A GRAMMAR OF NEW ITHKUIL

A CONSTRUCTED LANGUAGE

Introduction	4 Case Morphology	8 Adjuncts	12 The Writing System
1 Phonology	5 Verb Morphology	9 Referentials	13 Numbers
2 Morpho-Phonology	6 More Verb Morphology	10 Special Constructions	14 The Lexicon
3 Basic Morphology	7 Affixes	11 Syntax	Appendices

13.0 NUMBERS

The New Ithkuil system of numbers and counting is distinct from Western languages in two fundamental ways: it is centesimal (base one hundred) as opposed to decimal (base ten), and the numbers themselves are full formatives (i.e., nouns and verbs), not adjectives.

13.1 Features of a Centesimal Number System

Being a centesimal system of enumeration, the numbers from zero to 100 are considered autonomous units represented by single stems and written using single autonomous symbols. Beginning with the number 101, numbers are referred to by the number of hundreds plus the number of units, just as a decimal system, beginning with the number 11, refers to the number of tens plus the number of units. However, where a decimal system then shifts to a unit referring to 100 once “10 tens” is reached, a centesimal system proceeds to the number 10,000 before establishing a new unit reference (i.e., “100 hundreds”). Thus the number 3254, which in a decimal system is 3 thousands — 2 hundreds — 5 tens — 4 ones, in a centesimal system becomes 32 hundreds—54 ones, and would be only two digits when written (the single character representing 32, and the single character representing 54). The details of writing Ithkuil numerals are given below in Sec. 13.3.

After 100, separate unit numbers and symbols are assigned to the square of 100 (i.e. ten thousand, that being “100 hundreds”), then the square of that number, 100⁴ (100 million, i.e., 10,000 ten-thousands). The final unit is 100⁸, that is, 10 quadrillion or 100 million hundred-millions, the last number for which the language assigns a separate root and symbol. After ten quadrillion, numbers are referred to as multiples of lower sets, similar to saying in English “one trillion quadrillion” instead of the equivalent “one octillion.”

While the above system may seem awkward, it actually parallels Western base-ten numerals in terms of its systematization. For example, in a Western number like 456,321,777,123, each set of three numbers between the commas tells how many hundreds there are of a certain power of 1000 (i.e., there are 123 of 1000⁰, 777 of 1000¹, 321 of 1000², and 456 of 1000³, or in more common terms 123 ones, 777 thousands, 321 millions, 456 billions). The same exact system holds for New Ithkuil, except that the sets of numbers “between the commas” so to speak, is the number of ten-thousands, not thousands. Thus, if we were to rewrite the Western number 456,321,777,123 in such a system, it would be 4563,2177,7123 (i.e., 7123 of 10000⁰, 2177 of 10000¹, and 4563 of 10000², that being 7123 ones, 2177 ten-thousands, and 4563 hundred-millions).

13.2 Semantic Designations for Numerical Stems

The semantic roots for numbers in Ithkuil from 1 to 99 are based on roots for 1 through 10, to which the nine degrees of the TNX affix -rs are added. Each of the nine degrees of this suffix, when applied to one of the ten number-roots, corresponds to an additional multiple of ten.

0	1	2	3	4	5	6	7	8	9	10	100	100^2	100^4	100^8
-VR-	-LL-	-KS-	-Z-	-PŠ-	-ST-	-CP-	-NS-	-ČK-	-LẒ-	-J-	-GZ-	-PC-	-KẒ-	-ČG-

The following six number roots are used when needed to designate numbers beyond ten when needed for counting and mathematical operations involving non-decimal number bases up to base-16. They may also be used as “short-cut” substitutes for the standard decimal/centesimal forms using the TNX affix.

11	12	13	14	15
-CG-	-JD-	-ĻJ-	-BC-	-ŢẒ-

-rs	TNX Multiples of Ten (used with the number roots 0 thru 9 to create the numbers 11 through 99)
1	X plus 10
2	X plus 20
3	X plus 30
4	X plus 40
5	X plus 50
6	X plus 60
7	X plus 70
8	X plus 80
9	X plus 90

Whole numbers are full formatives signifying a set containing the particular number of members. The “simple” everyday counting system is base-100 (the mathematical sub-language will utilize base-12). Beginning with ‘two’, the Stem & Specification pattern is illustrated by the root -Z- ‘three’ below:

-Z- ‘THREE / TRINARY’ Associated Affix: 3XX
	STEM 1	STEM 2	STEM 3
BSC	(to be a) set or group of three entities; (to be) a trio	(to be) something manifesting three aspects / facets; to manifest trinariness; be trinary	(to be) the third entity/party in a group or sequence
CTE	(to be) a party/entity of whom/which there are three	(to be) the state of having three aspects/facets; to be trinary; to be tri-fold or tri-faceted	(to be) the state of being third in a sequence/group/pattern
CSV	(to be) a process which determines/identifies a set as being three in number; to count out to three; to determine that there are three of something	(to be) a process which determines/identifies an entity as having three aspects/facets; identify/determine that something is trinary/tri-fold/tri-faceted	(to be) a process which determines/identifies an entity’s sequential place in a sequence or group/pattern to be third
OBJ	(to be) one in a group or sequence of 3; to be one of 3	(to be) one of the aspects/facets of a trinary, tri-fold, tri-faceted entity	(to be) the entity/party whose numerical place in a sequence/group/pattern is third

Numbers from 11 through 99 are formed utilizing the TNX affix. Beginning with the number 101, numbers are formed as in Ithkuil-2011 using the comitative case and the coo affix. Having no multiples, the roots for ‘ZERO’ and ‘ONE’ have a different Stem & Specification pattern:

-VR- ‘ZERO / NULL’
	STEM 1	STEM 2	STEM 3
BSC	(to be) zero as the empty-set / a set having no members; to have no quantity or amount	(to be) the zero-dimension; to have geometrically no length, area or volume	(to be) the baseline “zero”-state or null-state in a sequence, hierarchy, gradient, pattern, etc.
CTE	(to be) a party/entity of whom/which there are no members	(to be) the state of having no substance/tangibility due to being zero-dimensional	(to be) the state of being the baseline “zero”-state or null-state
CSV	(to be) a set having no members; to have no (i.e., zero) members in a set	(to be) the process/act of determining/identifying zero-dimensionality	(to be) a process which determines/identifies an entity’s being the baseline “zero”-state or null-state
OBJ	(to be) a null value / a value for a parameter that is undefined and/or for which the expected or standard value(s) is/are inapplicable	(to be) an entity having zero-dimensionality; (to be) a Euclidean point; to have geometrically no length, area or volume, i.e., to be a Euclidean point	(to be) the entity/party in the baseline “zero”-state or null-state in a sequence, hierarchy, gradient, pattern, etc.

-LL- ‘ONE / UNITY’
	STEM 1	STEM 2	STEM 3
BSC	(to be) a set or group of one; to have one member	(to be) something (quasi-)indivisible, (quasi-)inseparable, unified, unitary, united, a union, a unit	(to be) the first entity/party in a group or sequence
CTE	(to be) a party/entity of whom/which there is only one	(to be) the state of having only one functional aspect/facet; to function/manifest as a unified whole or unit	(to be) the state of being first in a sequence/group/pattern
CSV	(to be) a process which determines/identifies a set as being one in number; to count out to one; to determine that there is only one of something	(to be) a process which determines/identifies an entity as having only one functional aspect/facet; to determine that an entity is a (quasi-)indivisible whole/unit	(to be) a process which determines/identifies an entity’s sequential place in a sequence or group/pattern to be first
OBJ	[same as CTE]	(to be) the party/entity having only one functional aspect/facet; to be an entity which functions/manifests as single unit	(to be) the entity/party whose numerical place in a sequence/group/pattern is first

The following affixes are based on the number roots; the meanings of their nine degrees are shown on the right. (Add a Type-3 TNX affix from above to create words such as “a set of thirty-four cats”.

-ks	XX2* [two]	1	being the #th member of a set per sequential physical arrangement
-z	XX3* [three]	2	being the #th member of a set per conventionalized/agreed-upon order
-pš	XX4* [four]	3	being the #th member of a set per hierarchical order
-st	XX5* [five]	4	being the #th member of a set per contextual/circumstantial order*
-cp	XX6* [six]	5	being/having #-number of members or instances/occurrences
-ns	XX7* [seven]	6	being/having at least #-number of members or instances/occurrences
-čk	XX8* [eight]	7	being/having #-number of parts/sections
-lẓ	XX9* [nine]	8	being/having #-number of nodes/hubs/connections/access points
-j	X10* [ten]	9	being/having #-number of hierarchical levels/tiers
-gz	XOH* [one hundred]	* i.e., the #th member of a set that does something or that something happens to
-pc	XTT* [ten thousand]
-kẓ	XTM* [100^4]	In addition to the above affixes based on number roots, the roots for ‘ZERO’ and ‘ONE’ also have affixes associated with the degree patterns to the right, to provide a means for saying, for example, “(set of) thirty-one cats” or “device having twenty parts”.
-čg	XTQ* [100^8]
-cg	X11* [eleven] (used in the context of a number-base higher than ten)
-jd	X12* [twelve] (used in the context of a number-base higher than ten)
-ļj	X13* [thirteen] (used in the context of a number-base higher than ten)	-zc	XX1 [one]
-bc	X14* [fourteen] (used in the context of a number-base higher than ten)	-vr	XXZ [zero]
-ţẓ	X15* [fifteen] (used in the context of a number-base higher than ten)
-vj	UHN Uncountably High Number (e.g., “zillions”, “a myriad of”, etc.)

13.3 Writing Numerals

When writing Ithkuil numerals, the core of the numeral is one of the ten characters for numbers 0 through 9, to which written extensions to the character indicate the number of tens to be added and/or the number of hundreds to be added. Diacritic marks then indicate the number of thousands (up to 9000) to b added. Consequently, written numbers from zero to 9999 are a single (composite) character. Numbers from 10,000 to 104 are similarly represented by a second number placed in front of the core “thousands” number.

Extensions to the bottom-left indicate the number of tens:

Extensions to the top-right of the above symbols indicate the number of hundreds:

Diacritics placed inside the top-left quarter of the 1-through-10 symbols indicate the number of thousands up to 9000:

Examples:

13.4 Using Numbers In Speech

Spoken numbers are formed from the above stems using both the PARTITIVE and COMITATIVE cases, as well as using the coordinative affix -Vň/1 (= -iň). The number of largest base units is shown by placing the base-unit term in the PARTITIVE. If this is then followed by another collection of smaller base units, that number of smaller base units is connected using the COMITATIVE case while the smaller base-unit term is again in the PARTITIVE. Single units (from 1 to 99) are connected by the coordinative affix when they are part of the number of hundreds or higher base-units.

It should be noted that when pronouncing numbers greater than 199, it is normal to omit the word gzalui (= the PARTITIVE of gzal ‘one hundred’) referring to the number of hundreds as long as this does not lead to ambiguity or a lack of clarity. This is equivalent to the custom in colloquial English of saying ‘three twelve’ for ‘three hundred (and) twelve.’ Examples:

ksalirsa  (gzalui)  walẓärs
literally: “42 (of hundreds) 29”
4229

cpalärsa  wapcui  wansorsë’i  (gzalui)  cpalörs
literally: “26 of ten-thousands with 97 (of hundreds) 66” = 26,9766
269,766

wallärsa  gzalui  wapcui
literally: “21 of hundred of ten-thousands”
21,000,000
[NOTE:  gzalui  is required in this example]

ksalorsa  gzalui  walẓorsiň  wakẓui  za’lëi  gzalui  zalëirsiň  wapcui  pša’lersëi  vralörs
literally: 
“72 of hundreds and 79 of hundred-millions with 3 of hundreds and 53 of ten-thousands with 3460” 
727,903,533,460

13.5 Dates and Times of Day

The SPT Affix is used in expressing the hour of day, day of the week, week of the month, month of the year, the year and the century. It is used with the number roots (usually Stem 3) to render, e.g., ‘the eighth hour of the day’, ‘the third day of the week (i.e., Wednesday)’ or ‘20th of May’, etc. Furthermore, each use of this affix can in turn be modified by a following Type-3 number affix (e.g., XX2, XX3, etc.) to enumerate the higher-ordered time-period named by the affix. For example, for the word wuksärsëirwa ‘22nd day of the month’, the SPT/5 affix -ëirw- can in turn be modified by a following Type-3 number affix, e.g., wuksärsëirwiasta ‘22nd of May’. Other Type-3 affixes may also be used in the same fashion, as per the third example below.

-rw/-ry	SPT Specified Points in Calendrical Time	Examples: · ‘the 15^th of March, 1969’ wustarsëirwiaza walẓarsa’o walẓörsürwë’i · ‘on Saturday’ wucpirwa’o · ‘on Saturday of next week’ wucpirwölţa’o · ‘the 21st century’ wullärsurya · ‘by the 34-second mark’ wupšersaryo’a Time of Day Using Degree 3 of the affix: · ‘8:52 a.m.’ wučkerwa ksalëirsoň [Note the use of the COO/7 affix on the 2nd word; the phrase is literally ‘eighth hour of the day and fifty-two (minutes)’ with the SPT/3 affix on the first word implying the possibility of a following number of minutes] · ‘8:52 p.m. and 33 seconds’ wuvrärserwa ksalëirsoň wazersarwë’i [literally: ‘twentieth hour of the day and fifty-two (minutes) with thirty-three seconds of a minute’]
1	second(s) of a/the minute
2	minute(s) of an/the hour
3	hour [and minutes] of the day, i.e., time of day
4	day of the week [1st day of week = Monday]
5	day of the month
6	week of the month
7	month of the year
8	year
9	century

13.5.1 Alternate Names of the Months

Name the first 4 months is via Degrees 1 through 4 of the Type-2 SEQ affix (-nt) attached to Stem No. 3 of -RḐ- (meaning ‘calendrical month’) to render words meaning ‘first month’, ‘second month’, ‘third month’, ‘fourth month’. Likewise, the last four months may utilize Degrees 6 through 9 of the same affix.

For the remaining months (and as alternates for the first four and last four months), use Degree 2 of the Type-2 XX(#) affixes. (Use non-decimal number base roots -CG- and -JD- for ‘11’ and ‘12’.) Thus:

January: wurḑainta / wurḑauzca May: wurḑaust September: wurḑounta / wurḑaulẓa

February: wurḑaunta / wurḑauks June: wurḑaucpa October: wurḑointa / wurḑauj

March: wurḑeinta / wurḑauz July: wurḑauns November: wurḑiunta / wurḑaucga

April: wurḑeunta / wurḑaupš August: wurḑaučka December: wurḑuinta / wurḑaujda

Introduction	4 Case Morphology	8 Adjuncts	12 The Writing System
1 Phonology	5 Verb Morphology	9 Referentials	13 Numbers
2 Morpho-Phonology	6 More Verb Morphology	10 Special Constructions	14 The Lexicon
3 Basic Morphology	7 Affixes	11 Syntax	Appendices