History: BPFK Section: MEX Operators

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Notes

  • When an expression is here said to be "undefined," it means that, in the absence of external information about the interpretation of the expression, it is semantically void, meaningless, and/or nonsensical.
  • Mathematical operators take one or more operands, which are here referred to by an 'x' subscripted with a number.
  • Operators defined here as "unary" are defined as operating on their first operand that is not equal to tu'o (referred to as "the operand" within the respective definitions), regardless of where it occurs in the operand list.
    • See also: CLL section 18.16, example 16.3 and the paragraph before it, in which the operand of va'a is in the x_1 place; CLL section 18.14, example 14.1, in which the operand of va'a is in the x_2 place.
  • If all operands of an operator are tu'o, the value of the expression is equal to tu'o.
  • If an operand described in an operator's definition is missing from the operator's operand list and there is no default value given, the missing operand shall be interpreted as tu'o.
  • If one or more extra operands not described in an operator's definition are present in the operator's operand list, they affect the evaluation in an undefined or unspecified manner analogous to sumti attached to a selbri with do'e.
  • Lojban mathematical operators are defined in terms of conventional mathematical operators, and thus there is an implicit type system used in determining the value of an expression. Mekso operands may be of any of the following types:
    • number — a "number" production in the formal grammar, usually corresponding to an element of the complex plane
    • array — an array, vector, ordered list/set, or tuple of one or more mekso values, constructed using jo'i
    • matrix — a composition of one or more arrays of numbers, all of the same size, constructed using pi'a or sa'i
    • function
  • If an operator is applied to operands of a type which its definition does not address, the expression is undefined.
  • The operand tu'o is polymorphic; its type in an expression is determined by its surrounding context.
  • Operator-specific type information (currently in the "Notes" subsections) needs to be included in the main definitions somehow without being too clunky.


Proposed Definitions And Examples


fu'u (VUhU0)

Current definition


unspecif operator — n-ary mathematical operator: elliptical/unspecified mathematical expression (mex) operator -

Proposed definition


elliptical/unspecified mathematical operator

Proposed keyword

  • unspecified operator

Examples of fu'u Usage

No examples in CLL

li pa fu'u pa du li re
1 'something' 1 is 2

Notes

  • This is the semantic analogue of zo'e, co'e, et alii.

ge'a (VUhU0)

Current definition

ge'a (VUhU0)
null operator — mathematical operator: null mathematical expression (mex) operator (used in >2-ary ops) --

Proposed definition


A null operator, in the sense that it returns all its arguments unchanged. Used as syntactic glue in poly-ary operators, such as to pass three arguments to a ternary operator in infix notation.

Proposed keyword

  • null operator

See Also

  • {tu'o} Null operand

Examples of ge'a Usage

papano bi'eju'u re gei pipanopano bi'eju'u re ge'a re
.1010 (base 2) × 2 ^ 110 (base 2) (CLL, 18.14.4)

fe'i (VUhU1)

Current definition


divided by — n-ary mathematical operator: divided by; division operator; (((a / b) / c) / ...) --

Proposed definition


The operation of division. Divides the first argument by all subsequent ones in a left associative manner: x1 / x2 / x3 ...

Proposed keyword

  • Division
  • Divides

Notes

  • The operands of fe'i are numbers, and a fe'i expression evaluates to a number.
  • When fe'i has only one operand, it is the identity function.
  • Distinct from the fraction slash {fi'u} in that it is an operator (and not of selma'o PA).

Examples of fe'i Usage

li na'u tanjo te'u vei pai fe'i vo ve'o du li pa
tan( pi / 4 ) = 1 (CLL, variation of 18.18.1)

pi'i (VUhU1)

Current definition


times — n-ary mathematical operator: times; multiplication operator; (((a * b) * c) * ...) --

Proposed definition

The operation of multiplication. Multiplies all the arguments together in a left associative manner: x1 * x2 * x3 ...

Proposed keywords

  • multiplication
  • multiply
  • times

Notes

  • When reducing a pi'i expression from left to right, the types of the two operands considered at each point must together take one of the following forms:
    • number and number — standard mathematical multiplication; evalutes to a number
    • number and matrix or matrix and number — scaling the elements of a matrix by a factor; evaluates to a matrix
    • number and array of numbers or array of numbers and number — scaling the elements of an array by a factor; evaluates to an array of numbers
    • matrix and matrix — standard matrix multiplication; the number of columns in the first matrix must equal the number of rows in the right matrix, otherwise the result is undefined; evaluates to a matrix
    • matrix and array of numbers — The operator sa'i is implicitly applied to the array, and the expression is then evaluated as a matrix-matrix multiplication, yielding a one-column matrix that is then implicitly converted to an array of the numbers in its column.
  • When pi'i has only one operand, it is the identity function.

Examples of pi'i Usage

li ci su'i vo pi'i mu du li cimu
3 + 4 × 5 = 35 (CLL, 18.5.8)

su'i (VUhU1)

Current definition

su'i (VUhU1)
plus — n-ary mathematical operator: plus; addition operator; (((a + b) + c) + ...) --

Proposed definition

The operation of addition. Adds all the arguments together in a left associative manner: x1 + x2 + x3 ...

Proposed keywords

  • plus
  • add
  • addition

See Also

  • {ma'u}

Notes

  • {su'i} is the operation of addition, addition from the plus sign that indicates a positive number {ma'u}

Examples of su'i Usage

li pa su'i pa du li re
1 + 1 = 2 (CLL, 18.5.1)
li ma'u pa su'i ni'u pa du li no
+1 + -1 = 0 (CLL, 18.5.5)

vu'u (VUhU1)

Current definition

minus — n-ary mathematical operator: minus; subtraction operator; (((a - b) - c) - ...) --

Proposed definition

The operation of subtraction. Takes the first argument as subtracts all subsequent ones in a left associative manner: x1 - x2 - x3 ...

Proposed keyword

  • subtract
  • minus

See Also

  • va'a
  • ni'u

Notes

This is the operator for subtraction. It is distinct from {va'a} which is the operator for negation, and also from {ni'u} which is a minus sign and is part of the number

Examples of vu'u Usage

li ci vu'u re du li ma'u
the-number 3 - 2 = some-positive-number (CLL, 18.8.14)
li tu'o va'a ny. du li no vu'u ny.
-n = 0 - n (CLL, 18.14.1)

fa'i (VUhU2)

Current definition

fa'i (VUhU2)
reciprocal of — unary mathematical operator: reciprocal; multiplicative inverse; 1/a --

Proposed definition

The operation of inverting. It is a unary operation. It returns the reciprocal of the given number: 1 / x1

Proposed keyword

  • inverse
  • reciprocal

See Also

  • {fi'u}

Notes

This is performs similar to {fi'u}. There are differences though. This is an operator and thus may take a mekso operand as an operator, whereas {fi'u} is part of PA and so is part a number.

Examples of fa'i Usage

li fa'i re du li re fa'i li fi'u re
Reciprocal of 2 = 2 inverse = 1/2

gei (VUhU2)

Current definition


exponential notation — trinary mathematical operator: order of magnitude/value/base; b * (c to the a power) --

Proposed definition

This operator is a trinary operator that mimics scientific notation. It is designed so that arguments can be omitted to give magnitudes quickly. gei x1 x2 x3 = x_2 * (x_3 ^ x_1). x_3 defaults to 10; so gei x1 x2 = x2 * 10 ^ x1. x_2 defaults to 1; so gei x1 = 10 ^ x1

Proposed keyword

  • scientific notation

Notes

  • The operands of gei are numbers, and a gei expression evaluates to a number.

Examples of gei Usage

li cinonoki'oki'o du li bi gei ci
300,000,000 = 3 * 10^8 (CLL, 18.14.2)

gei reno
10^20 (CLL, 18.14.3)

papano bi'eju'u re gei pipanopano bi'eju'u re ge'a re
.1010_2 * 2 ^ 110_2 (CLL, 18.14.4)

ju'u (VUhU2)

Current definition

ju'u (VUhU2)
number base — binary mathematical operator: number base; a interpreted in the base b --

Proposed definition

A binary operator that indicates the number base of its first argument: x1 is in base x2

Proposed keyword

  • base
  • number base
  • radix

Notes

For bases up to 16 Lojban has numerals you can use. But for bases greater than 16, use {pi'e} to separate the place values. {pi} is used as radix point in any base.

Examples of ju'u Usage

li pa no pa no ju'u re du li pa no
1010 base 2 = 10 (CLL, 18.10.1)

li daufeigai ju'u paxa du li rezevobi
ABC base 16 = 2748 (CLL, 18.10.2)

li pa pi'e re pi'e ci ju'u reno du li vovoci
1;2;3 base 20 = 443 (CLL, 18.10.7)

li pa pi'e vo pi ze ju'u reno du li re vo pi ci mu
1;4.7 base 20 = 24.35 (CLL, 18.10.10)

pa'i (VUhU2)

Current definition

pa'i (VUhU2)
ratio — binary mathematical operator: ratio; the ratio of a to b --

Proposed definition

A binary operator that returns the ratio of its two arguments: the ratio of x1 to x2.

Proposed keyword

  • ratio

Notes

No examples in CLL and none turned up searching lojban.org

Examples of ju'u Usage

li pa pa'i pa du li mo'e lo ninmu pa'i mo'e lo nanmu
the ratio 1 to 1 is equal to the ratio of men to woman

te'a (VUhU2)

Current definition

te'a (VUhU2)
to the power — binary mathematical operator: to the power; exponential; a to the b power --

Proposed definition

The binary operator of exponentiation: x1 to the power of x2

Proposed keyword

  • exponential
  • power

Examples of te'a Usage

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
(n + 1)(n + 1) = n^2 + 2n + 1 (CLL, 18.5.10)

cu'a (VUhU3)

Current definition

cu'a (VUhU3)
absolute value — unary mathematical operator: absolute value/norm |a| --

Proposed definition

The unary operator of taking an absolute value or norm: the absolute value of x1

Proposed keyword

  • absolute value
  • norm

Examples of cu'a Usage

li cu'a ni'u re du li re
|-2| = 2



Proposed Definition of de'o

de'o (VUhU3)
logarithm — binary mathematical operator: logarithm; log/ln a to base b; default base 10 or e --
    • Keywords:

Examples of de'o Usage

Proposed Definition of fe'a

fe'a (VUhU3)
nth root of — binary mathematical operator: nth root of; inverse power a to the 1/b power --
    • Keywords:

Examples of fe'a Usage

Proposed Definition of ne'o

ne'o (VUhU3)
factorial — unary mathematical operator: factorial; a! --
    • Keywords:

Examples of ne'o Usage

Proposed Definition of va'a

va'a (VUhU3)
additive inverse — unary mathematical operator: additive inverse; - a --
    • Keywords:

Examples of va'a Usage

Proposed Definition of pi'a

pi'a (VUhU4)
matrix of rows — n-ary mathematical operator: operands are vectors to be treated as matrix rows --
    • Keywords:

Examples of pi'a Usage

Proposed Definition of re'a

re'a (VUhU4)
transpose — unary mathematical operator: matrix transpose/dual; A* --
    • Keywords:

Examples of re'a Usage

Proposed Definition of ri'o

ri'o (VUhU4)
integral — trinary mathematical operator: integral of a with respect to b over range c --
    • Keywords:

Examples of ri'o Usage

Proposed Definition of sa'i

sa'i (VUhU4)
matrix of columns — n-ary mathematical operator: operands are vectors to be treated as matrix columns --
    • Keywords:

Examples of sa'i Usage

Proposed Definition of sa'o

sa'o (VUhU4)
derivative — trinary mathematical operator: derivative of a with respect to b of degree c --
    • Keywords:

Examples of sa'o Usage

Proposed Definition of si'i

si'i (VUhU4)
sigma summation — trinary mathematical operator: sigma summation of a using variable b over range c --
    • Keywords:

Examples of si'i Usage

Impact

History

Advanced
Information Version
Thu 21 of Aug, 2014 15:19 GMT Ilmen from 95.210.209.219 Fixing a typo 37
Thu 21 of Aug, 2014 15:17 GMT Ilmen from 95.210.209.219 Template + notes 36
Sun 08 of Jun, 2014 19:30 GMT mukti from 216.194.27.154 35
Tue 19 of Oct, 2010 01:02 GMT lindarthebard from 32.172.136.135 34
Wed 15 of Sep, 2010 08:12 GMT RossOgilvie from 210.84.42.221 Wrote up impact section 33
Wed 15 of Sep, 2010 07:48 GMT RossOgilvie from 210.84.42.221 Tidy up and some standardisation 32
Mon 13 of Sep, 2010 02:32 GMT RossOgilvie from 210.84.4.4 31
Mon 13 of Sep, 2010 02:32 GMT RossOgilvie from 210.84.4.4 sa'o 30
Sat 11 of Sep, 2010 05:16 GMT RossOgilvie from 220.239.170.184 29
Sat 11 of Sep, 2010 04:54 GMT RossOgilvie from 220.239.170.184 ri'o 28
Fri 10 of Sep, 2010 10:47 GMT RossOgilvie from 210.84.4.4 27
Fri 10 of Sep, 2010 10:36 GMT RossOgilvie from 210.84.4.4 26
Fri 10 of Sep, 2010 08:29 GMT RossOgilvie from 210.84.4.4 va'a 25
Fri 10 of Sep, 2010 08:10 GMT RossOgilvie from 210.84.4.4 ne'o 24
Fri 10 of Sep, 2010 06:00 GMT RossOgilvie from 220.239.170.184 fe'a 23
Fri 10 of Sep, 2010 05:50 GMT RossOgilvie from 220.239.170.184 de'o 22
Fri 10 of Sep, 2010 05:46 GMT RossOgilvie from 220.239.170.184 21
Thu 09 of Sep, 2010 07:23 GMT RossOgilvie from 210.84.4.4 did te'a 20
Thu 09 of Sep, 2010 07:18 GMT RossOgilvie from 210.84.4.4 19
Thu 09 of Sep, 2010 07:11 GMT RossOgilvie from 210.84.4.4 did pa'i 18
Thu 09 of Sep, 2010 05:01 GMT RossOgilvie from 172.17.36.41 did ju'u 17
Thu 09 of Sep, 2010 04:42 GMT RossOgilvie from 172.17.36.41 added a keyword to gei 16
Thu 09 of Sep, 2010 04:40 GMT RossOgilvie from 172.17.36.41 Several definitions 15
Wed 08 of Sep, 2010 04:01 GMT RossOgilvie from 172.17.36.37 Did fe'i 14
Tue 07 of Sep, 2010 14:30 GMT RossOgilvie from 220.239.170.184 Did ge'a 13
Tue 07 of Sep, 2010 14:18 GMT RossOgilvie from 220.239.170.184 Added boxes to the page 12
Tue 07 of Sep, 2010 14:13 GMT RossOgilvie from 220.239.170.184 Did vu'u 11
Wed 16 of Jun, 2010 14:02 GMT RossOgilvie from 220.239.170.184 Repealed earlier change (don't know what I was thinking) and minor formatting of CLL refererences 10
Sun 18 of Apr, 2010 14:17 GMT RossOgilvie from 220.239.170.184 9
Wed 14 of Apr, 2010 02:12 GMT Minimiscience from 129.21.72.20 Put boxes around the edited cmavo to make them stand out more 8
Wed 14 of Apr, 2010 02:05 GMT Minimiscience from 129.21.72.20 Described fe'i and pi'i 7
Wed 14 of Apr, 2010 01:49 GMT Minimiscience from 129.21.72.20 Added in a forgotten subsection heading 6
Wed 14 of Apr, 2010 01:48 GMT Minimiscience from 129.21.72.20 Began describing the implicit type system 5
Wed 14 of Apr, 2010 01:28 GMT Minimiscience from 129.21.72.20 Specified fu'u 4
Wed 14 of Apr, 2010 01:21 GMT Minimiscience from 129.21.72.20 Fixed formatting 3
Wed 14 of Apr, 2010 01:17 GMT Minimiscience from 129.21.72.20 Descibed "gei" and some general things about VUhU 2
Wed 17 of Nov, 2004 02:47 GMT rlpowell from 64.81.49.171 1