John Woldemar Cowan

“ 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. ”

“ 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. ”

“ Since “ju'u” is an operator of selma'o VUhU, it is grammatical to use any operand as the left argument. Semantically, however, it is undefined to use anything but a numeral string on the left. The reason for making “ju'u” an operator is to allow reference to a base which is not a constant.
There are some numerical values that require a “base” that varies from digit to digit. For example, times represented in hours, minutes, and seconds have, in effect, three “digits”: the first is base 24, the second and third are base 60. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ In general, the first place of the selbri specifies the result of the operator, and the other unfilled places specify the operands:
18.1) li na'u tanjo te'u vei pai fe'i re [ve'o] du li ci'i The-number the-operator tangent ( π / 2 ) = the-number infinity. tan (π/2) = ∞ “tanjo” is the gismu for “x1 is the tangent of x2”, and the “na'u” here makes it into an operator which is then used in forethought
The cmavo “ni'e” makes a selbri into an operand. The x1 place of the selbri generally represents a number, and therefore is often a “ni” abstraction, since “ni” abstractions represent numbers. ”

“ 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. ”

“ 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 from the right; multiple “bi'e” prefixes on a single operator are not allowed. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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”. ”

“ 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. ”

“ 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. ”

“ 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:
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. ”

“ Here is a computer floating-point number with a value of 40:
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 .10102 × 21102 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. ”