\]~ENGT SIGURD 
FR.OM NUMBERS TO NUMERALS AND VICE VERSA 
1. FROM MATHEMATICAL INTO LINGUISTIC REPRESENTATIONS - 
AND VICE VERSA 
Universally used decimal representations, such as 5; 200; 856763; 
189200000 are rendered (written, pronounced) differently in different 
languages. The number 5 is tllus five in English, fern in Swedish, cinq 
in French, ketsmala in Koyukon, biyar in Hausa and nga: in Burmese. 
Numbers can furthermore be written differently in different mathemat- 
ical systems. The number 5 is written V in the Roman system and 101 
in a system with base 2 (binary system). 
We will mainly be interested in the relations between the represen- 
tations of the decimal system and Swedish numerals. Numerals in some 
other languages, such as English, German, French, Danish, Burmese, 
Hausa and Urdu will be touched upon briefly. In particular we are inter- 
ested in rule systems (algorithms) that automatically convert mathe- 
matical representations into linguistic representations (numerals) or 
vice versa. Our interest in this area is part of a wider interest in Automat- 
ic Text Comprehension, 1 which in turn is part of the areas Automat- 
ical Language Translation and Artificial Intelligence. The study is 
focusing on problems that are due to the structure of numerals in (nat- 
ural) languages. The technical problems that turn up when conversion 
rules are to be implemented on computer are not treated in this explor- 
atory study. 
The practical applications of rule systems that convert mathemati- 
cal representations into numerals or vice versa are automatic devices 
or robots of various types, e.g. the "automatic stock market announc- 
er ", the "automatic cashier ", the "automatic accountant ", the 
1 This work is related to the project ATC (Automatic Text Comprehension) of The Institute of Linguistics, University of Stockholm, which is supported by Humani- 
stiska Forskningsr¢tdet. I am indebted to C. W. Welin, who is working on that project, 
for valuable comments. 
430 BENGT SIGURD 
" automatic telephone directory ". The automatic telephone directory 
would give you the number, if you furnish the name and address of a 
person. The system would have to identify the person by processes that 
correpond to optical and manual search in a directory. In the register 
of the directory the system would find the number and could pronounce 
it according to rules to be discussed later. 
The automatic directory would require phonetic identification pro- 
cedures that have not been developed fully yet, at least not for large 
inventories of possible messages. In societies where numbers play an 
increasing role - telephone numbers, registration numbers, addresses, 
social security numbers, bank account numbers, zip codes - systems 
that can identify and pronounce numbers are getting increasingly inter- 
esting. Current experiences suggest that any conversation with robots 
has to take place within heavy constraints. 
Further examples of machines that could use the rule systems under 
discussion would include warning systems, systems that read the values 
of meters (temperature, altitude, humidity) aloud. A system that reads 
the speed of the car aloud or tells you the distance of an approaching 
crossing etc. in a mild but distinct voice might have certain advantages 
in difficult traffic situations. Presently almost all warning systems, traf- 
fic signs and signals, use optical signals. If acoustic signals are used they 
are not speech signals, only rings or buzzes. The human voice and the 
human language may have a certain attraction to human beings. Imag- 
ine the "speaking alarm clock" which tells you the time and reads 
the temperature etc. in an attractive (re)male voice available as an option. 
In most of the applications mentioned so far pronunciation has been 
involved. But systems of a simpler kind, which rewrite mathematical 
(decimal) number representations as numerals (in letters) would cer- 
tainly also be useful. Banks and Post offices use numbers written both 
in mathematical and linguistic form, presumably because of the need 
for redundancy and security. This is at least the case in Sweden. It is 
required that a number, e.g. 1055 be written with letters as well 
as with figures. But these are almost our only opportunities for 
writing high numbers with letters. (Numbers are generally written 
with letters only if they are below ten or belong to the round numbers 
(see B. SIGURD, 1972)). Many persons hesitate when they are to write 
numbers with letters (words). Swedes would for instance hesitate bet- 
ween ett tusen ferntiofem, ett-tusen-femtiofem, ett(t)usenfemtiofem, and tu- 
senfemtlofem. Fortunately the banks do not require any special spelling 
(only that you add Kronor after). 
FROM NUMBERS TO NUMERALS AND VICE VERSA 431 
There are many previous studies related to the problems under fo- 
cus here, e.g. CORSTIUS (1968) where marly general problems are treat- 
ed starting from Dutch, and C.-CH. ELERT (1970) where the morpho- 
logy of Swedish numerals is treated. We have used presentations and 
analyses of numerals in various languages. But it is only for Swedish 
that our rules are outlined in any detail. The other languages are treat- 
ed to indicate what problems one would have to face in an automatic 
translation system for numerals. A system that translates between math- 
ematical representations and representations in Swedish, English, 
French, German (and a couple of other languages) will be implemented 
on computer in the near future. 
We will work with written forms in this study. Unfortunately it 
is not the case that such representations could be run through a speech 
synthesizer giving beautiful pronunciation. Nor are there any speech 
recognition devices that can render large or infinite inventories of spo- 
ken numerals as written ones presently. A forthcoming study by DE 
SERVA-LEIT.~O will shed light on the phonetic problems. Till then this 
examination of some of the fundamental problems may be of some 
importance. Some issues of interest to theoretical linguistics will also 
turn up. 
2. MATHEMATICAL AND LINGUISTIC REPRESENTATIONS OF NUMBERS 
Representations of numbers using digits (figures) will be called 
mathematical representations, while representations using letters (sounds) 
organized in morphemes, words and phrases as other linguistic material 
will be called linguistic representations. The number 123 is now written 
in mathematical representation. The equivalent linguistic representa- 
tion in Swedish is (ett)hundratjugotre. There are often alternative repre- 
sentations within a language. The number 123 could thus also be ren- 
dered: ett-tv,~-tre, tolv-tre, or ett-tjugotre. It is above all in technical con- 
texts (telegraphy, telephony) that the latter types are used. The diffe- 
rence has to do with difference in division. In the first (normal) case the 
whole row of digits is taken as a unit and the highest numeral (position 
word) offered by the language (hundra) is used giving: (ett)hundra- 
tjugotre. In the other (technical) cases the series of digits is divided into 
smaller groups which are treated separately. In the extreme case each 
digit is treated separately. 
We will call the different divisions of the group of digits different 
fusion. We can show the differences as follows: 123, 12 + 3, 1 + 23, 
432 BENGT SIGURD 
1 + 2 + 3. The first case implies total fusion, since the whole series 
of digits is taken into consideration. The last case implies no fusion; 
cases in between may be characterized as partial fusion. We might 
call the case where the digits are treated separately spelling, since this 
is the word used when a word is decomposed in letters and each letter 
is spoken separately. In Swedish we could devise a new word siffrering 
(" digitalization ") as an equivalent of bokstavering and stavning which 
mean rendering each letter by its name. Spelling (bokstavering), e.g. 
rendering the word bad as be-a-de, is often used for names and other 
words with low redundancy. Military forces would use special letter 
names with more redundancy, in Swedish for bad: Bertil, Adam, David. 
The following is a table showing mathematical and linguistic repre- 
sentations of numbers. The linguistic representations are Swedish nu- 
merals according to different fusion. We are mainly interested in the 
relations between decimal representations and linguistic ones, but for 
some lower numbers binary and t(oman representations are given. 
We will touch on the problem of pronouncing binary representations 
briefly. The relations between Koman mathematical representations 
and Latin numerals are interesting but very complicated, and we will 
not suggest any conversion rules for those representations here. 
Mathematical Linguistic 
representation representation 
Normal (total Technical (partial 
Koman Decimal fusion) or no fusion) 
I 2 tvd tvd(a) 
XI 11 elva at-at 
XII 12 tolv ett-tvd 
CCCIX 309 rrehundranio tre-noll-nio 
4378 fyratusentre- fyra-tre-sju-dtta, 
hundrasjuttio- fyrtiotre-sjuttio- 
dtta dtta 
121457 etthundratjugo- ett-tvd-ett-fyra- 
entusen fyra- fem-sju, tolv-fjorton- 
hundrafemtlo- femtiosju, etthundra- 
sju tjugoett-fyrahundra- 
femtiosju 
123456789 etthundratjugo- ett-tvd-tre-fyra-fem- 
tre millioner sex-sju-dtta-nio, ett- 
fyrahundrafemtio- hundratjugotre-fyra- 
sex tusen sju- hundrafemtiosex-sju- 
hundradttionio hundradttionio 
Binary 
10 
1011 
1100 
FROM NUMBERS TO NUMERALS AND VICE VERSA 433 
The fundamental difference between fused and non-fused expres- 
sions is the presence of position words: -tio(-ton), hundra, tusen, million ... 
in fused forms. The etymology of such words might be far from the 
decimal system. It is well known that there often are rival systems, 
above all the 20-system (vigesimal), the 12-system (cf. dussin, gross) 
and the 5-system (quinary). Many ways of grouping items may seem 
natural. 
There are no position words associated with the positions of binary 
representations, but considering the popularity of the binary system 
in the computer age one might suggest some, e.g. duo (or pair) for po- 
sition 2 (counting from the right), quartet for position 3 (23) and octet 
for position 3 (28). Using these position words (group names) the number 
11 (eleven), which is 1011 in binary form would be: (one) octet one 
duo one. We might alternatively construct new position words for the 
positions of binary numbers. One way would be to use the decimal 
words rio, hundra etc., changing them, by substituting b for the first 
letter, into bi(o), bunclra, busen, billion, billiard ('). In this system 12 
(twelve), which is 1100 in binary form, would be: busenbundra. Since 
binary representation tends to get very long, an enormous number 
of position words would be needed. We refrain from developing fur- 
ther binary numerals. 
The choice of fusion form is apparently depending on the use of the 
number. The number 121457 is a telephone number. Such numbers 
are often divided into 2-groups, but 3-groups or separate pronunciation 
of each figure also occurs. It is a well-known fact that such high tele- 
phone numbers are difficult to get through correctly. In practice one 
introduces pauses at strategic points to facilitate communication. 
We will not discuss the communicative and mnemonic advantages 
of different systems here. Let us just mention that there are at least 
some cases, where the totally fused numerals give shorter expressions, 
e.g. en million compared to ett-noll-noll-noll-noll-noll-noll. The last 
example with all the zeros (noll) is probably difficult to get through 
over the telephone, since one might easily lose count. Position words 
have the advantage of indicating how far to the left we are. They are 
in a way the equivalents of the zeros used as position fillers in mathemat- 
ical representations (zeros do not vary according to position). _As is 
obvious from the table above there are cases where partial or no fusion 
gives shorter expressions than expressions based on total fusion. It is 
often said that since there are infinitely many numbers and each number 
has a name (word, numeral) in the language, there are infinitely many 
28 
434 ~rNGX $IGURD 
words in the language. If numbers can be infinitely long, words can 
be infinitely long too. 
Decimal fractions are rarely rendered by words, if written: three- 
point-fourteen looks strange, and so does the Swedish equivalent with 
komma instead of point: tre-komma-fjorton. The figures after the decimal 
point (komma) are generally given separately, in particular, if there is 
a long row of decimal figures. For instance, if the value of pi is to be 
given by more than 2 decimals one turns to separate pronunciation of 
the figures immediately: compare 3,14 and 3,14159. For practical pur- 
poses it seems the rule that figures after the decimal point (komma) 
are pronounced without fusion is sufficient, and we will not discuss the 
matter further. 
The year 1718 is pronounced sjuttonhundraarton not ettusensjuhun- 
draarton (just as in English). We will say that total fusion is applied, but 
not maximum fusion, since the maximum position word (tusen) of- 
fered by the language is not used. In Swedish the normal way of pro-- 
nouncing 1066 would be (ett)tusen-sextiosex, but tiohundrasextiosex might 
be heard (equivalent to the English ten sixty-six except for the lack of 
position word in English). Non-maximum fusion is only allowed in 
(1000)1100-1900. The year 2000 is pronounced dr tvd tusen (not tjugo 
hundra). The year 2384 is naturally renderd as tvdtusentrehundradttiofyra. 
The reason for not allowing non-maximum fusion above 19 has to do 
with a concept of "primary " numerals. The primary numerals in 
Swedish (as in many other languages) include (words for) 1-9, the base 
number 10, 11-19 (the -teens, in Swedish: -tontalen: the first two are 
irregular in Swedish as in all the other Germanic languages; further 
details below). 
The pronunciation of dates is paralleled by the pronunciation of 
prices. A price, e.g. 1900 may be rendered as nitton-hundra (nineteen 
hundred) or ettusen niohundra (one thousand and nine hundred). The price 
2300 (Kronor) may only be pronounced as tv8 tusen tre hundra (not 
tjugotrehundra). There is, however, an interesting alternative way of 
pronouncing prices. 2300 could be given as tvd-och-tre, 1900 could be 
given as ett-och-nio. A (used) car could thus have the price 1300, which 
may be rendered as trettonhundra, ett(t)usentrehundra or ett-och-tre. The 
contexts (or hidden units) are important. The expression tvd-och-tre 
could perhaps mean the same as tvd millioner trehundratusen (2300000) 
kronor, (tvd-komma-tre millioner), but not 230 kronor, nor 23 kronor. 
It might mean tvd kronor och tre i~re (2,03 kronor). But ambiguities of 
this sort are rare in practice. In order to avoid ambiguity one might 
FROM NUMBERS TO NUMERALS AND VICE VERSA 435 
pronounce the zero in such cases: tvd-octl-nolt-tre (with stress on nol O. 
Fortunately most prices of the type under discussion are unambiguous, 
e.g. tvd-och-tio (2,10), sexton-och-sjuttiofem (16,75). 
Numbers play an extremely important r61e in modern society. As 
the set of numbers often is restricted due to context, function etc. re- 
duced ways of expression, or sublanguages, develop. A certain row of 
figures or numerals may be completely unambiguous among used car 
dealers, although it is ambiguous from the point of view of the total 
language, or means something different in another sublanguage. De- 
tailed studies of the use of figures and numerals in different functions, 
contests and surroundings would certainly be rewarding. 
3. FROM MATHEMATICAL INTO SWEDISH REPRESENTATION. 
GENERAL CONSIDERATIONS 
Separate pronunciation of each figure is no great problem, and we 
will concentrate on total fusion in the following. We begin by presen- 
.ting the following table where position numbers and position words 
associated with positions are to be found. The term position word, 
of course, reflects our looking at the numerals from the point of view 
of the decimal positional representations. The words could also be called 
group names or number measures. For reasons of space we have not 
included any position word higher than milliard. 
example: 666 
Position number 10 9 8 7 6 5 4 3 2 1 6a6261 
position value 10g 10 s 10 v 106 105 104 108 102 101 100= 1 6.103 +6.101 
+ 6.10 o 
position word mill- mill- tu- hun- tio/ sexhundrasex- 
lard ion sen dra -ton tiosex 
There is no position word for unit in Swedish. One might use 
stycken (pieces, units), but presumably since position word for units 
is redundant it is generally left out. As a name for the whole set Swedes 
use ental, but it is not possible to use en in the numerals, perhaps because 
en lacks plural. There is no point in stating that en is there in the under- 
lying form but obligatorily deleted. (The numeral en could then be 
derived by deleting the numeral en before the position word en, which 
is an optional rule in many cases: (ett)tusen, (ett)hundra. But this seems 
unnatural sophistication.) 
436 BENGT SIGURD 
The steps between the higher position words are 108 (1000). We 
will see examples of languages with different steps later, such as Urdu. 
The only genuine position words are the first three. They are also Ger- 
manic, but they have not always been associated with the decimal sys- 
tem as is done now. Many Swedes would hesitate, when it comes to 
position words higher than million. Some might also have a feeling 
that languages use milliard, billion etc. differendy (which is also the case). 
The number 666 is represented in three different ways above. Let 
us call the normal decimal representation: positional decimal, or just 
decimal as usual. The representation 6.10 ~ + 6.101 + 6.100 may be 
called: analytic decimal, or just analytic representation, since it analyzes 
the number into the terms to be added. The terms do not have to be 
in a certain order or take certain positions. The representation sexhun- 
drasextiosex is called (fused) linguistic representation as before. 
Our purpose is to derive linguistic representations from decimal 
ones but the relations between the analytic (decimal) representation 
and the linguistic one are also of interest. We will compare generative 
grammars for the different types of representations briefly. 
Decimal representations, the input or output of our conversion rul- 
es, may be generated by the following simple generative rules, where 
N is the number, d is digit. The number of digits in the strings (n) may 
be infinite. 
(1) N- d (d) (d)... 
d ~1, 2, 3, 4, 5, 6, 7, 8, 9, 0 
If numbers such as 000 and 007 are not to be permitted, we have to 
state that the first d must be # 0. 
The analytic decimal representations may be generated by the fol- 
lowing rules, where (a) and (b) are variants with different order. 
(2) N~ 
d---~ 
l d.10 q- d.101 -I- d.10L,.d.10 '~ (a) 
d.10"... + d.10~ + d.101 + d.10° (b) 
1,2,3,4,5,6,7,8,9,0 
The b-variant can easily be derived from the decimal representa- 
tion by inserting powers of 10 according to position number. We might 
perhaps consider the b-variant as a semantic interpretation of the deci- 
mal representation. Rules converting decimal representations into ana- 
lytic ones would then seem related to the interpretative rules suggested 
FROM NUMBERS TO NUMERAES AND VICE VERSA 437 
in transformational grammar. The analytic representations would be 
the readings or the meaning of the decimal expressions. 
The " generative semantics" approach would then be to start from 
the analytic expressions and derive the normal decimal expressions. 
This can be done by deleting the powers of 10 from the b-variant, 
where the order then carries the information about the powers. The 
a-variant gives the units first and has to be reversed by a transformation 
in order to fit the standard decimal representation. Changing the de- 
cimal representation by mentioning units first would have some advan- 
tages in communication - presently the listener does not know anything 
about positions if numbers are given by pronouncing each digit sepa- 
rately as observed by C.-CH. ErERT (1970). Note, however, that the 
units are given first (in the left-most place) in many languages in the 
teens (tontalen), and in some languages eveal in higher numbers, e.g. 
German: ein-und-zwanzig. 
It is, of course, easy to generalize the rules (2) to cover systems with 
any base and any arrangement of the terms. 
Rules for Swedish numerals have been suggested in C.-CH. ELERT 
(1970). They are, however, too surface oriented for our purposes. We 
need "deep structure" rules for the numerals which are as close as 
possible to the decimal representations. Let us first suggest the follow- 
ing rules, covering numbers as great as millions. 
N-+ (d million) (d kundratusen) (d tiotusen) (d tusen) (d hundra) (d tio) (d) 
d --~ en, tvd, tre, fyra, fern, sex, sju, dtta, nio 
The coverage of the first rule can be increased by adding brackets 
with the proper content or by introducing recursion (e.g. N instead 
of d before millioner) substitution rules, such as tusen millioner ~ milliard, 
tusen milliarcler-+ billion (etc.). In order to derive surface structures 
from such deep representations we need the following main types of 
rules. 
1) Morphophonemic rules, which change sju tio into sjuttio, fyra 
into fjor before ton (fjorton) etc. 
2) Reordering rules which change the order between units and 
tens in the teens (tontalen). Our deep structure would generate en tio 
fyra for 14, which has to be changed into fjorton with the units before 
ton, the representative of rio. (the en is deleted according to 3.) 
438 BEN'GT SIGURD 
3) Deletion rules, which obligatorily delete en before tto and 
optionally delete ett before hundra and tusen. We have used the com- 
pound position words: tiotusen, hundratusen, for positions 5, 6. Since the 
tusen only occurs in those compounds if there is no other tusen to the 
right in the string we must delete tusen in other cases. 
4) Concord rules, which give the plural millioner etc. at proper 
places and the neuter form ett before neuter nouns such as hundra and 
barn "child ". 
It is possible to improve the deep structure rules suggested in various 
ways. Arguments could also be found in the pronunciation and ortho- 
graphic rendering of the numerals. The following rules are established 
with such arguments in view. 
(3b) 
N~ (G million) (G tusen) (G) 
G-~ (d hundra) (d rio) (d) 
d ~ en, tvd, tre, fyra, fern, sex, sju, dtta, nio 
In this approach hundreds, tens and units are treated as a group 
(constituent) which may occur alone, before tusen, million(er), mil- 
liard(er) etc. The difference between rules (3a) and (3b) is perhaps best 
shown by the tree diagrams for an example, such as sex millioner sex 
hundra sex rio sex tusen sex hundra sex tio sex (Fig. 1). 
The following arguments support analysis according to 3b (cf. also 
C.-CH. EI.~RT, 1970). 
1) Position words are spaced by 103 in Swedish (not necessarily 
in all languages) and for positions in between the fundamental group 
position words (hun&a, tio) are used. 
2) It corresponds to the division into 3-groups often made in 
decimal representations (666 666 666). M Swedish space is used; in Eng- 
lish comma is available for the task, since point is used in decimal 
fractions. 
3) Although the phonetic details are by no means clear, it seems 
the proper rules can be formulated easily within this framework. The 
main rule seems to be the following: The last (rightmost) d (simple 
number) within the group ((3) is always given the main stress. The 
others in the group are reduced accordingly. The G could perhaps best 
be treated as an attributive (Adjective Determiner) within an NP. The 
N within the NP is tusen, millioner, etc. (As observed above a head for 
the last group is lacking.) In high numbers several NP's are coordinat- 
FROM NUMBERS TO NUMERALS AND VICE VERSA 439 
(3a) 
N 
d million d hundra d tio d tusen d hundra d tio d 
(tusen) (t.sen) 
sex millioner sex kundra sex tio sex tusen sex kundra sex tio sex 
(3b) 
N 
d million d hundra d tlo d tusen d hundra d tio d 
sex millioner sex hundra sex rio sex tusen sex hundra sex tio sex 
Fig. 1. Deep structure trees for the number 6 666 666: sexmillioner sexhundrasextiosextusen 
sexhundrasextiosex according to two different solutions (3a and 3b). 
ed. The constituents within G may also be considered as coordinated 
NP's. Some languages show the conjunction (and) between certain 
NP's. The phonetic properties of numerals will be treated in greater 
detail in a study by Dr SrRPA-LErr.~o (to appear). 
4) The orthographic conventions for numerals are by no means 
clear. The parts rio/ton are treated (phonologically) as suffixes and 
always joined with the preceding number. The units are similarly joined 
with a preceding term rio (sextiosex), perhaps also with preceding hundra 
(sexhundrasex). As for higher terms it is hard to know: cases are difficult 
to come by. (Sexmillionersex or sex millioner sex?) The simple numeral 
440 BENGT SIGURD 
before hundra (the coefficient) may or may not be joined with hundra 
(sex hundra or sexhundra). The tens may or may not be joined with the 
preceding position word: sexhundra sextio or sexhundrasextio. In brief, 
the situation is unclear and detailed studies of usage are needed. Hand- 
books are conspicuously silent. 
Many persons probably join all numerals and write them as one long 
word as a radical solution in this situation. If conventions are to be 
introduced, they could easily be formulated within the tree structure 
suggested by (3b). A simple rule would be the following:join all words 
within the group G, but no other elements. This rule would introduce 
spaces where there are deep branchings in the tree. We would get the 
following division for our example: sex miUioner sexhundrasextiosex 
tusen sexhundrasextiosex. But it is possible to formulate other conven- 
tions if desired. Some languages use the hyphen (--) between certain 
constituents, as we will see. 
The relations between analytic representations and the deep struc- 
tures described by rules 3a and 3b are fairly simple. Conversion rules 
operating on analytic representations would have to replace the powers 
of 10 by proper position words, replace the digits by proper simple 
numerals etc. Taking the non-analytic decimal representation as input 
to conversion rules, implies inserting the proper position words, but 
except for that there is litde difference, We will now focus our interest 
on automatic conversion schemes. 
4. CONVERSION RULES CHANGING MATHEMATICAL INTO SWEDISH 
REPRESENTATIONS 
We are now ready to test conversion rules written as generative 
rules or instructions of different kinds. We will discuss alternative solu- 
tions at some points. We distinguish between two blocks of rules: 
I) fusion rules, which insert position words and introduce syntactic 
structure; 
II) lexical rules, which substitute words (simple numerals) for fi- 
gures (digits). 
Ia) Fusion rules. 
Count number of digits in decimal representation and mark positions. 
The (maximum) number of positions (digits) is determined by the position 
of the left-most digit which is not zero (~ (3). 
FROM NUMBERS TO NUMERALS AND VICE VERSA 441 
If 
10- digits 
7-9 digits 
5-6 digits 
4 digits 
3 digits 
2 digits 
then insert 
milliarder after pos. 
10, milliard if 1 in 
pos. 10 and higher 
positions empty. 
millioner after pos. 
7, million if 1 in pos. 
7 and higher post. empty. 
tusen after pos. 4. 
hundra after post. 3, if 
1 in pos. 4 not 0 in pos. 3, 
else tusen after pos. 4. 
hundra after pos. 3. 
+ tio after pos. 2 
Apply the rules until all groups of digits 
have been dissolved (interspersed by po- 
sition words). Determine positions anew 
for each group of digits. 
Examples 
25000000000 ~ 25 milli- 
arderO00000000, 21000- 
000000 -+ 21 milliarder.. 
1000000000 -+ 1 milliard.. 
16000000 ~ 16 millioner.. 
1000000 -+ 1 millionO00.. 
234000 --~ 234tusenO00 
1632-+ 16 hundra 32 
1000 -+ 1 tusen 000 
2400 -+ 2 tusen 400 
458 ~ 4 hundra 58 
96 ~ 9 + rio 6 
13~1 +rio3 
12-~ 1 q- rio 2 
121457 ~ 121tusen457 
lhundra21tusen457 
lhundra 2 + tioltusen457 
lhundra 2 + tioltusen 4 
hundra 5 + tio 7 
1000000000000000000 
l O00000000milliarderO00- 
000000 --~ lmilliardOOOO- 
O0000milliarderO00000000 
Normally the maximum position word is inserted, but as mention- 
ed before prices and years may be different, if between 1100-1999, which 
is taken into account in the rules. Instead of recursive application of the 
rules, compound position words such as tiotusen, hundratusen could be 
inserted (under certain conditions). 
As an alternative we might use the following procedure, which 
has some advantages when rules of pronunciation and rules for sepa- 
rating and combining numerals are to be formulated. It is based on 
deep structure rules 3b (above) which assumes the division of high num- 
bers into 3-groups (in Swedish often marked by space already). 
Ib) Fusion rules. 
Count number of digits and divide into 3-groups starting from right. 
Count number of 3-groups. Mark the groups starting from right. 
442 
After 
group 4 
group 3 
group 2 
Within 
group 
Distinguish positions 
1, 2, 3 within group 
counting from the 
right. Mark positions 
Z, y, X. 
II) Lexical rules. 
Substitute: 
2 + rio ~ tjugo 
1 + rio 2-+ tolv 
1 + rio 1 -+ elva 
1 + rio 3...9 ~ 3...9 + ton 
1 + tio ~ rio 
t tusen I ett / -- hundra 
1 ~ ~ en (neuter nouns) 
2 ~ tvd 
3__.. l tret / __l + tio + ton 
tre 
fjor / -- + ton 
4-+ fyr \] -- + tio 
fyra 
5  fem 
6 ~ sex 
7-+ l I + tio sjut / ~ + ton 
sju 
BENGT SIGURD 
insert 
milliarder after the group, 
if the group is > 1, else 
insert milliard 
millioner after group, if 
group is > 1, else million 
tusen 
insert 
hundra after third digit (x), if it is ~ 0 
+ rio after second digit (y), if it is ~ 0 
Example 
1 + tio 4 ~ 4 + ton (~fjorton) 
ett tusen ett hundra 
en million 
(1 + rio + 3 ~ 3 + ton ~) 3 + ton 
tret + ton 
4 + tio ~fyrtio/f~rtio 
FROM NUMBERS TO NUMERALS AND VICE VERSA 443 
l a(de)r I-- -t-ton 
8~ dt /-- + tio 
dtta 
l _ I+ ,,° 9 -~ \[ + ton nio 
0 -+ O (deletion) 
(1 + tio 8 -->)8 + ton --> arton 
Apply the rules in order, repeatedly if necessary, until all digits have 
been processed. 
As observed before, the elements tio/ton are always joined with the 
preceding number and treated as suffixes (-ton numbers take the grave 
accent). This phenomenon is related to the lengthening of t in rio/ton 
(sjutton, sjuttio, nitton etc.). We have treated this as allomorphic choice 
(not by phonological rules) assuming allomorphs of tre, sju, nio ending 
in t. In a traditional generative grammar (meeting demands of eco- 
nomy and simplicity) the doubling of t should be handled by the same 
rule (dental gemination) that gives bott from bo + t (participle of bo) 
and bldtt from bld + t (neuter of bld). We have introduced + rio and 
+ ton in our rules with the plus to be used (as internal juncture) for 
special purposes. There might be better ways of handling the situation; 
it is depending on how pronunciation and orthographic conventions 
are to be handled. We have some use of the plus sign, but have to de- 
lete it in some cases. In the rule that deletes en (ett) before rio (to avoid 
the impossible ett-tio) we also delete the plus sign. This gives the result- 
ant tio the right status, and it is not treated as a suffix in e.g. hundratio. 
Notice that the o in tio can only be deleted (optionally) in the suffix 
tio (femti(o), sjutti(o)), not in cases such as hundratio where tio is stressed. 
The fusion rules do what human beings do when they face a string 
of digits. System (Ia) corresponds roughly to a procedure, where the 
person counts the number of digits to get an idea how many tens, hun- 
dreds, thousands, ten thousands, hundred thousands, millions etc. are 
in the number, the order of magnitude. Fusion rules Ib correspond to 
what is done by a person who divides the string of digits into 3-groups 
(if this is not done already), counts the number of 3-groups to find out 
whether the number is in the thousands, millions, milliards etc. Estima- 
tion of numerousness is easily handled in a glimpse if the number is 
five or less. Longer strings need counting. Division into groups of 
three digits is a practical method, which facilitates determination of 
the size of the number (see B. SICURD, 1972). 
444 BENGT SIGURD 
5, FROM SWEDISH INTO MATHEMATICAL REPRESENTATION 
Let us now outline a system which is the reverse of the previous 
conversion system. Again we distinguish between two blocks of rules: 
I) lexical rules which identify numerals and substitute digits for 
them; 
II) fusion rules which identify the position words tusen, hundra etc.; 
delete these words using their information for placing the digits in 
proper positions. 
We might call these rules positioning rules. In addition a zero inser- 
tion rule is needed. Zeros must be inserted at all empty places in the 
decimal representation. We will apply the lexical rules first, but the 
reversed order or mixed order may have advantages. 
I) Lexlcal rules. 
Substitute: Example 
tjugo ~ 2 tio tjugofem ~ 2 rio fern (4 2 tio 5) 
tolv-+ 1 tio 2 tolvtusen ~ 1 tio 2 tusen 
elva ~ 1 tio 1 
3...9 ton ~ 1 tio 3...9 
tio ~ 1 rio, if not 1...9 before 
en I ett 
tvd---> 2 
tre(t) ~ 3 
frr(a) ( 
f~r ~ 4 
fl"or 
fem--~ 5 
sex ---> 6 
sju (t) --~ 7 
a(de)r 
dtta --~ 8 
dt 
nio --~9 
nit 
--~ 1, except after tus 
(as in tusen) 
(tretton---~) 3 ton--~ 1 tio 3 
femhundratio --~ femhundra 1 rio 
(--> 5 hundra 1 rio) 
trettio ~ 3 tio 
fjorton --~ 4 ton (--~ 1 tio 4) 
sjutton ~ 7 ton (~ 1 rio 7) 
arton--> 8 ton (--> 1 tio 8) 
dtta --~ 8 rio 
nittionio -~ 9 rio 9 
Apply the rules in order until all substitutable items have been changed. 
The examples show, within brackets; previous or following applica- 
tions of rules. 
FROM NUMBERS TO NUMERALS AND VICE VERSA 445 
II) Fusion rules (two alternatives a, b). 
,7) Place digit 
in position if it precedes (position 
word) 
1 no position word 
2 rio 
3 hundra 
4 tusen 
5 tio...tusen 
6 hundra...tusen 
7 million(er) 
8 tio...million(er) 
9 hundra...million(er) 
10 rnilliard(er) 
Insert 0 in empty positions 
Example (with pos. 
index) 
2 tio --* 2,. 
2 tusen --~ 24 
3 tio 4 tusen 
3544 (~ 34000) 
5 tusen 6 ~ 5461 ~ 5006 
b) Place in 
group o(), 
4 
3 
2 
1 
if preceding (position 
word) 
milllarde(er) 
million(er) 
tusen 
no position word 
1 tio 5 tusen-+ 
~(1 tio 5)3 
Place within group 
r z) 
in position 
i(z) 
2 (y) 3(x) 
if preceding 
no position word 
rio 
hundra 
Insert 0 in empty positions 
300 000 000 
Rules lib operate on the 3-groups mentioned above and derive the 
position of the groups and the position of the digits within the groups 
in two steps. 
We will not dwell on the details any more. Implementation on 
computer will determine which analysis is preferable taking the prop- 
erties of the computer, programming language, storage etc. into 
account. 
446 BENGT SIGURD 
6. OUTLINES OF CONVERSION RULES FOR SOME LANGUAGES 
6.1. English. 
Decimal representations could be considered as an interlingua in 
an automatic translation system between several languages. Transla- 
tion between two related languages, such as Swedish and English (or 
all the Germanic languages) could be handled without using decimal 
representations, since word-for-word translation (and a few additional 
rules) would suffice. This will be obvious from the following exami- 
nation, but we will not discuss such conversion rules for pairs of natu- 
ral languages. 
I) Fusion rules for English could operate just like in Swedish. 
Within 3-groups we assume that hundred is inserted after position 3, 
+ ty after pos. 2, etc. 
II. Lexical rules. 
Substitute: 
1 + ty 1 -+ eleven 
1 + ty 2 -+ twelve 
1 + ty 3...9 -+ 3...9 q- teen 
1 +ty-+ten 
1 --+ one/a 
2 --+ t twen/ ~ + ty 
two 
l +ty thir\] m q- teen 
3 -+ 1 three 
4 -+ i for/-- + ty 
four 
5 ~ l fif\[--five +teen 
6-.six 
7 ~ seven 
l eigh/- + ty 
8 -+ eight 
9 --, nine 
0 ~ ~ (deletion) 
FROM NUMBERS TO NUMERALS AND VICE VERSA 447 
English has the same complication in the position word for position 
2 as Swedish (and the other Germanic languages). In Swedish the ele- 
ments tio (alone), ti(o) in e.g.femtio and ton in e.g.femton are assumed to 
be manifestations of the same position (word). Intuitively most Swedes 
certainly associate the first two (tio and -ti(o)). In English the manifes- 
tations ten and -teen seem the closest. 
English has a hyphen between units and preceding tens (twenty- 
five; twenty-one thousand). This may be handled by inserting a hyphen 
before units (the last position) in 3-groups, if tens are preceding. Simi- 
larly and must be inserted before tens or units preceded by htmdreds, 
thousands etc. (250630: two hundred and fifty thousand six hundred and 
thirty). 
Notice that one/a before hundred, thousand and higher words must 
not be deleted in English. We refrain from giving any rules for the 
choice between one and a. 
English is known to have special habits in telephone numbers, e.g. 
the use ofo (ou) for zero and double x as in:five ou double one for 5011. 
Such idiosyncrasies must be taken into account when constructing 
automatic recognition systems. 
6.2. German. 
1 ein(s) 11 e/f 10 zehn 
2 zwei 12 zwi~If 20 zwanzig 
3 drei 13 dreizehn 30 dreissig 
4 vier 14 vierzehn 40 vierzig 
5 fiinf 15 fiinfzehn 50 fiinfzig 
6 sechs 16 sechzehn 60 sechzig 
7 sieben 17 siebzehn 70 siebzig 
8 acht 18 achtzehn 80 achtzig 
9 neun 19 neunzehn 90 neunzig 
100 (ein) hundert 21 einundzwanzig 
1000 (ein) tausend 22 zweiundzwanzig 
1000000 elne million 175 (ein) hundert (und) fiinfundsiebzig 
The position words in German are million, tausend, hundert, -zig/-zehn. 
German shows perfect similarity between the number for 10 and the 
suffix in the teens (zehn). There is little allomorphic variation. We note 
the exceptions for 11, 12. We take zwan as an allomorph ofzwei, just 
as we took twen as an allomorph of two in English. One might go fur- 
ther and consider zw8 in German, twe in English to in Swedish as allo- 
448 BENGT SIGURD 
morphs in zwb'If, twelve, tolv respectively. If so e/f, eleven, elva may be 
analyzed to contain a variant of the word for 1 and something like 
-l(e)v-, This is etymologically correct (the last root being related to 
Swedish lgimna, English leave). But both intuitively and technically 
this seems unnecessary. 
The conjunction und is obligatory in German between units and 
tens (the units preceding the tens), optional between hundreds and low- 
er terms. The unit word ein may optionally be deleted before hundert 
and tausend. It is easy to imagine the rules for German and we refrain 
from giving further details. 
6.3. French. 
1 un, une 11 onze 10 dix 
2 deux 12 douze 20 vingt 
3 trois 13 treize 30 trente 
4 quatre 14 quatorze 40 quarante 
5 cinq 15 quinze 50 cinquante 
6 six 16 seize 60 soixante 
7 sept 17 dix-sept 70 soixante-dix 
8 huit 18 dix-huit 80 quatre-vingts 
9 neuf 19 dix-neuf 90 quatre-vingt-dix 
100 cent 21 vingt et un 
1000 mil(le) 81 quatre-vingt-un 
1000000 million 91 quatre-vingt-onze 
101 cent un 
1876 dix-huit cent soixante-seize, rail huit cent soixante-seize 
Hyphen is used extensively between tens and units and between the 
elements in vigesimal numerals (e.g. quatre-vingt-dix). A connective et 
is used, but only before un after the numerals for: 20, 30, 40, 50, 60, 70. 
French has a morpheme ante for tens, but certain features must be asso- 
ciated with a vigesimal system. In the numbers for 11-16 one might 
distinguish a morpheme ze, which is in fact etymologically related to 
dix (E. Bouacmz, 1950, p. 68). French has thus three completely dif- 
ferent manifestations of the position element for ten: ante, ze, dix, dis- 
tributed somewhat irregularly. 
We give an outline of the conversion rules for French. 
i) 
Insert + ante, cent, mille... 
FROM NUMBERS TO NUMERALS AND VICE VERSA 449 
ii) 
Substitute: 
1 + ante 1..6 -+ 1..6 + ze 
1 +ante~dix 
7 + ante (1...9) --+ 6 + ante-1 + ante (1...9) 
8 +ante~4--2 +ante 
9 + ante (1...9) -+ 4-2 + ante (1...9) 
J. ---> un, f/ne 
dou \[ -- + ze 2--> 
deux 
tr / -- + ante 
3-+ trei /-- + ze 
trois 
quator / -- + ze 
4 ~ quar /-- +ante 
quatre 
I quin \[-- +ze 
5 -+ cinqu/ -- + ante 
cmq 
l sei / ~ + ze 
6 ~ soix /-- +ante 
slx 
7 --> sept 
8 --> huit 
9 --> neuf 
Example 
1 +ante 5-->5 +ze 
72 ~ 7 + ante 2 -+ 6 + ante 
1 + ante 2 ~ 6 + ante- 
2 + ze -~ soixantedouze 
There is great morphophonemic variation in these numerals as a result 
of the operation of various sound laws. In languages where the numerals 
are based on a different non-decimal system, such as the vigesimal or 
quinary, one might use a kind of adaptive rules between the decimal 
and the vigesimal, quinary etc. representations, which are suitable as 
underlying forms for the numerals. Since French only shows some frag- 
ments of a different system this is hardly necessary for French. The 
first block of rules within the lexical block can take care of irregulari- 
ties. These rules apply to more than one item. 
6.4. Danish. 
1 en, et 11 elleve 10 ti 
2 to 12 tolv 20 tyve 
3 tre 13 tretten 30 tredive 
29 
450 BENGT SIGURD 
4 fire :1.4 fjorten 40 fyrre (tyve) 
5 ibm 15 femten 50 halvtreds (indstyve) 
6 seks 16 seksten 60 tres (indstyve) 
7 syv 17 sytten 70 halvfjerds (indstyve) 
8 otte 18 atten 80 firs (indstyve) 
9 ni 19 nitten 90 halvfems (indstyve) 
100 et hundrede 1893 atten hundrede og treoghalvfems 
1000 (et) tusind(e) 
1000000 en milton 
Danish uses the conjunction og (and) between units and tens and be- 
tween tens and hundreds. Units precede tens. The lexical rules would 
have to include the following: 
1 + ti 3...9 ~ 3...9 + ten 
2...9 q- ti 1...9 --~ 1...9 og 2...9 + ti 
1 + ti -~ ti 
2 + tl -+ tyve 
3 + ti ~ tredive 
4 + ti~fyrre (tyve) 
5 "t- ti--~ halvtreds (indstyve) 
6 q- ti ~ tres (indstyve) 
7 q- ti -+ halvfjerds (indstyve) 
Example 
1 + ti 4 ~ 4 + ten (~fjorten) 
2 + ti 7 ~ 7 og 2 q- ti (-+ syvogtyve) 
It is tempting i:o do something more insightful to catch the vigesimal 
character of the later Danish numerals. Such rules should include the 
following facts. For 50, 60, 70, 80, 90 (y) the number of multiples of 
20 (x) is determined, using the simple equation (4) y = x.20. 
The resulting value of x is rendered by the usual numerals (with ~ 
some minor morphophonemic changes), unless it is a fraction, such 
as 2.5, 3.5, 4.5. A fraction such as 2.5 is not rendered by to (og en)halv, 
but by halvtre. If z is the value before halv we can apply the following 
rule 
(5) z halv ~ halv (z + 1) 
This rule is also used e.g. in telling the time in Swedish. Instead of tvd 
och en halv Swedes say halv tre, when it is half past two. What the rule 
does is to change an additive representation into a subtractive. 
FROM NUMBERS TO NUMERALS AND VICE VERSA 451 
The multiplication sign is rendered by sinds in the Danish numerals 
and special reduction rules apply to give the short forms. Fortunately 
new numerals (more like the Swedish) are catching on in Denmark. 
6.5. Burmese. 
1 ti', ta- (tahku) 11 hse.ta 10 tahse, hse 
2 hni', hna- 12 hse.hna 20 hnahse 
3 thoun: 13 hse.thoun: 30 thoun:ze 
4 lei: 14 hse.lei: 40 lei:ze 
5 nga: 15 hse.nga: 50 nga:ze 
6 hcau ~ 16 hse.hcau' 60 hcau'hse 
7 hkun(-ni',-ha') 17 hse.hkun (-ha') 70 hkun(na)hse 
8 hyi' 18 hse.hyi 80 hyi'hse 
9 kou: 19 hse.kou: 90 kou:ze 
1 tahku 101 taya.ti' 
10 tahse, hse 654 hcau' ya.nga :ze.lei : 
100 taya 1965 tahtaun.kou : ya.hcau'hse.nga : 
1000 tahtaun 123456789 hse. hnagadei thoun :dhan : lei :dhein : 
10000 tathaun : nga :dhaun : hcau'htaun, hkun-naya. 
100000 tathein : hyi'hse.kou : 
1000000 tathan : 
10000000 tagadei 
The signs ' , : and . mark word tones. Aspirated voiceless stop 
or fricative changes into unaspirated voiced stop or fricative. This rule 
explains the variation between hse : ze for tens, and between hku :gu for 
units. 
Burmese numerals are well adapted to the decimal system. Restric- 
ted knowledge in Burmese makes us hesitate but Burmese numerals 
seem very simple. They look almost as if they were constructed at the 
writing-table. The morphological analysis of the words is evident, there 
are no exceptions, such as our elva and tolv. The morphemes for 1 
and 2 have morphophonemic variation (special forms for compound- 
ing), but except for that there is little morphophonemic variation. 
The correspondences are direct, and conversion rules should be simple 
to write. We will only make some further comments. 
Burmese seems to have (unanalysable, non-compound) position 
words for high numbers, such as: 10000, 100 000, 1 000 000, 10 000 000 
(unless we are mistaken). Burmese also seems to have a word for unit: 
hku. In Burmese the unit before 10 in the expression for 10 may option- 
452 BBNGT $1GURD 
ally be deleted. We have noted the obligatory deletion of 1 before 10 
in several languages and the optional deletion of 1 before 100, 1000 in 
some languages. Burmese uses hse for + ty as well as + teen (and ten). 
This has the effect that some words, such as the words for 18 and 80 
are minimal pairs: hse.hyi(18): hyi hse(80), at least from a segmental 
point of view. We do not have such minimal pairs in Swedish because 
10 is represented by ton after the units in the teens, and by ti(o) before 
the units in the other cases. We note that traditional morphemic ana- 
lysis cannot reasonably treat ton and ti(o) as allomorphs in Swedish, since 
they contrast as infemtio :femton (cf. C.-CI~. Er~RT, 1970, p. 154). 
6.6. Hausa. 
1 daya 11 (goma) sha daya 10 goma 
2 biyu 12 (gorna) sha biyu 20 ashirin 
3 uku 13 (goma) sha uku 30 talatin 
4 hudu 14 (goma) sha hudu 40 arba'in 
5 biyar 15 (goma) sha biyar 50 hamsin 
6 shida 16 (goma) sha shida 60 sittin 
7 bakwai 17 (goma) sha bakwai 70 saba'in 
8 takwas 18 (goma) sha takwas 80 tamanin 
9 tara 19 (goma) sha tara 90 tis'in/casa'in 
21 ashirin da daya 99 sasa\[n da tara 
100 dari 200 daribiyu 300 dari uku 
1000 dubu/alif 2000 dubu biyu 
100000 dubu dari 1100 dubu da dari 
It is clear from this table that the Hausa numbers contain various 
complications from our point of view. The morpheme for 10 (goma) 
may optionally appear in the teens. If it is deleted, it seems sha would 
have to take its place and be considered as a variant (manifestation) of 
10. We cannot identify any simple numerals in the words for 20, 30 
etc. We might take in as a representative of 10, but the rest has to be 
treated as a suppletive allomorph of 2, 3, 4 etc. We note that the mul- 
tiplier (cocfi%icnt) is placed after its position word in numbers such as 
2000, 100 000. (Do attributives follow their heads in Hausa?) Note 
that, as a consequence, the only difference between the words for 
100000 and 1100 is the word da. Our knowledge of Hausa is restricted 
and we will not procede any further in this analysis. 
FROM NUMBERS TO NUMERALS _AND VICE VERSA 453 
6.7. Urdu. 
1 ek 11 gyar8 10 di~s 21 ykkis 
2 do 12 bar8 20 bis 22 bais 
3 tin 13 ter8 30 tis 23 teis 
4 car 14 c~iwd8 40 calis 24 c~wbis 
5 pa~c 15 piindr~ 50 p~cas 25 p~ccis 
6 che, chiiy 16 solii 60 sath 26 ch~bbis 
7 sat 17 siitrtl 70 siittiir 27 siittis 
8 ath 18 iJiitthar~ 80 i~ssi 28 ~tthais 
9 n~w 19 wnnis 90 niivve 29 wni~ttis 
31 yk#ttis 41 yktalis 100 (ek) si~v 
32 b~ttis 42 bealis 1000 (ek) hSzar 
33 tetis 43 tetalis 100000 (ek) lakh 
34 cSwvtis 44 ciiwalis 10000000 (ek) kiiror 
35 p@vtis 45 p~yntalis 854697253 pycasi k~ro C cheatis lakh 
36 ch~ttis 46 chealis si~ttanve hiizar do si~v trepiin 
37 s~y~tis 47 siiy~talis 
38 iiftis 48 iictalis 
39 wntalis 49 wncas 
We give these numerals just to show what complications may occur 
in languages. Rules for Urdu would have to take much morphopho- 
nemic variation into account. A morpheme is occurs in 20, 30, 40 and 
a teen morpheme r8 can a/so be attested, but there are many exceptions. 
The units go before the morpheme for 10 in the teens and in 21, 22 etc., 
but the elements seem merged. We note the existence of position words 
for high numbers spaced 102 , not 103 as in Swedish and English. 
7. CONCLUSION 
• After this brief survey of numerical systems, we can state that con- 
version between different representations does not cause great problems. 
We have mainly worked on written forms and disregarded the pho- 
nological problems. Solving the problems in connection with conver- 
sion between spoken numerals and decimal representations is very 
important One might even accept ad hoc solutions, e.g. using prere- 
corded position words and primary numerals, which are combined 
according to the rules indicated above - a synthesis based on "prefab " 
elements. 
454 BENGT SIGURD 
We might sum up the problems we have faced when writing con- 
version rules as follows: 
1) The fit to decimal representation is rarely as good as in Bur- 
mese. Deviations concern the use of other base numbers (5, 20, 12) 
and different order. Special adaption rules may be used in such cases. 
2) The order between multiplier (coefficient) and position words 
varies (as does the order between attributive and head in noun phrases). 
In Hausa 5 000 is (the equivalent of) thousand five, while European lan- 
guages have (the equivalent of)five thousand. The order between units, 
tens, hundreds, thousands, millions etc. only varies between units and 
tens. Many languages have the units first in the teens, some have the 
units first in all the tens. 
3) Languages rarely have exactly the same element in the word 
for 10, the teens and the " ties ". The variation is clearly useful and may 
have a functional explanation, as well as a historical. In Swedish the 
words for 15 (femton) and 50 (femtio) would have coincided, if 10 was 
manifested in the same way in both numbers. The complication only 
occurs if units are preposed as in Swedish. 
4) There are clear tendencies to delete redundant material in the 
sublanguage that numerals constitute. The words for 1 and unit are 
often deleted, but conventions vary between languages. 
5) Beside position words and fundamental numerals some lan- 
guages use additive (or subtractive) morphemes and morphemes mean- 
ing multiplication (" times "). But most languages rely heavily on 
order arrangements, which are very economic. 
6) Sporadic occurrences of fractions have been attested, e.g. in 
Danish. Such cases need special treatment. If they are isolated, lexical 
rules may take care of the matter without any insightful (and etymolo- 
gically suggestive) rules, at least for practical purposes. 
7) The conventions for joining numerical morphemes vary very 
much between languages and so does the use of hyphen. The rules can 
probably always be explained on the basis of (assumed) constituent 
structure. 
8) The pronunciation of numerals above all long numerals can 
presumably be predicted on the basis of segmental structure and tree 
structure. We have not investigated this area in any detail, however. 

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T. G. BAtt~Y, Teach yourself Urdu, Lon- 
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E. BouacI~z, Precis historique de phon& 
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BitteR, THORN, Fransk sprdktdra, Stock- 
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W. S. CORNYN, D. H. P~ooP, Beginning 
Burmese, New Haven, 1968. 

H. BRANT CORSTIUS, Grammars for num- 
ber names, The Hague, 1968. 

C.-CH. ELrRT, Riikneordens morfologi, in 
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V. HAMMARBERG, S, ZETTERSTROM, En- 
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H. HJORTn, S. LIDE, Fdrkortad tysk gram- 
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A. HOW~IDY, Concise Hausa Grammar, 
London, 1953. 

V. D. HYMrS, Athabaskan Numeral Sy- 
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N. N1etsEN, P. LmD~G.~RD HJORrH, 
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1959. 

Z. SAr.ZMAN, A Method for Analyzing Nu- 
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