A GRAMMAR OF NEW ITHKUIL
A CONSTRUCTED
LANGUAGE
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, 100^{4} (100
million, i.e., 10,000 tenthousands). The final unit is 100^{8}, that
is, 10 quadrillion or 100 million hundredmillions, 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 baseten 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 1000^{0},
777 of 1000^{1}, 321 of 1000^{2}, and 456 of 1000^{3},
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 tenthousands, 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 10000^{0}, 2177 of 10000^{1}, and 4563 of 10000^{2},
that being 7123 ones, 2177 tenthousands, and 4563 hundredmillions).
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 numberroots,
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 nondecimal number bases up to base16. They may also be used as “shortcut” 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 base100 (the mathematical sublanguage will utilize base12). 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 trifold or trifaceted 
(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/trifold/trifaceted 
(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, trifold, trifaceted 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 Ithkuil2011 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 emptyset
/ a set having no members; to have no quantity or amount 
(to be) the zerodimension; to
have geometrically no length, area or volume 
(to be) the baseline “zero”state or
nullstate 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 zerodimensional 
(to be) the state of being the
baseline “zero”state or nullstate 
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 zerodimensionality 
(to be) a process which
determines/identifies an entity’s being the baseline “zero”state or
nullstate 
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
zerodimensionality; (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 nullstate 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 Type3 TNX affix from above to create
words such as “a set of thirtyfour 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/agreedupon 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) thirtyone cats” or “device having twenty
parts”. 


čg 
XTQ* [100^{8}] 



cg 
X11* [eleven] (used in the
context of a numberbase higher than ten) 



jd 
X12* [twelve] (used in the
context of a numberbase higher than ten) 



ļj 
X13* [thirteen] (used in the
context of a numberbase higher than ten) 

zc 
XX1
[one] 

bc 
X14* [fourteen] (used in the
context of a numberbase higher than ten) 

vr 
XXZ
[zero] 

ţẓ 
X15* [fifteen] (used in the context
of a numberbase 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 bottomleft indicate the number of
tens:
Extensions to the
topright of the above symbols indicate the number of hundreds:
Diacritics placed
inside the topleft quarter of the 1through10 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 baseunit 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 baseunit 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 baseunits.
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
tenthousands with 97 (of hundreds) 66” = 26,9766
269,766
wallärsa gzalui wapcui
literally: “21 of hundred
of tenthousands”
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 hundredmillions with 3 of hundreds and 53 of tenthousands
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 Type3 number affix (e.g., XX2, XX3, etc.) to enumerate the higherordered timeperiod 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 Type3 number affix, e.g., wuksärsëirwiasta ‘22nd of May’. Other Type3 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 34second
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 fiftytwo
(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 fiftytwo (minutes) with thirtythree 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 Type2 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 Type2 XX(#) affixes. (Use nondecimal 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