THE SEARCH FOR THE PERFECT NUMERAL SYSTEM, WITH PARTICULAR REFERENCE TO SOUTHEAST ASIA


THE SEARCH FOR THE PERFECT NUMERAL SYSTEM,
WITH PARTICULAR REFERENCE TO
SOUTHEAST ASIA

Dra. Wan Anayati, MA

Abstract

Apabila kita mengadakan perbandingan antar-bahasa, kita boleh saja beranggapan bahwa bahasa yang satu lebih atau kurang ideal dibandingkan bahasa yang lain. Sistem penghitungan dalam bahasa merupakan salah satu cara dimana kita dapat melakukan perbandingan semacam itu. Dengan mengambil contoh-contoh pada sejumlah bahasa di Asia Tenggara, penulis beranggapan bahwa sistem penghitungan seharusnya mencerminkan bagaimana angka-angka ditulis dengan figur. Kini, cara desimal untuk menulis angka-angka hampir bersifat universal. Oleh karena itu, penulis berpendapat bahwa sistem penghitungan yang ideal dalam budaya penghitungan yang moderen seharusnya juga desimal. Dalam makalah ini penulis akan memfokuskan pada perkalian dan penjumlahan dalam sistem penghitungan desimal.


1.  INTRODUCTION
The title of this article - which should not be interpreted without a certain element of humor - is taken from that of a book by Umberto Eco (Eco 1997). While I doubt whether one can evaluate any language as a whole as being more or less ideal than any other, such a comparison may be possible in certain very restricted parts of the structure of a language, and the numeral system is one area where one might be able to make such a comparison. In the following sections, I will try to do this with respect to selected Southeast Asian languages. In the final section, however, I will express a number of caveats with respect to the notion of “ideal” numeral system, suggesting that even here the notion of “ideal” may not be so easy to define.
I will make a number of assumptions in the definition of an “ideal” numeral system. First, I will assume that the numeral system should mirror closely the way numbers are written by means of figures. In the present-day world, the decimal way of writing numbers is almost universal, so I will assume more specifically that an “ideal” numeral system in a modem, numerate culture should be decimal - though this does not exclude the possibility that in other scenarios a numeral system with some other base would have been “ideal”, a concrete example being the ancient Mayan system, which both linguistically and in its notational system was vigesimal (base 20). Incidentally, even those Southeast Asian cultures which optionally or obligatorily use their own notation for figures, such as Burmese, Khmer, and Thai, nonetheless have a decimal system for this notation.
A decimal system is characterized ideally by the following features. The numerals 1 - 9 are expressed by distinct morphemes. There is a distinct morpheme for 10, and products of 10 are expressed by a conventionalized means of indicating multiplication, as in Indonesian dua puluh ‘20, i.e. two ten, i.e. 2 x 10’, with the convention that a smaller number followed by a larger number is to be interpreted as multiplication. Numerals in between products of 10 are expressed by a conventional means of indicating the addition of the remainder to the power of 10, as in Indonesian tiga puluh sembilan ‘39, i.e. three ten nine, i.e. 3 x 10 + 9’, with the convention that a larger numeral followed by a smaller number is to be interpreted as addition. Ideally, a decimal system should also have a systematic way of expressing powers of 10 (‘exponentiation’), and indeed this is found to some extent in the international system for higher powers of 10 (bi-Ilion, tri-ilion, quadr-illion, etc.). However, no language seems to use such a system without exception for the lower powers of 10, so that we rather find portmanteau forms like English hundred for 102, thousand for 1O3, etc. I will not discuss further the expression of powers of 10.
In this paper, I will concentrate on multiplication and addition in a decimal numeral system. There are other features that should surely be imposed on an “ideal” numeral system, such as expressibility (i.e. the possibility of expressing any number) and absence of ambiguity, and indeed I have discussed such features in Comrie (1997), but I will not discuss them further in the present context.
As a starting point, one might ask how English and other major European languages fare in terms of such an “ideal” system. Basically they operate in terms of transparent multiplication and addition, with addition proceeding from higher to lower powers of 10, as in English three thousand five hundred and six ‘3506 (i.e. 3 x 1000 + 5 x 100 + 6)’, there are nonetheless a rather large number of deviations from this pattern. In English, the forms of the l0s use not the element ten but rather -ty, and there are some morphophonological irregularities of combination (cf. five but fif-teen). The teens again do not use the element ten, but rather a different suffix -teen, and invert the usual order by having the unit before this suffix, as in six-teen ‘16, i.e. 6 + 10’, again with some morphophonological irregularities (e.g. fif-teen), and with at least one complete irregularity, namely eleven ‘11’, which is synchronically unanalyzable. Other irregularities found in other major European languages include a partial foray into vigesimalism in French, where 80 is expressed as quatrevingts, literally ‘four-twenties, i.e. 4 x 20’, an unexpected portmanteau form for 40 in Russian (sorok), and consistent inversion of the tens and units in German and Dutch (ein-und-zwanzig and een-en-twintig respectively for 21, lit. ‘one- and-twenty’).
Note finally, before we proceed to Southeast Asia, that I am not concerning myself explicitly with how multiplication and addition are expressed, so long as there is a consistent way of indicating them, as in the analysis of the Indonesian forms given above. Of course, an “ideal” system should operate consistently, so that English is inconsistent in sometimes requiring expression of the unit in multiplication (e.g. a/one hundred ‘100’), and sometimes not (e.g. ten ‘10’), while French is inconsistent in sometimes requiring overt expression of addition (e.g. vingt-et-un ‘21, lit, twenty-and-one’), sometimes disallowing it (e.g. vingt-deux ‘22, lit, twenty-two’).
In the Southeast Asian languages considered below, the basic system is consistently decimal, with (covert) indication of addition and multiplication as in the Indonesian example analyzed at the beginning of this section. Nonetheless, individual languages have more or fewer deviations from this “ideal” system, as illustrated in the following sections.



2.  INDONESIAN
[The source of my data on Indonesian is Sneddon (1996: 184 - 185).] Indonesian comes close to the ideal system outlined in section 1. The basic structure is as follows: A smaller numeral followed by a larger numeral is interpreted as multiplication; a larger numeral followed by a smaller numeral is interpreted as addition; all multiplications are carried out before addition. This can be seen in example (1):
(1) dua ribu enam ratus tiga puluh sembilan
            two thousand six hundred three ten nine
            ‘2639 (i.e. 2 x 1000 + 6 x 100 + 3 x 10 + 9)’

To this general pattern, there are only two exceptions. One is essentially morphophonological, in that the numeral 1 in multiplication is expressed as the prefix se- rather than the separate word satu in the powers of 10 from 10 through 1000, and optionally in the case of ‘million’, e.g. se-ribu ‘1000 (i.e. 1 x 1000)’; both prefix and separate word are attested elsewhere in the languages, so this is making use of an already existing set of forms. The second is that the formation of the teens falls outside the general pattern, so that ‘10 + n’ is expressed as n belas, e.g. tiga belas is ‘13 (i.e. three teen)’; the form for 11 combines this with the morphophonological property just noted to give sebelas.
3.  MANDARIN CHINESE
[The source of my data on Mandarin is Yip and Rimmington (1997: 11—12) and Chao (1968: 567—575).] Although Chinese is gographically an East Asian rather than a Southeast Asian language, its presence in the area, especially through cultural influence on other languages - Thai, for instance, has borrowed most of its numerals from Chinese - justifies at least a brief treatment in the present context, especially as Chinese, here illustrated by Mandarin, fits well into the general Southeast Asian pattern, close to the “ideal system”. The general pattern is essentially as given above for Indonesian, as illustrated in (2):
(2) wŭ- băJi yi-shi èr
            five-hundred   one-ten      two
            ‘512 (i.e. 5x 100 + l x 10 + 2)’
(Note that this pattern also extends to the teens. It would also be possible to omit the morpheme yì ‘1’ of yi-shi.)
The main exception to this regular pattern concerns variant forms of the numeral 2, namely èr and liǎng. (A further complication is discussed in section 8.) In numerals, generally, the variant èr is used. However, hang is an optional variant before products of powers of 10 from 100 upwards, i.e. 200 can be either êr- bai or liäng-bffi ‘two-hundred’. In addition, the numeral 1 undergoes tone sandhi, so that it has falling tone in (2), but level tone in isolation.

4.  THAI
[The source of my data on Thai is Smyth (2002: 172—174).] Thai follows essentially the same pattern as Indonesian, as can be seen in example (3):
(3) sẻẻη phan hâa róẻy sìi sip cèt
            two      thousand   five      hundred     four     ten       seven
            ‘2547 (i.e. 2 x 1000 + 5 x 100 + 4x 10 + 7)’
This same pattern extends to the teens, which are thus formed regularly, as in (4):
(4) sip sẻẻη
            ten       two
            ‘12 (i.e. [1 x] 10 + 2)’
(Note that Thai does not express 1 before 10, although it optionally does so before 100 and higher powers of 10, e.g. (n)phan ‘1000 (i.e. 1 x 1000)’.) Nonetheless, there are two striking deviations from the ideal pattern set out in section 1: First, whenever I appears as the final element of an additive compound numeral, in place of the word nfzj ‘1’ one finds rather èt, as in (5):
(5) sii sip èt
            four ten one
            ‘41 (i.e. 4 x 10 + 1)’
            Second, 20 is expressed using a different word for 2, as in (6) [cf. (4) above]:
(6) yii sIp
      two ten
      ‘20 (i.e. 2 x 10)’
Moreover, when 20 is followed by a unit in an additive construction, the combination as given in (6) may optionally be reduced to yǐip, although the full form as in (6) is also possible.

5    KHMER
[The source of my data on Khmer is Jacob (1998: 81—83).] Khmer presents a rather larger number of departures from the ideal system as presented in section 1. First, the numerals 6—9 are expressed as if in a quinary system, i.e. 6 is expressed as 5 + 1, as in (7):
(7) pram-muẻy
            five-one
      ‘6 (i.e. 5 + 1)’
However, the quinary system plays no part in multiplication (i.e. there are no forms interpreted as ‘n x 5’), nor in exponentiation (i.e. there are no morphemes interpretable as ‘125’, or more generally 5”).
            The word for 10 is dap, and the teens are formed regularly, as in (8):
(8) dap-pii
            ten-two
            ‘12 (i.e. 10 + 2)’
However, the products of 10 from 30 to 90 are expressed using morphemes borrowed from Thai. Thus, although 3 is by and 10 is dap, the form for 30 is as given in (9):
(9) saam-sap
            three-ten
            ‘30 (i.e. 3 x 10)’
Although this is sometimes described by saying that the words for the tens are not synchronically analyzable in Khmer, this is not strictly speaking correct, since the recurrent element -sap is found in all of the tens 30-90 and is thus synchronically an irregular allomorph of the word for 10. Likewise a form like saam- is more appropriately treated synchronically as an irregular allomorph of the word for 3. These forms for the tens have a synchronically transparent internal structure. Beyond this, the word for 20 is completely irregular, namely mephiy. Note that products of the higher powers of 10 are formed regularly, thus giving rise to combinations like (10):
(10)      pram-muзy-rccy saam-scp-budn
            five-one-hundred three-ten-four
            ‘634(i.e.(5+ 1) x 100+3 x 10+4)’
In other words, Khmer illustrates basically the same kind of structure as in the other cited Southeast Asian languages, but with rather more deviations,

6.  VIETNAMESE
            [The source of my data on Vietnamese is Thompson (1987: 184 - 190).] The basic forms in Vietnamese follow the same pattern as we have already seen in other Southeast Asian languages, with a decimal system using multiplication and addition, as in example (11):
(11)      ba mu’o’i bô’n three ten four
            ‘34(i.e.3x 10+4)’
However, as in some of the other languages considered, there are some morphophonological changes, in Vietnamese concerning tone. When used as the multiplicand to express the tens, the word much ‘10’ has the high level tone (no diacritic in Vietnamese orthography), while in isolation it has the low level tone, indicated orthographically by means of a grave accent, i.e. muoi ‘10’, and this form is also used in teens, which are formed regularly, as in (12):
(12)      muoi    mqt
            ten       one
            ‘11 (i.e. 10 + 1)’
The numeral 1 in isolation, and also following an unmultiplied power of 10 (i.e. in 11, 101, 1001, etc.), has the sharp falling tone, indicated orthographically by a subscript dot, e.g. mot ‘1’; but after other products of powers of 10 its tone is rising, indicated orthographically by means of an acute accent, as in (13):
(13)      hai       mu’oi  
            two ten one
            ‘21 (i.e. 2 x 10 + 1)’
            In addition, there are certain contractions that are frequent at least in the spoken language. Thus, for 20 followed by a unit, in addition to the full form just given there is also a contracted form as in the alternative form ham mô’t ‘21’; 30 behaves similarly. But for 40 upwards, in compound numerals involving addition of a unit the abbreviation is rather through omission of the word for 10, as in (14):
(14)      bô’n     (mu’o’i)     chin
            four     ten             nine
            ‘49 (i.e. 4 x 10 + 9)’
7.  BURMESE
[The source of.my data on Burmese is Comyn and Roop (1968: 30-32, 355- 356); tones are marked by means of one of the four symbols{=;. ‘} after the syllable.] Burmese also evinces the same basic system as in other Southeast Asian languages, as can be seen in example (15):
(15)      hyi’-ya.      hcau’-hse. thoun:
            eight-hundred six-ten three
            ‘863, i.e. 8 x 100+6x 10 +3’
Departures from this regular system concern morphophonological alternations, some (but not all) of which are paralleled elsewhere in the language. For instance, in some of the tens the element -hse’‘10’ is voiced to -ze, e.g. nga:-ze ‘50, lit, five-ten’. This same element also changes its tone when followed by a unit, as in the expression for 60 in (15); the same is true of expressions for the other powers of 10 when followed by a lower power of 10 (including a unit). The numerals 1, 2, and 7 change their last vowel when multiplying a power of 10, and also lose their tone, so that 2 is hni’, but 20 is hna-hse, lit. ‘two-ten’; this vowel change also occurs when these numerals precede numeral classifiers, i.e. it is not idiosyncratic to the formation of numerals.

8.  PARADISE LOST ... AND REGAINED
The presentation of material on Southeast Asian numeral systems suggests that they come close to the “ideal” numeral system proposed in section 1, certainly much closer than most major European languages. But a little more thought suggests putting the numeral systems of languages of Southeast Asia in a somewhat different perspective, especially once one starts considering the departures from the “ideal” system.
            First, numeral systems are heavily cultural objects, and cultural pressures can override “ideal” structure. We saw this in section 5, where the Khmer numerals for the tens are borrowed from Thai, thus giving rise to synchronically irregular allomorphs that reflect diachronically Thai equivalents of the usual Khmer morphs. A perhaps even more striking example is seen slightly outside our geographical area. In Tok Pisin, the English-lexified lingua franca of most of Papua New Guinea, the traditional system was “ideal”, with the exception of a few morphophonological alternations, as in (16) - (17):
(16)      wan-pela          ten       tu
            one-SUFFIX   ten       two
            ‘l2 (i.e. l x 10 + 2)’
(17)      tri-pela       ten       tri
            three-SUFFIX ten three
            ‘33 (i.e. 3 x 10 + 3)’
However, in current usage, these traditional numerals are replaced by their standard English equivalents (in local pronunciation and spelling), with all the idiosyncrasies of standard English, i.e. twelv, teti-tri, respectively (Mihalic 1971: 20).
            Second, in a numerate culture, making frequent use of numerals, there is some advantage to having shorter forms, especially where this does not lead to ambiguity. In this way one can account for the contracted forms for 20 in Thai and Vietnamese (in the latter language, also 30). A prelude to contraction can be seen in various morphophonological alternations, which also serve economy of pronunciation. A further contraction may be observed in Thai and Vietnamese - and perhaps some other languages, although shorter descriptions even of Thai and Vietnamese often fail to note the contraction under consideration. In both languages, a numeral like 2200 can be expressed simply by saying ‘two thousand two’, with the convention that the apparent unit in fact refers to the next lower power often, i.e. here 2 x 1000 + 2 x 100. Thai forms are discussed by Noss (1964: 110—ill); a Vietnamese example is given in (18):
(18)      hai       ngàn hai
            two thousand two
Now, this contraction actually gives rise to an ambiguity, since (18) can mean not only 2200 but also 2002 (but not 2020, since the contraction is only possible where the final component is of the next lower power of ten relative to the preceding element). If it is necessary to express unequivocally 2200, then one can add the word for 100; in Vietnamese at least, putting íé ‘and’ before the final element unequivocally indicates 2002 (and the morphophonology of 1 means that there is always a distinction between contracted 1001, with sharp falling tone on the element ‘1’, and 1100, with rising tone on this element.
            Finally, a radically different form, whether an irregular combination or a portmanteau morph, can serve to quickly identify a particular numeral. Thus, for instance, the portmanteau Russian numeral sorok ‘40’ is readily identifiable as such, without any need to identify morphemes for ‘4’ and ‘ten’. This might account for the other irregularities noted above, such as the formation of the teens in Indonesian, or Khmer 20, and also other occasional anomalies, such as the alternative form for 500 in Vietnamese: nô’a ngàn, literally ‘half thousand’. Language is always an arena of tension between the competing forces of clarity (favoring more extended formulations) and economy (favoring more concise formulations), and our characterization of the “ideal” numeral system in section 1 pays attention only to the former of these. Typologically, Southeast Asian numeral systems seem to place more emphasis on clarity, but economy is never completely absent.

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