Lojban Reference Grammar: Chapter 18

PLEASE NOTE: THIS IS AN OLD VERSION. The current version is linked from The Complete Lojban Language.

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Chapter 18
lojbau mekso: Mathematical Expressions in Lojban

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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

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 1.1 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.

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:

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.

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:

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 8.

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

(In some cultures, a comma is used instead of a period in the symbolic version of Example 3.2; ``pi'' is still the Lojban representation for the decimal point.)

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

Example 3.3 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:

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
	 $.35142857142857 ...$

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

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

(In some cultures, spaces are used in the symbolic representation of Example 3.7; ``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:

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

4. Special numbers

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.  $1.61803 ...$ )

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:

4.1)	ma'u ci'i
	 $+$ ¥

4.2)	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).

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

However, many combinations are as yet undefined:

4.6)	pa pi re pi ci
	1.2.3

4.5)	pa ni'u re
	1 negative-sign 2

Example 4.5 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 Sections 8 to 12.

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 5.1, 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:

5.2)	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

5.3)	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 5.2 would produce

5.4)	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:

+1 + -1 = 0">

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
	 $5 + 1 = 3 + 3$

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 5.6), 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 5.6, 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:

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
	 $3 + 4 × 5 = 23$

Is the Lojban version of Example 5.7 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:

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
	 $3 + 4 × 5 = 35$

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 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
	 $3 + 4 × 5 = 23$

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.

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:

(n + 1)(n + 1) = n^2 + 2n + 1">

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
	 $(n + 1)(n + 1) = n$ ² + 2n + 1

There are several new usages in Example 5.10: ``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 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)

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''

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:

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

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 6.1, when there are two consecutive numbers or lerfu strings.

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

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, j ``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.

... 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
	 $z = f(x)$

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 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. 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

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 17.6 came to grief because I forgot this rule.

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:

x < 5">

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

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: p = x = z">

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

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 8.1 might be true, whereas Example 8.2 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:

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

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'':

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.

``da'a'' is a related cmavo meaning ``all but'':

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

Example 8.12 is similar in meaning to Example 8.3.

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.

(The use of ``da'a'' means that Example 8.13 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 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.

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:

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:

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''.

9. Approximation and inexact numbers

The following cmavo are discussed in this section:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

is a meaningful claim.

Like the numbers in 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

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:

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

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:

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

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:

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

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 to make ``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:

10.5)	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:

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

Carefully note the difference between:

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

which is equal to ten, and:

10.9)	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 decimal fractions:

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

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

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

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 four members of MOI, each of which serves to create number-based selbri with specific place structures.

``-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:

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.

In Example 11.3, 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.''

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.

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.

``-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.

``-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:

``-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. Here is an example:

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:

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:

11.12)	ta ny.moi le'i mi ratcu
	that is-nth-of the-set-of my 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. It is often used to make a name into a predicate:

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).

A whole ``me'' construction can have a member of MOI added to the end to create a complex mekso selbri:

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.

Note: the elidable terminator ``boi'' is not used between a number and a member of MOI. As a result, the ``me'u'' in Example 11.14 could also be replaced by a ``boi'', which would serve the same function of preventing the ``pa'' and ``moi'' from joining into a compound.

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 Chapter 19 for more on Lojban questions.)

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?

``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:

12.3)	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.)

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:

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

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:

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}

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:

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

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:

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
	 $-n = 0 - n$

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:

3 × 10^8">

14.2)	li cinonoki'oki'o
		du li bi gei ci
	the-number three-zero-zero-comma-comma
		equals the-number eight scientific three.
	 $300,000,000 = 3 × 10$ ⁸

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:

10^20">

14.3)	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:

.1010_2 × 2^{110_2}">

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:

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

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:

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 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''.

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 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
	 $30,000,000 = 3 × 10$ ⁸

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 ``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.

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.

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:

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  $2 × 2 = 4$ .

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  $2 × 2 = 4$ .

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  $ax$ ²  + bx + c = 0,
		then  $x =-b ±$ Ö(b²  - 4ac)
			-----------------------
				 $2a$

Note the mixture of styles in Example 17.6: 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 17.6 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:

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

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

17.8)	li pimu ga'o mi'i ke'i pimu
	the-number 0.5 plus-or-minus 0.5

which 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]

Here we have the sum of a number and an interval, which produces another interval centered on the number. As Example 17.9 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:

x_{b,d}">

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

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:

``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:

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

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:

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

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.

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:

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''

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:

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.

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

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:

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

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, it would number each section with ``mo'o''.

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 Chapter 10.

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

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.

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:

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.

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:

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

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:

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.

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:

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.

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

21.6)	la zemei poi gunta
		la tebes. pu nanmu
	those-named-the Sevensome who attack
		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.

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:

22.1)	bize
	eight seven
	87

Example 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
	 $4 × 20 + 7$

Example 22.2 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:

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''.

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:

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.

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 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

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 $...$ pi'i times $...$ vu'u minus $...$ fe'i divided by $...$ 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 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 $a*$

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

Decimal digits:

: 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

Hexadecimal digits:

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

Special numbers:

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

Number punctuation:

: 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

Indefinite numbers:

: 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

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

	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)

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.

18.1)	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
	 $tan(pi/2) =$ ¥

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)

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

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.

11.11)	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.

11.10)	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.

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

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.

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

8.1)	mi catlu pa prenu
	I look-at one person

8.2)	mi catlu ro prenu
	I look-at all persons

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

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

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.

1.1)	 $3x + 2y$

Chapter 18lojbau mekso: Mathematical Expressions in Lojban

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4. Special numbers

11. Special mekso selbri

17. Logical and non-logical connectives within mekso

18. Using Lojban resources within mekso

19. Other uses of mekso

24. Complete table of VUhU cmavo, with operand structures

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

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

Chapter 18
lojbau mekso: Mathematical Expressions in Lojban

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