Most cultures use a base 10 system since we have 10 fingers for counting. Noting that fingers and toes combined give 20 digits, we can appreciate why the Mayans devised a base 20 system instead. Computers, knowing only 'on' and 'off' as states for a single bit have an internal counting scheme with base 2. As Tolkien tells us in Appendix D of LOTR the Eldar preferred to reckon in sixes and twelves as far as possible. Mathematically, there is no difference between bases. The number 10 is not in any way special, whereas real mathematical properties of numbers (e.g. being a prime number) are true in any base. But of course the words for numbers in a language crucially depend on the chosen base.
Unique numbers are only needed up to the base unit. Observe e.g. how in English 'four' reappears in 'fourteen' (the first series above the base unit). As soon as we reach multiples of the base unit, elementary numbers are used to count it (where the base unit may appear as reduced affix, cf. 'four-ty' instead of '**four-tens'). Single digit numbers are then just added, cf. 'four-ty one'.
Of course there are many exceptions: English 'eleven' and 'twelve' are unique labels, they do not make use of the base unit. The construction scheme of the first series above ten is different from higher numbers, instead of '**ten-four' (like 'twenty-four') 'four-teen' is found. Other languages may have different exceptions - while Finnish has no unique labels for 11 and 12, the scheme to create 12 'kaksitoista' from 10 'kymmenen' is still different from how to create 22 'kaksikymmenta-kaksi' from 20 'kaksikymmenta'.
Languages also follow different rules when going to higher units. When 10 times 10 is reached, a new label 'hundered' is needed, but with this label one could go all the way up to 100 times 100 - instead, a new label is provided already at 10 times 100 - 'thousand'. Japanese is more logically consistent in introducing the next unit at 10.000. English than stays consistent in waiting to 1000 times 1000 to introduce 'million', but the next unit 'billion' is not introduced at a million times a million but at 1000 times a million already. Nevertheless, languages tend to get more regular for larger numbers.
In creating numerals for his Elvish languages, Tolkien did take all of the above points into consideration and it is interesting to compare the structure of the Elvish numerals with the English ones. There has been considerable debate whether Elves use decimal or duodecimal counting, and as we will see in the following, both schemes are realized, but his final decision seems to be that the decimal system is more common, although that contradicts the statement published in LOTR.
In the first sentences, 'ten' appears as a counting unit for 'warriors'. Then, 'first', the ordinal (adjectival) form of 'one' is used as an adverb specifying the verb 'to appear'. 'three' in this sentence is used as a pronoun, it stands for 'warriors'. In the next sentence, 'the three' again is a pronoun, but 'two' painted on the shields uses the number as a noun, refering to the number two. 'second', adjectival form of 'two' is then used as an adjective to describe 'army'. Finally, 'one' appears as a pronoun, but it isn't counting 'warrior' but rather used to denote a part of a group - 'one of the warriors' is different from 'one warrior'.
It can be seen that numbers are rather complicated grammatical entities. While English makes some distinction (i.e. between cardinal and ordinal numbers) explicitly, (cf. 'four' and 'fourth'), it doesn't mark others, there is no word for the 'number four'. This is different in Finnish where one has 'neljä' as 'four' but 'nelonen' as 'number four'. Japanese adjusts the number word according to the thing being counted. While 'ichi, ni, san' is 'one, two, three', once we count small round things the Japanese gets 'ikko, niko, sanko'; if we count long things (like scrolls) we find 'ippon, nihon, sanbon'; if we count flat things 'issatsu, nisatsu, sansatsu' and so on.
English joins the counted object in plural, cf. 'three elves', but other languages find different solutions: Finnish joins the counted obejct in partitiivi singular, i.e. 'kolme suomalaista' lit. 'three of type 'Fin''.
We cannot establish the precise grammatical role of the numerals in Tolkien's Elvish languages in all conceptual stages, but in the instances where we get such information, we observe an extremely rich and complicated system which is very far from being based just on English.
In the following, the focus will be more on the structure of the number system, especially how larger numbers are formed from smaller units, and on what we know about the grammar of numerals. This article will not primarily cover Tolkien's ideas with regard to the origin of the number words and their derivation. Such information can be found e.g. in .
|9.||olme, olmet (PE12:69)|
|20.||atwen, uiwen (PE12:33)|
|100.||tuksa pīnea or lempea (PE12:95)|
The emerging scheme for numbers above 10 is quite interesting. If we consider atwen, uiwen 20 (PE12:33) along with the prefix at-, att-, atta- 'bi-' (ibid.), we would assume that -wen somehow marks the counting unit 10 and 'twenty' is just two times the counting unit in a decimal system. However, olwen 36 shows the same suffix while it is evidently based on ole 'three'. In can be explained as three times the counting unit in a duodecimal system where the base is 12 and 3 times 12 equals 36. However, we have to assume that -wen now marks units of 12 instead of 10. This is essentially confirmed by otwen 84 (PE12:71) which would seem to involve otso 7 (ibid.) times the base 12 which yields 84. The solution to the apparent puzzle is found with tuksa which usually means 144 (the counting unit times itself), but when combined with the additional pīnea or lempea (the latter renders as 'decimal' (PE12:52)) means 100, i.e. again the counting unit times itself, only with a different base. Thus, one can count 'normal' or lempea, and we must assume that atwen then represents a number in the lempea counting scheme.
A puzzle is the number leminkainen 23 (PE12:52). Its first part clearly is the element lemin 'five' (ibid.), but since 23 is a prime number, it cannot be five times the multiple of any integer base. The only conclusion is that the five has to be added to another number, thus #kainen 18? But if so, 18 is not a special number in either the decimal or the duodecimal system - should we thus assume that there was also a base 18 counting scheme in Tolkien's mind? Unfortunately, we learn nothing about the usage of these numbers.
We can infer how to form multiples from PE12:56 where a suffix -lukse '-times' appears along with the form attalukse '*two times'.
|1.||min 'one, first' (PE11:57) er 'one single' (PE11:32)|
|8.||uvon, uvin and (deleted) †ungin (PE11:75)|
|1000.||mothwen, moth(in) 'flock' (PE11:58)|
Apart from min/mir there is not much similarity between the numerals of Goldogrin and Qenya. In Goldogrin, we meet for the first time the clear distinction between two different words for 'one' - while min denotes the first in a series and implies that other elements follow, er denotes a single element and implies that no others are following. This distinction is already hinted at in the QL, cf. the element eresse 'singly, only, alone' (PE12:36), although no numeral is derived.
It seems difficult to decide if the system glimpsed in these numbers is decimal or duodecimal. The similarity between ungin and unthos might indicate that the latter number involves the 'eight' and is thus decimal. mothwen 'thousand' would clearly indicate decimal counting, whereas beleth(os) is characteristic for a duodecimal system.
Interestingly enough, belethos seems to incorporate the elemtn beleg 'great' (PE11:22) and hence presumably literally means 'a large number'. We will see below that the idea that a large counting unit developed from a word for 'large number' frequently recurs in Tolkien's writings, however his definition of precisely what a large number is changes somewhat over time.
As for other numbers and related forms, one finds a prefix gwi- 'bi, twi-' (PE11:45), the ordinal obin 'second, next' (PE11:62), a few words for 'half', elfeg 'half (adj.)' (PE11:32) or lemfin, lemfa (PE11:53) together with elef 'half (n.)' (PE11:32) or lemp (PE11:53) (i.e. there are two stems associated with 'half') and finally ungra '8th' (PE11:75).
The sole example for the usage of numbers remains Tolkien's remark that 'half a' is expressed using the noun elef, hence basgorn elef which presumably translates '*half a loaf of bread' (PE11:53). It seems impossible to draw substantial conclusions from this form.
A table of cardinal numbers is found in PE14:49f:
|1.||er 'a single' min 'one, first'|
|2.||satta 'both' (satto in PE14:82), yúyo 'two'|
|6.||enqe or enekse|
|10.||kea (adj.), kai, kainen (part.)|
|18.||tolkea or hualque|
|21.||min yukainen (minya yukainen in PE14:82)|
|70.||otsokainen (also occasionally okkainen (PE14:83)|
|100.||tuksa, or when not multiplied keakai(li)|
|110.||kea tuksa or minqekainen|
|120.||yukainen tuksa or yunqekainen|
|200.||yúyo tuksa, yutuksa (PE14:83)|
|300.||nelde tukse (PE14:83)|
|1000.||tuksakainen or tuksainen or húme (orig. great number) (maite in PE14:83)|
|'million'||mindóra (sóra 'a very great number' in PE14:83)|
|'2 million'||yundóra (yundóre is 'billion' in PE14:83)|
|'3 million'||neldóra (neldóre is 'trillion' in PE14:83)|
|'*4 million'||kantóra (kantóre is 'quadrillion' in PE14:83)|
Clearly, the system displayed here is decimal. As apparent from nelde 'three' and neldekainen '30', the first counting unit is kainen. This is denoted as part[itive] of kai and agrees well with the partitive ending -inen seen in PE14:46. Thus, literally neldekainen ought to be translated '*three of tens'. The form kea is marked as adj[ective]. It is used to form the numbers 13-19 in combination with the basic numbers, so nelkea perhaps ought to be read literally as '*tenish three'.
Like in English, 11 and 12 have special names not following the scheme. Interestingly enough, also 18 has a special name hualque which appears to refer to its role as 2 times 9 where 9 is hue. Thus, while the system itself is decimal, there are remnants of other counting schemes to be found as well.
Unlike in English where the larger counting unit comes first, Qenya seems to have the smallest unit first, so is 'twenty-one' translated as min yukainen, not as **yukainen min and 'hundred and twenty' as yukainen tuksa - chances are 'hundred and twenty-one' would render as *min yukainen tuksa.
Like in English, Qenya switches to a new counting unit húme at 10 times 100, although logically such a unit is not yet needed. Again as in English, the next unit is then reached with a million mindóra. Here we see the most remarkable difference between the manuscript and the typescript grammar - while yundóra, combining the elementes 'two' and 'million' means 'two million' in the manuscript grammar, the meaning changes to 'billion' in the typescript grammar with the shift (pluralization?) to yundóre.
The usage of the numerals is also explained in quite some detail. We learn that 'all numerals precede the qualified noun' (PE14:50) and that 'all [numerals] may be employed alone as nouns or pronouns (...) except kea and those ending in -kea, -kainen. kea may be used as a pronoun but the abstract used (...) is kai' (PE14:50). What this presumably means is that while both 'three [elves]' (pronoun) and 'number three' (abstract noun) translates as nelde, 'ten [elves]' is kea while 'number ten' is kai.
With regard to the relation of numeral and counted object, we find quite complex rules (PE14:50):
These rules even seem to have some influence on the construction of the number system itself, cf. yúyo tuksa 200 (PE14:83) with nelde tukse 300 (ibid) - while in the first case tuksa is joined in singular, it is in the second case joined with plural. There are, however, differences between the typescript and the manuscript version of the grammar, in the latter we find 2000 as yúyo húmi (i.e. with plural for singular húme).
Finally, we also see again the idea that a word which originally meant 'great number' húme or 'very great number' sóra acquired actual numerical values in the development of the language.
The Early Qenya grammar provides also a table of ordinal numbers along with the information that all these are adjectives:
|2nd||potsina 'next, following' etya 'other' (erya in PE14:82)|
|*10th||kaiya, keatya (keanya in PE14:82)|
|*11th||minqetya (minqenya in PE14:82)|
|*12th||yunqetya (yunqenya in PE14:82)|
|*13th||nelkeatya, nelkaiya (nelkeanya in PE14:82)|
|14th||kankaiya, kankeanya (PE14:82)|
|18th||hualtya, hualqetya, hualtya, hualqenya, tolkaiya (PE14:82)|
|*21th||min-yukainenya, minya, yukainen(ya)|
|'millionth'||mindóratya, mindóranya (mindorinya in PE14:83)|
The forms are apparently created with the endings -ya, -tya and -nya (the latter seen to replace -tya in the typescript grammar) from the cardinal numbers after loss of some final elements, so nel-de <-> nel-ya or tol-to <-> tol-ya. Usually the first syllable is preserved. The only 'irregular' form within this scheme appears to be potsina 'next, following'.
Another list covers the use of multiples. We find:
|'once'||eru, eresse, ellume|
|'first'||min, minyallume (PE14:84)|
|'thrice'||nel, nellume, neldellume|
|'four times'||kan, kantallume|
For the small numbers, the multiple is found by the number stem, i.e. the part that is also used with an ending to form the ordinal numbers. Alternatively and for higher number, -llume (probably simply translating 'time[s]') is appended to the full numeral, contractions occur however if the stem ends with -l, cf. neldellume and nellume.
The usage of these forms is illustrated with the sentence hue yullume i hualqe 'twice nine is eighteen' (PE14:51) which is so far the only example for Elvish use of mathematics.
Another table on PE14:51 illustrates the formation of fractions (used as nouns and pronouns) from the basic numbers:
|1/6||enekto, enqetto (enqesto in PE14:84)|
|1/7||otsotto (otsonto in PE14:84)|
|1/9||huetto (huesto in PE14:84)|
|1/10||keatto (kesto in PE14:84)|
|1/11||minqetto (minqesto in PE14:84)|
|1/13||nelkeatto (nelkesto in PE14:85)|
|1/100||tuksatto (tuksanto in PE14:85)|
|1/1000||maisitto (maisinto in PE14:85)|
|1/1.000.000||mindoritto (mindórinto in PE14:85)|
Apart from the words kaina 'whole' and lempe 'half' (which seems to be related to the stem of 'five') the fractions are usually formed with the endings -sto, -tto or -nto where -sto consistently replaces -tto in the typescript grammar. Note that the form maite has the stem maisi- (PE14:83), thus explaining the form maisitto well as maite combined with an ending -tto.
A table of fractions used as adjectives is also found on PE14:51:
|'half'||lenya, lempea (lemya in PE14:84)|
|1/6||enektya, enqetya (enqestya in PE14:84)|
|1/7||otsotya (otsontya in PE14:84)|
|1/9||huetya (huestya in PE14:84)|
|1/10||keatya (kestya in PE14:84)|
|1/11||minqetya (minqestya in PE14:84)|
|1/13||nelkestya, nelkeastya (PE14:85)|
|1/100||tuksatya (tuksantya in PE14:85)|
|1/1000||maisitya (maisintya in PE14:85)|
For the most part, these adjectival forms can be explained by replacing the noun endings -tto > -tya, -sto > -st(y)a and -nto > -ntya.
|1. (one single)||ERE-||er||LR:356|
|3.||NEL-, NÉL-ED-||nelde||neledh later neled||LR:376|
|5.||LEP-, LEPEN, LEPEK||lempe||lheben||LR:368|
|7.||OT-, OTOS, OTOK||otso||odog||LR:379|
While the existence of roots like RÁSAT- '12' or KHOTH- '144' would seem to indicate that a duodecimal system is intended here, we may note that no actual numerals in either Qenya or Noldorin are given for '12' and that a language may well have special words for 12 or 144 (like English in fact) without using an actual duodecimal counting scheme. The available evidence does not allow to judge this point.
The list does not add substantial new ideas - there are still two different roots associated with 'one' dependent on context, and many Qenya numbers are not much changed from their shape in Early Qenya.
The presence of variant roots, cf. TOLOTH, TOLOT or analogical formations, cf. neledh later neled suggests that Tolkien had at this point a rather clear scheme in mind what the number words in Noldorin and Qenya should be, thus root elements were apparently devised accordingly. The impression created is markedly different from what we see in later works where Tolkien struggled considerably to bring published root derivatives and number words in agreement within his phonological development.
The two numerals in this sentence, lheben 'five' and neledh 'three' are readily found in the Etymologies. However, the sentence is interesting as it allows to study the usage of cardinal numbers in Noldorin. lheben here is used to count 'feet'. teil here is an unlenited plural. As Noldorin nouns in genitive are lenited (cf. Noldorin Compounds ) this is in fact not to be interpreted as partitive genitive 'five of type feet' but rather as the normal way of combining a number with a noun. Thus, just like in Early Qenya, the numeral preceeds the noun. Particular of Noldorin, the noun remains unlenited and is (different from the typical situation in Early Qenya) pluralized. If nouns should be pluralized in general or if this is only true for certain numbers cannot be inferred.
neledh exemplifies the usage of numerals as pronouns - apparently here it signifies 'three [persons, dwarves]' who can go through the gate. Its verb gar '*can' is in singular, indicating that at the numeral is seen as a unit in spite of the fact that several persons are meant.
Most puzzling however is nelchaenen - removing an ending -en we are left with a candidate for a cardinal number #nelchaen. Given that the roots for 'ten' are KAYAN and KAYAR-, #caen can plausible be explained as 'ten', but then the number seems to be 30 rather than 31. Carl F. Hostetter has suggested an explanation in VT31:31 having to do with the fact that the preposition used in the phrase is uin 'from' rather than erin 'on the' as for the other two phrases and thus a date 30 days from the beginning of the month may well be on the 31st day of the month dependent on how days from a date are counted.
If so, #nelchaen is a rare example of a compound number published after the Early Qenya grammar. The construction principle seems to be very similar to nel(de)kainen in PE14:49 if the internal consonant mutation characteristic of Noldorin/Sindarin is taken into account. Thus, we note that this number is definitely given in a decimal counting scheme which presumably is not very different in structure from the one found in Early Qenya. However, it is up to guesswork what larger counting units at this stage might be.
In WJ:388, the Sindarin name of the Petty-dwarves is given as Levain tad-dail or Tad-dail, in WJ:389 the corresponding Quenya name Attalyar is given and translated as 'Bipeds'. Quite evidently, these forms involve Q: atta and S: tad 'two'. Writers have sometimes argued based on Tad-dail that in Sindarin (unlike in Noldorin) words following a number are lenited. But there is no evidence for this here: Tad-dail does not translate 'two feet' but 'biped' - judging from the Quenya form even #tail is not a pluralized 'foot' but rather the Sindarin cognate of #talya '*footed' and tad-dail is an adjective 'two-footed'. The fact that it appears unlenited in Levain tad-dail does not imply that it cannot be adjective - about half of the Sindarin adjectives attested in phrases are unlenited, thus there is no reason to expect lenition in the first place.
While discussing the river Lefnui (Levnui) appearing on the map of Middle-Earth, Tolkien noted that his intended derivation as 'fifth' did not go together with Sindarin phonology and the fact that he wanted to derive 'five' and 'finger' from the same root LEP (for obvious reasons). It is probably correct to say that the average reader of LOTR does not even notice the Lefnui, is much less aware of the fact that it should mean 'fifth' and even less bothered by the fact that the name doesn't agree with Sindarin phonology. Not so Tolkien.
Since there was no way standard phonology would yield the name from LEP and Tolkien at that time wasn't prepared to simply write down an alternative root LEM, he first considered an analoguous derivation. But to show why this particular form was changed by analogy, he first had to derive all the other Sindarin ordinal numbers. It is to this phonological problem that we owe a list of stem forms and cardinal number words (mainly for Sindarin, but also some Telerin and Quenya forms are discussed):
The list of stems is fairly similar to those seen in the Etymologies, notable exceptions are KWAYA, KWAY-AM for 'ten', leading to Sindarin pae instead of Noldorin caer found in the Etymologies and YUNUK(W) for 'twelve', making the root RÁSAT- obsolete. Another difference is that the monosyllabic number words have now acquired a long vowel.
Since the main purpose of the discussion of numerals was to discuss how levnui could arise, the list of ordinal numbers extends to other languages as well. Here we see the first complete sequence of numerals in Telerin.
|1st||mein, main, minui||minya||minya|
|2nd||taid, tadui||tatya, attea||tatya|
|3rd||neil, nail, nelui||nelya, neldea||nelya|
Perhaps not surprisingly, the Sindarin ordinal numbers have the ending -ui, leading to the desired feature that levnui could inherit this ending by analogy from the other numerals in spite of its different expected phonological development. This is markedly different from the ordinals seen in the King's Letter which were all characterized by the ending -en.
Quenya ordinals are apparently marked by an ending -ea, Telerin ordinals by -ya which replace a final consonant if necessary.
What is outlined here does not really resemble a duodecimal counting system, but at this time LOTR Appendix D with the statement the Eldar preferred to reckon in sixes and twelves as far as possible was published already, so Tolkien felt he had to give some kind of clarification. What appears in this essay is
But already in Common Eldarin, the multiples of three, especially six and twelve, were considered specially important, for general arithmetic reasons, and eventually besides the decimal numeration a complete duodecimal system was devised for calculations, some of which, such as the special words for 12 (dozen), 18, and 144 (gross), were in general use. (VT42:24)
Thus, while the Elves had a duodecimal system for calculations, only some of its words were in general use. However, I fail to see what 'general arithmetic reasons' would lead to a preference for a base 6 system.
A good starting point is probably provided by a (rather) complete list of numerals from 1 to 12 in Sindarin, Telerin and Quenya found in VT48:6:
|1.||er, min||er, min||er, min|
There is no real surprise in the Sindarin forms. In this list, monosyllabic numerals are short (as in the Etymologies and different from the list in 'Rivers and Beacon Hills' shown above) and 'three' appears as a longer form neleð (again as in the Etymologies) rather than the shortened nêl. Probably the most remarkable fact is that 'five' in Quenya now appears as lemen instead of lempe - this seems to indicate that Tolkien finally considered a derivation of 'five' from an alternative stem LEM after all. However, immediately contradicting this list, VT47:10 rather has 'five' as S: leben Q: lempe T: lepen.
The same page also lists an alternative Quenya name of 'five' and 'ten' - Q: makwa 'five' maquat 'pair of fives, ten' (VT47:10), based on counting through all fingers of one or two hands, are mentioned along with the name maquanotie 'decimal system of counting' (VT47:10). In VT48:11 the word Q: kaistanótie 'decimal system' is provided instead.
Various Common Eldarin stems can be found throughout the texts. Especially the numerals above 10 leave little doubt that Tolkien considered a base 10 number system here:
There is some overlap to a list of Quenya numerals above 12 found in VT48:21:
Interesting here are the alternative forms in which the element quai 'ten' appears in front position. This seems to reverse a tradition in which the smallest digit is always mentioned first. An interesting alternative to the number 13 is provided in VT47:15 where we learn that Q: yunquenta 13 is derived as '12 and one more' using an element enta 'one more' (ibid).
However, there is almost no trace of a duodecimal system. Even Q: yunque 'twelve' is not a particularly exceptional formation given that yu- has a long tradition as a stem for 'two' and quean is the word for 'ten'. As in 'Rivers and Beacon Hills of Gondor', Tolkien felt that he had to give explanations:
At a (probably) later period the Eldar now provided with a numeral system firmly based on the manual 'decimals' 5,10 became interested in sixes, and a word for 6 times 2 (12) was already devised before the end of the Common Eldarin period. (VT47:16)
In spite of their later predominant (...) interest in and use of six-twelve (as group units) they did not develop a complete duodecimal nomenclature, though they invented (after the common Eldarin period for numbers above 12) special names for the multiples of six times six. Of these, 18 and 24 were also in daily use, as well as the 'gross' 144 and 72 half-gross. (VT47:17)
Thus, while the Elves preferred to calculate in sixes and twelves (as stated in LOTR) they did not create the nomenclature for this system. It stands to reason that there must be a duodecimal tengwar mode available for calculations, or any other system of writing duodecimal numbers, because it certainly is exceedingly difficult (and useless) to do doudecimal calculations using decimal words and numbers.
The last item of considerable interest is a list of Quenya fractions found in VT48:11 which are derived with the help of the root element SAT 'divide':
|1/3||nelesta, neldesta, nelta, nelsat|
|1/4||kanasta, kasta, kansat|
|1/7||otosta, osta, otsat|
|1/8||tolosta, tosta, tolsat|
|1/9||neresta, nesta, nersat|
As Patrick Wynne remarks in his analysis of these forms, the similarity with the fractions found in the Early Qenya Grammar is remarkable.
Exceptions are late variant forms such as yunquenta 'thirteen' or the inverted quainel 'thirteen' found in the late texts. But it is unclear if those represent more than just passing ideas. On the other hand, the structure of the Goldogrin number system is very different from the forms seen in Noldorin. While min and cant would be recognized by a student of Sindarin, the same is not true for the other Goldogrin numerals.
Evidence for a true duodecimal number system can only be found in the Qenya Lexicon. Although Patrick Wynne mentions unpublished duodecimal number tables in VT47:42, no such table is published, and later published writings by Tolkien dealing with numerals show either clear evidence for a decimal system or are consistent with the assumption. How should we reconcile this with the rather strong statement that the Eldar preferred to reckon in sixes and twelves as far as possible in Appendix D in LOTR? Presumably Tolkien liked the idea of having relic words for 12, 18 or 144 in the language, and he devised the Elvish reckoning of time in duodecimal counting, but he was not actually prepared to convert the whole numeral system to duodecimal; even less so when he had the idea that numerals and finger names should be derived from the same stems. So, we are to assume that Elves did use duodecimal counting in the reckoning of time and when dealing with mathematical problems, but not in everyday language, but some words of everyday language were influenced by the duodecimal forms. We know precious little about Elvish mathematics, but it stands to reason that there must have been a system for writing down duodecimal numbers to allow such computations.
The Early Qenya Grammar describes in detail the quite complicated grammatical rules governing numeral and the counted object. Precious little is known or can be inferred from later texts. It is quite possible that the usage of numerals in Quenya or Sindarin is of similar complexity, with numeral-dependent case and number requirements and such like - if so, we simply do not know and possibly never will. However, judging from the complexity seen in the Early Qenya grammar, it would be unreasonable to assume that the grammar of numerals in later Quenya or Sindarin must necessarily be simple.
 'The Gnomish Lexicon' by J.R.R. Tolkien, publised in Parma Eldalamberon 11, edited by Christopher Gilson, Carl F. Hostetter, Patrick Wynne and Arden R. Smith
 'Early Qenya Grammar' by J.R.R. Tolkien, published in Parma Eldalamberon 14, edited by Carl F. Hostetter and Bill Welden
 'The Etymologies' by J.R.R. Tolkien, published in 'The Lost Road and other Writings', edited by Christoper Tolkien
 'J.R.R. Tolkien - Artist & Illustrator' by Wayne G. Hammond and Christina Scull
 Compounds in the Noldorin of the Etymologies by Thorsten Renk
 'The Rivers and Beacon-Hills of Gondor' by J.R.R. Tolkien, published in Vinyar Tengwar 42, edited by Carl F. Hostetter
 'Eldarin Hands, Fingers & Numerals' and related writings by J.R.R. Tolkien, published in Vinyar Tengwar 47 and 48, edited by Patrick H. Wynne
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