'u nu'a su'i re du li vo the-number two (the-operator the-selbri plus) two equals the-number fourwhere the ``na'u nu'a'' cancels out, leaving a truthful bridi.

19.7) pamai firstly 19.8) remai secondly 19.9) romai all-ly lastly 19.10) ny.mai nth-ly 19.11) pasomo'o nineteenthly (higher order) Section 19

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.

A few other points:

21.1) 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.

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

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

```
21.4) 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.
```

21.5) 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.

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

22.1) bize eight seven 87Example 22.1 is mathematically correct, but sacrifices the spirit of the English words, which are intended to be complex and formal.

```
22.2) vo pi'i reno su'i ze
four times twenty plus seven
$4\; \times \; 20\; +\; 7$
```

22.3) mo'e voboi renomei su'i ze the-number-of four twentysomes plus seven

In Example 22.3, ``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''.

22.4) vo pi'e ze ju'u reno four ; seven base 20 $47$_{20}

Overall, Example 22.3 probably captures the flavor of the English best. Example 22.1 and Example 22.2 are too simple, and Example 22.4 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.

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 Chapter 17) 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 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 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 24) XI subscript flag

The operand structures specify what various operands (labeled $a$, $b$, $c$,

- su'i
- plus
$(((a\; +\; b)\; +\; c)\; +...)$
pi'i times
$(((a\; \times \; b)\; \times \; c)\; \times ...)$
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
$b\; \times \; (c$[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 b'th 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 summation summation of a using variable b over range c va'a negation of/additive inverse $-\; a$ re'a matrix transpose/dual $a*$

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

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

- pai, ka'o, te'o, ci'i pi, imaginary i, exponential e, infinity

- pi, ce'i, fi'u decimal point, percentage, fraction (not division) rafsi: piz, cez, fi'u (from frinu; see Section 20)
- pi'e, ma'u, ni'u mixed-base point, plus sign (not addition), minus sign (not subtraction)
- ki'o, ra'e thousands comma, repeating-decimal indicator
- ji'i, ka'o approximation sign, complex number separator

- ro, so'a, so'e, so'i, so'o, so'u, da'a all, almost all, most, many, several, few, all but rafsi: rol, soj, sor or so'i, sos, sot, daz
- su'e, su'o at most, at least rafsi: su'e, su'o
- me'i, za'u less than, more than
- no'o the typical number

Subjective numbers:

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

Miscellaneous:

- xo, tu'o number question, null operand

```
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]
rafsi: mem, mei
moi x1 is the (n)th member of set x2
when ordered by rule x3 [by standard x4]
rafsi: mom, moi
si'e x1 is an (n)th portion of mass x2
[by standard x3]
rafsi: none
cu'o event x1 has probability (n) of occurring
under conditions x2 [by standard x3]
rafsi: cu'o (borrowed from cunso; see Section 20)
```