1. Introductory

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

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.

2. Lojban numbers

	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.

3. Signs and numerical punctuation

	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

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

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

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:

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

	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. $1.61803...$)

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
	$$À0 (a transfinite cardinal)

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

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.

5. Simple infix expressions and equations

	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.
	$1\; +\; 1\; =\; 2$

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:

5.2)	le ci prenu
	the three persons

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-person

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
	$5\; +\; 1\; =\; 3\; +\; 3$

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:

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
	$3\; +\; 4\; \times \; 5\; =\; 35$

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 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,

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
	$3\; +\; 4\; \times \; 5\; =\; 23$

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 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 5.10 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.

6. Forethought operators (Polish notation, functions)

	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
	$sum(1,2,3)\; =\; 6$

Forethought mekso can use any number of operands, in Example 6.1 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:

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''
	$p\; +\; -n\; +\; z\; =\; x$

... ku'e compared with parenthesization"> ... ve'o contrasted with operator ... ku'e"> ... ve'o, contrasted with operator ... ku'e in Polish notation"> ... ku'e in Polish notation, contrasted with vei ... ve'o"> In Example 6.2, 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:

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

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. Examples 6.1 to 6.3, respectively, with explicit ``pe'o'' and ``ku'e'':

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. Other useful selbri for mekso bridi

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.

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

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

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

8. Indefinite numbers

	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 persons

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:

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

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.

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 persons

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'':

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.

As mentioned in Section 3, ``ma'u'' and ``ni'u'' are also legal numbers, and they mean ``some positive number'' and ``some negative number'' respectively.

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).

9. Approximation and inexact numbers

	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)

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

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

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

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

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

If no number follows one of these cmavo, ``pa'' is understood: therefore,

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

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

10. Non-decimal and compound bases

	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.

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'':

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 20

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

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.

	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

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.

``-moi'' creates ordinal selbri. The place structure is:

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.

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

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.

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

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.

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:

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.

12. Number questions

	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

13. Subscripts

	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  = x1  + x2

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.

14. Infix operators revisited

	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 × 21102

	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)
	$(1,2)\; +\; (3,4)\; =\; (4,6)$

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)

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)

16. Reverse Polish notation

	fu'a	FUhA	reverse Polish flag

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 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:

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

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:

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
	$30,000,000\; =\; 3\; \times \; 108$

	.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

In addition, A+BO and A+KE constructs are allowed for grouping logically connected operands, and ``ke ... ke'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 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.

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.

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:

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(b2  - 4ac)
			-----------------------
				$2a$

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:

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 0.5

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

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

x_{b,d}">

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

	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
	$tan(pi/2)\; =$¥

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.

	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

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

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

19.3)	li ni'umu cu nu'a va'a li ma'umu
	the-number $-5$ is-the-negation-of the-number $+5$

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 19.4, to which Example 19.5 is one correct answer:

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 four

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

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
	87

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

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

	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

	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)