Ithkuil: A Philosophical Design for a Hypothetical Language

Update: The Ithkuil Grammar Is Now Available in Book Form

Chapter 12: The Number System

The 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. This has already been discussed briefly in Section 4.5.1 regarding the PARTITIVE case. This section will examine the numerical system in greater detail.

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 Section 12.5.

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 Ithkuil 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 may seem unwieldy or even arbitrary, 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 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).

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 Slot XI derivational affix -Vk 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. This is illustrated in Table 64 below.

Table 64: The SSD Affix with Numerals

The addition of a particular degree of this affix to one of the ten indicates that the root number is added to that multiple of ten. For example, the stem ksal ‘two,’ plus the seventh degree affix -Vk/7, gives ksalok ‘seventy-two.’ Because there is no root corresponding to ‘zero’ (see Sec. 12.3 below), each multiple of ten is constructed using stem mřal ‘ten’ plus one of the above suffixes. Thus, the numbers 20, 30 and 40 are respectively mřalik, mřalök and mřalek, but the numbers 22, 32 and 42 are ksalök, ksalek, and ksalîk. This pattern only operates up to the nineties, as there is a separate autonomous root for 100, -ň-.

Since numbers are formatives in Ithkuil, not adjectives as in most Western languages, holistic stem No. 1 is a formative signifying a set containing a number of members corresponding to that particular root. Thus, the formative ksal, translatable as ‘two,’ actually means ‘a set of two; a duo / to be a duo.’ In turn, the two complementary derivatives of each stem denote its multiple and its fraction respectively. This is illustrated below for both Form I and II using the roots -ks-, TWO, and -ns-, meaning SEVEN:

For -ks-, TWO:

1. ksal / aksál ‘a set of two, a duo; to be two in number’

COMPLEMENTARY DERIVATIVES:
oksal / oksál: ‘twice the number of something; to double, to multiply by two’
âksal / âksál: ‘a half; to halve, to be or make half, to divide by or in two’

2. eksal / eksál ‘to be or make dual; having two uses or aspects; bi-; twofold’

COMPLEMENTARY DERIVATIVES:
öksal / öksál: ‘two times (i.e., iterations), twice; to be/do/make twice’
êksal / êksál: ‘to be of or make into two parts; bifurcate(d)’

3. uksal / uksál ‘the second one in a sequence; to be or make second (in a sequence)’

COMPLEMENTARY DERIVATIVES:
ûksal / ûksál: ‘to the second power, squared; to square, raise to the 2nd power’
ôksal / ôksál: ‘to the negative second power, the inverse square; to divide by the square of’

For -ns-, SEVEN:

1. nsal / ansál ‘a set/group of seven, a septet; to be seven in number’

COMPLEMENTARY DERIVATIVES:
onsal / onsál: ‘7 times the number of something; to multiply by 7; septuple’
ânsal / ânsál: ‘a seventh; to be or make a 7th part of something, to divide by 7 or into 7 parts’

2. ensal / ensál ‘to be or make seven-faceted; having 7 uses or aspects; septi-; sevenfold’

COMPLEMENTARY DERIVATIVES:
önsal / önsál: ‘7 times (i.e., iterations); to be/do/make 7 times’
ênsal / ênsál: ‘to be of or make into 7 parts; separate(d) into 7 parts’

3. unsal / unsál ‘the seventh one in a sequence; to be or make 7th (in a sequence)’

COMPLEMENTARY DERIVATIVES:
ûnsal / ûnsál: ‘to the 7th power; to raise to the 7th power’
ônsal / ônsál: ‘to the negative 7th power; to divide by the 7th power of’

In addition to the above-described roots, there is the root -ll-, ONE/UNITY. As this root has no multiples, its semantic designations follow a unique pattern. NOTE: The INFORMAL versus FORMAL distinction in this root (i.e., Form I versus Form II of each stem) distinguishes between a focus on non-duplication/singularity for the INFORMAL, and indivisibility/unity for the FORMAL:

1. llal / allál ‘a single entity; to be one in number’

COMPLEMENTARY DERIVATIVES:
ollal / ollál: ‘to be indivisible, whole, a single unit; unitary; to unify’
âllal / âllál: ‘to be (an) individual, a distinct entity in itself; to individualize’

2. ellal / ellál ‘a lone entity, something alone; an entity in solitude, something/someone isolated; be alone; to isolate; be in solitude’

COMPLEMENTARY DERIVATIVES:
öllal / öllál: ‘something/someone lonely; be or make lonely’
êllal / êllál: ‘something/someone independent, self-sufficient, singular (i.e., without need of, connection to, or dependency on others); be or make independent, self-sufficient, singular’

3. ullal / ullál ‘something/someone unique, the only one; to be or make unique’

COMPLEMENTARY DERIVATIVES:
ûllal / ûllál: ‘a sole entity, the only one available or able (in terms of sufficiency or applicability to the context)’
ôllal / ôllál: ‘something/someone one-of-a-kind, unparalleled, without equal or peer (in terms of uniqueness of characteristics)’

The Ithkuil numerical roots as described in the section above are as follows:

Ithkuil has no word for “zero” nor is it conceptualized as a numerical category. Instead any appropriate formative may take the PTW suffix in first-degree -iss ‘no amount of’ or the EXN suffix in first-degree -ib ‘no…at all’ in terms of degree or extent to create negative expressions that convey the idea of an absence of a numerical entity or quantity. In many cases, simply the negative of whatever formative is under discussion may be used.

As for handling the concept of zero as a null placeholder when writing Ithkuil numbers, this is addressed in Section 12.4 below.

Writing Ithkuil numerals is somewhat similar to writing numbers in Western languages (i.e., “Arabic” numerals), in that the interpretation of a number as a different power of 100 (analogous to interpreting single Arabic numerals as either ones, tens, hundreds, thousands, etc.) is based on its sequence within the entire number. However, there are two aspects of writing Ithkuil numbers that are quite different from Arabic numbers.:

1. Ithkuil does not employ a symbol for zero. Instead, Ithkuil employs separate autonomous symbols for each power of 100 (100, 10,000, 100 million, etc.) each of which operates as the appropriate placeholder instead of zero. To illustrate what this means by analogy, pretend that “@” is an autonomous symbol for 27 (since Ithkuil numbers from 1 to 99 each have a separate symbol), “&” is a symbol for 100, “#” is a symbol for 10,000 and there is no symbol 0 (zero). The numbers 2700, 2705, 327, 22700 and 4,270,027 would then be written @&, @5, 3@, 2@&, and 4@#@ respectively. (NOTE: In actual practice, numbers which contain the “hundred” symbol, here represented as “&,” normally place a dot above or below the adjacent numeral and dispense with the &, indicating that the number so marked is to be multiplied by 100.
   
   
2. Since Ithkuil is a base-100 system, numbers do not become two digits in length until the hundreds, do not become three digits in length until the ten thousands, do not become four digits in length until the millions, etc.

One must also remember that in terms of left-to-right orientation, numbers follow the boustrophedon mode the same as the Ithkuil script (see Sec. 11.3.2). Similarly to Western languages, small non-compound numbers can be written using either their numerical symbols or written out in script (as in English “12” versus “twelve”).

The following table gives the Ithkuil numerical symbols along with their morphological stems. The symbols for 11 through 99 consist of the symbols for 1 through 9 plus various extensions added to their lower left “trailing” line indicating the particular 10-group to be added.

Table 65: ITHKUIL NUMERICAL STEMS AND WRITTEN SYMBOLS

llal	ksal	ţkal	pxal	sţal	cqal	nsal	fyal	xmal	mřal
1	2	3	4	5	6	7	8	9	10

extensions:

+10	+20	+30	+40	+50	+60	+70	+80	+90

ňal	zmal	pstwal	čkhal
100	10 000	100 000 000	10 000 000 000 000 000

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 in Ithkuil to omit the word ňial (= the PARTITIVE of ňal ‘one hundred’) referring to the number of hundreds. This is equivalent to the custom in colloquial English of saying ‘three twelve’ for ‘three hundred (and) twelve.’ The difference is that in Ithkuil, this omission of the word for ‘hundred’ is the preferred option, the word ňial being used only in larger numbers for clarity’s sake.

These principles are illustrated by the following examples:

ksalîk  (ňial)  xmalök
literally: “42 (of hundreds) 29”
4229

cqalök  zmial  nseuluk  (ňial)  cqalûk
literally: “26 of ten-thousands with 97 (of hundreds) 66” = 26,9766
269,766

llalök  ňial  zmual
literally: “21 of hundred of ten-thousands”
21,000,000
[NOTE: ňial is required in this example]

ksalok  ňial  xmalokiň  apstwial  ţkeul  ňial  ţkalakiň  zmual  pxeulek  mřalûk
literally:
“72 of hundreds and 79 of hundred-millions with 3 of hundreds and 53 of ten-thousands with 34 60”
727,903,533,460

We have already seen that when numbers are used to indicate how many of a certain noun there are, the noun must appear in the PARTITIVE case, since the number itself is functioning as the “head” of the numerical expression (e.g., English “12 boxes” being constructed in Ithkuil as a “12-set of a box” or perhaps more appropriately a “box-dozen”). Another syntactical consequences of numbers being full formatives is when a number functions as a label or overt identifier, as in the English sentence You’ll find him in Room 216. Such usage of numbers is not primarily sequential (which would involve the equivalent of “ordinal” numbers such as ‘fourth,’ ‘twenty-sixth’, etc. equivalent to stem No. 3 of each number root) but rather organizational (e.g., as in the three-dimensional array of room numbers in a hotel). Ithkuil handles such organizational labeling using either the COMPARATIVE case (see Sec. 4.5.32) or the ESSIVE case (see Sec. 4.5.9) depending respectively on whether the enumeration of the noun in question is to distinguish it from other enumerated nouns versus merely identifying the noun by a numerical name. Examples:

açtál  ksou’lik
‘room’-OBL    ‘twelve’-CMP
‘the room marked “12”’ OR ‘Room 12’ OR ‘Room No. 12’ [i.e., as opposed to being some other room]

açtál  ksealik
‘room’-OBL    ‘twelve’-ESS
‘the room marked “12”’ OR ‘Room 12’ OR ‘Room No. 12’ [identifying reference only]

Lastly, when numbers comprising multiple number-stems are declined for case, configuration, extension, etc., rather than writing out the entire number “long-hand,” the number symbol is used, preceded by the carrier stem ep- (see Sec. 9.3) which carries the appropriate declensions. This use of the carrier stem applies even to single-stemmed numbers when writing, in order to allow use of the number symbol instead of writing it out. In such cases involving single-stemmed numbers, the carrier stem is not pronounced (rather, the numerical stem bears the pronounced declensions); it is there only as a written indicator of the morphological declensions/derivations to be applied to the number stem.

Home		Introduction	4 Case Morphology	8 Adjuncts	12 The Number System
FAQs		1 Phonology	5 Verb Morphology	9 Syntax	List of Abbreviations
Links of Interest		2 Morpho-Phonology	6 More Verb Morphology	10 Lexico-Semantics	The Lexicon
		3 Basic Morphology	7 Suffixes	11 The Writing System	Texts

For those who would like a copy of the Ithkuil Grammar in book form, it is now available! And while you’re at it, you can check out the novel I co- wrote with my twin brother Paul, also now available! (It’s a political thriller/science fiction story that explores the philosophical implications of quantum physics, and features Ithkuil as a “para-linguistic” interface to a quantum computer.)

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