There are some numerical values that require a =93base=94 that varies f= rom digit to digit. For example, times represented in hours, minutes, and s= econds have, in effect, three =93digits=94: the first is base 24, the secon= d and third are base 60. To express such numbers, the compound base separat= or =93pi'e=94 is used:

Both =93pi=94 and =93pi'e=94 can be used to express large-base decimal = fractions:

When the quantifier preceding any MOI cmavo includes the subjective num= bers =93rau=94, =93du'e=94, or =93mo'a=94 (enough, too many, too few) then = an additional place is added for =93by standard=94. For example:

This place is not normally explicit when using one of the subjective nu= mbers directly as a number. Therefore, =93du'e ratcu=94 means =93too many r= ats=94 without specifying any standard.

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

The cmavo =93me=94 (of selma'o ME) has the function of making a sumti i= nto a selbri. A whole =93me=94 construction can have a member of MOI added = to the end to create a complex mekso bridi:

The cmavo =93me=94 (of selma'o ME) has the function of making a sumti i= nto a selbri. A whole =93me=94 construction can have a member of MOI added = to the end to create a complex mekso selbri:

It is perfectly possible to use non-numerical sumti after =93me=94 and = before a member of MOI, producing strange results indeed:

Previous=

Non-decimal and compound bases

Number questions

diff --git a/18/14/index.html b/18/14/index.html index cbfd95e..5e29b08 100644 --- a/18/14/index.html +++ b/18/14/index.html @@ -49,21 +49,21 @@ The =93tu'o=94 fulfills the grammatical requirement for= a left operand for the i Why are the arguments to =93gei=94 in reverse order from the conventional = symbolic notation? So that =93gei=94 can be used in forethought to allow ea= sy specification of a large (or small) imprecise number:

 14.3)  gei reno
        (scientific) two-zero
        10²⁰

Note, however, that although 10 is far and away the most common exponent b= ase, it is not the only possible one. The third operand of =93gei=94, there= fore, is the base, with 10 as the default value. Most computers internally = store so-called =93floating-point=94 numbers using 2 as the exponent base. = (This has nothing to do with the fact that computers also represent all int= egers in base 2; the IBM 360 series used an exponent base of 16 for floatin= g point, although each component of the number was expressed in base 2.) He= re is a computer floating-point number with a value of 40:

 14.4)  papano bi'eju'u re gei pipanopano bi'e=
ju'u re ge'a re
        (one-one-zero base 2) scientific (point-one-zero-one-zero base 2) w=
ith-base 2
-       .1010 =D7 2^110₂
+       .1010₂ =D7 2^110₂

Previous=

Subscripts

Vectors and matrices

diff --git a/18/19/index.html b/18/19/index.html index 06bcd91..8016189 100644 --- a/18/19/index.html +++ b/18/19/index.html @@ -83,21 +83,21 @@ where the =93na'u nu'a=94 cancels out, leaving a truthf= ul bridi 19.9) romai all-ly lastly =20 19.10) ny.mai nth-ly =20 19.11) pasomo'o nineteenthly (higher order) -The difference between =93mai=94 and =93mo'o=94 is that =93mo'o=94 enumera= tes larger subdivisions of a text. Each =93mo'o=94 subdivision can then be = divided into pieces and internally numbered with =93mai=94. If this chapter= were translated into Lojban, it would number each section with =93mo'o=94.= (See Chapter 19 for more on these words.) +The difference between =93mai=94 and =93mo'o=94 is that =93mo'o=94 enumera= tes larger subdivisions of a text. Each =93mo'o=94 subdivision can then be = divided into pieces and internally numbered with =93mai=94. If this chapter= were translated into Lojban, each section would be numbered with =93mo'o= =94. (See Chapter 19 for more on these words.)

A numerical tense can be created by suffixing a digit string with =93ro= i=94. This usage generates tenses corresponding to English =93once=94, =93t= wice=94, and so on. This topic belongs to a detailed discussion of Lojban t= enses, and is explained further in Chapter 10.<= /p>

Note: the elidable terminator =93boi=94 is not used between a number an= d a member of MAI or ROI.

Previous=

Using Lojban resources within mekso

Next diff --git a/18/20/index.html b/18/20/index.html index 8b3c38c..f7c1bcf 100644 --- a/18/20/index.html +++ b/18/20/index.html @@ -19,21 +19,21 @@

Miscellany

lojbau mekso: Mathematical Expressions in L= ojban

The Lojban Reference Gram= mar

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 unders= tand these declarations.

The declaration is made in the form of a metalinguistic comment using = =93ti'o=94, a member of selma'o SEI. =93sei=94, the other member of SEI, is= used to insert metalinguistic comments on a bridi which give information a= bout the discourse which the bridi comprises. The format of a =93ti'o=94 de= claration has not been formally established, but presumably would take the = form of mentioning a mekso operator and then giving it either an absolute n= umerical precedence on some established scale, or else specifying relative = precedences between new operators and existing operators.

The declaration is made in the form of a metalinguistic comment using = =93ti'o=94, a member of selma'o SEI. =93sei=94, the other member of SEI, is= used to insert metalinguistic comments on a bridi which give information a= bout the discourse which the bridi comprises. The format of a =93ti'o=94 de= claration has not been formally established, but presumably would take the = form of mentioning a mekso operator and then giving it either an absolute n= umerical precedence on some pre-established scale, or else specifying relat= ive precedences between new operators and existing operators.

In future, we hope to create an improved machine parser that can unders= tand declarations of the precedences of simple operators belonging to selma= 'o VUhU. Originally, all operators would have the same precedence. Declarat= ions would have the effect of raising the specified cmavo of VUhU to higher= precedence levels. Complex operators formed with =93na'u=94, =93ni'e=94, o= r =93ma'o=94 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 =93commonly assumed precedences=94 built= into it (selectable by a special =93ti'o=94 declaration) that would match = mathematical intuition: times higher than plus, and so on.

Previous=

Other uses of mekso

Miscellany

diff --git a/18/22/index.html b/18/22/index.html index e52f590..ca783ea 100644 --- a/18/22/index.html +++ b/18/22/index.html @@ -20,39 +20,39 @@

lojbau mekso: Mathematical Expressions in L= ojban

The Lojban Reference Gram= mar

22. Four score and seven: a mekso problem

Abraham Lincoln=92s Gettysburg Address begins with the words =93Four sc= ore and seven years ago=94. This section exhibits several different ways of= saying the number =93four score and seven=94. (A =93score=94, for those no= t familiar with the term, is 20; it is analogous to a =93dozen=94 for 12.) = The trivial way:

-22.1)  bize
+22.1)  li bize
        eight seven
        87

Example 22.1 is mathematically correct, but sacr= ifices the spirit of the English words, which are intended to be complex an= d formal.

-22.2)  vo pi'i reno su'i ze
+22.2)  li vo pi'i reno su'i ze
        four times twenty plus seven
        4 =D7 20 + 7

Example 22.2 is also mathematically correct, but= still misses something. =93Score=94 is not a word for 20 in the same way t= hat =93ten=94 is a word for 10: it contains the implication of 20 objects. = The original may be taken as short for =93Four score years and seven years = ago=94. Thinking of a score as a twentysome rather than as 20 leads to:

-22.3)  mo'e voboi renomei su'i ze [DOES NOT P=
ARSE]
+22.3)  li mo'e voboi renomei te'u su'i ze
        the-number-of four twentysomes plus seven

In Example 22.3, =93voboi renomei=94 is a sumti = signifying four things each of which are groups of twenty; the =93mo'e=94 a= nd =93te'u=94 then make this sumti into a number in order to allow it to be= the operand of =93su'i=94.

Another approach is to think of =93score=94 as setting a representation= base. There are remnants of base-20 arithmetic in some languages, notably = French, in which 87 is =93quatre-vingt-sept=94, literally =93four-twenties-= seven=94. (This fact makes the Gettysburg Address hard to translate into Fr= ench!) If =93score=94 is the representation base, then we have:

-22.4)  vo pi'e ze ju'u reno
+22.4)  li vo pi'e ze ju'u reno
        four ; seven base 20
        47₂₀

Overall, Example 22.3 probably captures the flav= or of the English best. Example 22.1 and Example 22.2 are too simple, and E= xample 22.4 is too tricky. Nevertheless, all four examples are good Loj= ban. Pedagogically, these examples illustrate the richness of lojbau mekso:= anything that can be said at all, can probably be said in more than one wa= y.

Previous=

Miscellany

diff --git a/18/3/index.html b/18/3/index.html index ce65cf1..6064b87 100644 --- a/18/3/index.html +++ b/18/3/index.html @@ -82,21 +82,21 @@ Note that the =93ra'e=94 marks unambiguously where the = repeating portion =931428 1,023,004 In the same way, =93ki'o=94 can be used after =93pi=94 to divide fractions= into groups of three:

 3.9)   pi ki'o re re
        point comma two two
        .022
=20
 3.10)  pi pa ki'o pa re ki'o pa
        point one comma one two comma one
-       .101012001
+       .001012001

Previous<= /div>

Lojban numbers

Special numbers

diff --git a/18/5/index.html b/18/5/index.html index bc83bf4..2f70233 100644 --- a/18/5/index.html +++ b/18/5/index.html @@ -88,21 +88,21 @@ Is the Lojban version of Example 5= .7 true? No! =933 + 4=20 Here we calculate 3 + 4 first, giving 7, and then calculate 7 =D7 5 second= , leading to the result 35. While possessing the advantage of simplicity, t= his result violates the design goal of matching the standards of mathematic= s. What can be done?

There are three solutions, all of which will probably be used to some d= egree. The first solution is to ignore the problem. People will say =93li c= i su'i vo pi'i mu=94 and mean 23 by it, because the notion that multiplicat= ion takes precedence over addition is too deeply ingrained to be eradicated= by Lojban parsing, which totally ignores semantics. This convention essent= ially allows semantics to dominate syntax in this one area.

(Why not hard-wire the precedences into the grammar, as is done in comp= uter programming languages? Essentially because there are too many operator= s, known and unknown, with levels of precedence that vary according to usag= e. The programming language =92C=92 has 13 levels of precedence, and its li= st of operators is not even extensible. For Lojban this approach is just no= t practical. In addition, hard-wired precedence could not be overridden in = mathematical systems such as spreadsheets where the conventions are differe= nt.)

The second solution is to use explicit means to specify the precedence = of operators. This approach is fully general, but clumsy, and will be expla= ined in Section 20.

The third solution is simple but not very general. When an operator is = prefixed with the cmavo =93bi'e=94 (of selma'o BIhE), it becomes automatica= lly 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 =D7 5 =3D 23

-is a truthful Lojban bridi. If more than one operator has a =93bi'e=94 pre= fix, grouping is to the right; multiple =93bi'e=94 prefixes on a single ope= rator are not allowed. +is a truthful Lojban bridi. If more than one operator has a =93bi'e=94 pre= fix, grouping is from the right; multiple =93bi'e=94 prefixes on a single o= perator are not allowed.

In addition, of course, Lojban has the mathematical parentheses =93vei= =94 and =93ve'o=94, which can be used just like their written equivalents = =93(=94 and =93)=94 to group expressions in any way desired:

 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 (=93n=94 plus one) times (=93n=94 plus one)
             equals the-number n-power-two plus two-times-=93n=94 plus 1.
        (n + 1)(n + 1) =3D n² + 2n + 1

There are several new usages in Example 5.10: = =93te'a=94 means =93raised to the power=94, and we also see the use of the = lerfu word =93ny=94, representing the letter =93n=94. In mekso, letters sta= nd for just what they do in ordinary mathematics: variables. The parser wil= l accept a string of lerfu words (called a =93lerfu string=94) as the equiv= alent of a single lerfu word, in agreement with computer-science convention= s; =93abc=94 is a single variable, not the equivalent of =93a =D7 b =D7 c= =94. (Of course, a local convention could state that the value of a variabl= e like =93abc=94, with a multi-lerfu name, was equal to the values of the v= ariables =93a=94, =93b=94, and =93c=94 multiplied together.)

The explicit operator =93pi'i=94 is required in the Lojban verbal form = whereas multiplication is implicit in the symbolic form. Note that =93ve'o= =94 (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 =93bi'e= =94 (also marked by square brackets) is not necessary to get the proper gro= uping, but it is included here for symmetry with the other one.

diff --git a/18/6/index.html b/18/6/index.html index a2e61ff..241ba9c 100644 --- a/18/6/index.html +++ b/18/6/index.html @@ -38,21 +38,21 @@ fy. BY letter =93f=94

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 exac= tly 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 h= ave a variable number of operands. The preferred form of expression in such= cases is the use of =93forethought operators=94, also known as Polish nota= tion. In this style of writing mathematics, the operator comes first and th= e 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) =3D 6

Note that the normally elidable number terminator =93boi=94 is required af= ter =93pa=94 and =93re=94 because otherwise the reading would be =93pareci= =94 =3D 123. It is not required after =93ci=94 but is inserted here in brac= kets for the sake of symmetry. The only time =93boi=94 is required is, as i= n Example 6.1, when there are two consecutive numb= ers 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 amb= iguous circumstances? The usual Lojban solution is employed: an elidable te= rminator, namely =93ku'e=94. Here is an example:

Forethought mekso can use any number of operands, in Example 6.1, three. How do we know how many operands there are in am= biguous circumstances? The usual Lojban solution is employed: an elidable t= erminator, namely =93ku'e=94. Here is an example:

 6.2)   li py. su'i va'a ny. ku'e su'i zy du l=
i xy.
        The-number =93p=94 plus negative-of(=93n=94) plus =93z=94 equals th=
e-number =93x=94.
        p + -n + z =3D x

where we know that =93va'a=94 is a forethought operator because there is n= o operand preceding it.

=93va'a=94 is the numerical negation operator, of selma'o VUhU. In cont= rast, =93vu'u=94 is not used for numerical negation, but only for subtracti= on, as it always has two or more operands. Do not confuse =93va'a=94 and = =93vu'u=94, which are operators, with =93ni'u=94, which is part of a number= .

In Example 6.2, the operator =93va'a=94 and th= e terminator =93ku'e=94 serve in effect as parentheses. (The regular parent= heses =93vei=94 and =93ve'o=94 are NOT used for this purpose.) If the =93ku= 'e=94 were omitted, the =93su'i zy=94 would be swallowed up by the =93va'a= =94 forethought operator, which would then appear to have two operands, =93= ny=94 and =93su'i zy.=94, where the latter is also a forethought expression= .

Forethought mekso is also useful for matching standard functional notat= ion. How do we represent =93z =3D f(x)=94? The answer is:

diff --git a/18/7/index.html b/18/7/index.html
index c85ff7f..f8afc00 100644
--- a/18/7/index.html
+++ b/18/7/index.html
@@ -35,21 +35,21 @@
      du          x1 is identical to x2, x3, x4, ...
      dunli       x1 is equal/congruent to x2 in/on property/quality/dimens=
ion/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, equa=
l or less]
      dubjavmau   x1 is greater than or equal to x2      [du ja zmadu, equa=
l or greater]
      tamdu'i     x1 is similar to x2                    [tarmi dunli, shap=
e-equal]
      turdu'i     x1 is isomorphic to x2                 [stura dunli, stru=
cture-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, numb=
er-near]
+     na'ujbi     x1 is approximately equal to x2        [namcu jibni, numb=
er-near]
      terci'e     x1 is a component with function x2 of system x3

Note the difference between =93dunli=94 and =93du=94; =93dunli=94 has a th= ird place that specifies the kind of equality that is meant. =93du=94 refer= s to actual identity, and can have any number of places:

 7.2)   py. du xy.boi zy.
        =93p=94 is-identical-to =93x=94 =93z=94
        p =3D x =3D z

Lojban bridi can have only one predicate, so the =93du=94 is not repeated.

Any of these selbri may usefully be prefixed with =93na=94, the contrad= ictory negation cmavo, to indicate that the relation is false:

--=20 You received this message because you are subscribed to the Google Groups "= BPFK" group. To post to this group, send email to bpfk-list@googlegroups.com. To unsubscribe from this group, send email to bpfk-list+unsubscribe@googleg= roups.com. For more options, visit this group at http://groups.google.com/group/bpfk-l= ist?hl=3Den.