India's Unique Place
in the World of Numbers and Numerals
Shrikant G. Talageri
[This
article was intended to be purely informative, and to serve no purpose other
than pointing out three uniquely special features of Indian numbers and
numerals. But, inevitably, I found one more strong piece of evidence to confirm
the OIT (Out-of-India Theory of Indo-European Origins) which is so irrefutably
established by the textual, linguistic and archaeological evidence.]
Numbers and
numerals form an important, even a vital, part of life. This point is too
elementary to waste time here in elaborating why it is so. The fact is that
India occupies a unique place in the whole world in the field of numbers and
numerals, and the purpose of this article is to place these unique features on
record:
A. The
Concept of Numbers.
B. The Written
Numeral System.
C.
Indo-Aryan Numbers.
A. THE CONCEPT OF NUMBERS
The very
idea of numbers is something which seems very natural to us. But is it really
so? Incredible though it may appear to us, there are some very primitive or
simple languages in the world which do not even have concepts or names for
numbers up to ten (the number of fingers on a human pair of hands, which
would seem to be the natural base for counting). This is mainly so among the
Australian aboriginal languages. The following examples of some Australian
aboriginal language numbers will make this clear:
Nunggubugu:
1.
anyabugij.
2. wulawa.
3.
wulanybaj.
4. wulal
wulal.
There are no
number words beyond four.
Kamilaroi:
1. mal.
2. bular.
3. guliba.
4. bular
bular.
5. bular
guliba.
6. guliba
guliba.
There are
specific number words for numbers up to three, and the same words are combined
to produce numbers from four to six, at least.
Gumulga:
1. urapon.
2. ukasar.
3. ukasar
urapon.
4. ukasar
ukasar.
5. ukasar
ukasar urapon.
6. ukasar
ukasar ukasar.
This
language has specific number words for one and two, and these are combined to
produce numbers from three to six.
A related
language Mabuiag has similar words from one to six:
1. urapun.
2. okosa.
3. okosa
urapun.
4. okosa
okosa.
5. okosa
okosa urapun.
6. okosa
okosa okosa.
And then
there is a number word for seven and any other number after 7:
7 or 7+.
ras.
There are
apparently a few rare languages in the extreme isolated portions of areas
within Papua-New Guinea, interior Africa and Patagonia (the southern half of
South America) which have similar structures in which the numbers do not go
beyond six or ten and are based on words from one to three.
But India
represents the world in microcosm: here we have the simplest and most
primitive number system in the world even as late as the twentieth-twenty-first
century CE, as well as the most developed and elaborate number
system in the world even as early as during the Vedic period (extending back beyond
3000 BCE, or, even as per the now discredited AIT version of Vedic history,
to 1200 BCE).
The simplest
and most primitive number system in the world (the word "primitive",
it must be emphasized here, is not a deprecatory word) is found in the
Andaman islands in India. Formerly (as per older colonial records, and I have
myself quoted them in an earlier article) it was believed in fact that the
Andamanese languages had numbers only for one and two. However, it appears this
is not so.
The
following are the number words in the Aka-bea-da (Greater Andamanese)
language, which only has words from one to five:
1: obatul.
2: ikpaurda.
3: edarobai.
4: eijipagi.
5: arduru.
But an even
simpler and more primitive form of number system, the simplest and most
primitive form in the world, is found in the Onge language, which
has numbers only from one to three, and any number above that is represented by
a word ilake which does not mean "four" but specifically means
"many":
1: yuwaiya.
2: inaga.
3: irejidda.
On the other
hand, as early as the Vedic Samhitas, we had words in India for very high
numbers. The Yajurveda, for example, in the course of a hymn (Yaj. 17.2),
casually lists the following words for numbers from ten (10^{1} or
10) to one trillion (10^{12} or 1,000,000,000,000):
10^{1}:
daśa.
10^{2}:
śata.
10^{3}:
sahasra.
10^{4}:
ayuta.
10^{5}:
niyuta.
10^{6}:
prayuta.
10^{7}:
arbuda.
10^{8}:
nyarbuda.
10^{9}:
samudra.
10^{10}:
madhya.
10^{11}:
anta.
10^{12}:
parārdha.
It is
obvious that while, for ritual purposes, the enumeration in this hymn stops at
10^{12}, logically there is clearly an understanding of the infinite
nature of this mathematical series and of the idea that these are just the
first steps in an infinite series of numbers each being a multiple of the
previous number by ten. This becomes apparent from countless references and
number words in the ancient Vedic and Sanskrit texts, but most particularly in
certain texts which play with mathematical ideas. For example, the Lalitavistara,
a Buddhist text, actually describes an even more elaborate system (where some
of the above words from the Yajurveda are now replaced by other words, and all the
names are given in multiples of hundred. Here in fact some of the above words,
like ayuta and niyuta, are given higher values):
10^{3}:
sahasra.
10^{5}:
lakṣa.
10^{7}:
koṭi.
10^{9}:
ayuta.
10^{11}:
niyuta.
10^{13}:
kaṅkara.
10^{15}:
vivara.
10^{17}:
akṣobhya.
10^{19}:
vivāha.
10^{21}:
utsāṅga.
10^{23}:
bahula.
10^{25}:
nāgabala.
10^{27}:
tiṭilambha.
10^{29}:
vyavasthānaprajñāpti.
10^{31}:
hetuhila.
10^{33}:
karaphū.
10^{35}:
hetvindriya.
10^{37}:
samāptalambha.
10^{39}:
gaṇanāgati.
10^{41}:
niravadya.
10^{43}:
mudrābala.
10^{45}:
sarvabala.
10^{47}:
visaṁjñāgati.
10^{49}:
sarvasaṁjña.
10^{51}:
vibhūtaṅgamā.
10^{53}.
tallakṣaṇa.
The text
does not stop there: it points out that this is just the first of a series of nine
counting systems that can be expanded geometrically, and then goes on to
mention the names of the culmination points of each of the nine systems (starting with
the number 10^{53 }above, as tallakṣaṇa,
dhvajāgravatī, dhvajāgraniśāmaṇī, vāhanaprajñapti, iṅgā, kuruṭu, kuruṭāvi,
sarvanikṣepa and agrasārā), culminating in a large number, 10^{421}, or one followed by
421 zeroes! This text, and many other Sanskrit
texts, go even further in indulging in flights of fantasy involving even higher
numbers. The point is not whether such incredibly high numbers could
possibly serve any practical purpose: obviously they could not! The point is
that the ancient Indian theoretical concept of numbers had a vision which was
limitless.
India therefore occupies a unique
position in the world: on the one hand, it has even in the
twenty-first century the Onge language with no number words of its own beyond
three (i.e. the simplest number system in the world), and on the other, it
had even in ancient times:
a) number words for numbers as high as
10^{53}, and, in theory, even as high as 10^{421}, and in further theory, going into
unimaginably and fantastically high numbers beyond even that;
b) the concepts of zero, finite numbers and infinity (and, in Jain
texts, even different categories of what are now called transfinite numbers);
c) the concept of fractions (found
even in the Rigveda, in the Puruṣa sūkta, Rig. X.90.3,4);
d) the concept of negative numbers.
[All this is apart from the highly
developed state of almost every branch of Mathematics in ancient India].
B. THE WRITTEN NUMERAL SYSTEM
Numbers (at
least till three) are found in every language in the world. A written numeral
system is something different from the mere concept of numbers. The numeral
system used all over the world today is the system invented in India. In
popular parlance, this is often described as follows: "India invented/contributed
the zero". But this is an extremely haphazard statement, at least when
it comes to the importance of India in the history of numerals: the zero was
also (at much later dates) independently invented in ancient Mesopotamia and
Mexico (the Mayans). Also, it is quite a silly way of putting it. It sounds like some old-time fable: all the ancient civilizations of the
Old World got together and decided "let us invent/contribute numbers".
China announced that it was contributing the numbers one, four
and six. Egypt announced it was contributing two, three
and nine. Mesopotamia announced it was contributing five, seven
and eight. India, a little slow off the mark, was left with
nothing to contribute. Then, the Indian representative had a brilliant idea: he
immediately invented the zero, and announced "we contribute zero"!
The fact is,
zero is just one essential part of the whole of the present day decimal
numeral system which is used all over the world and which was
invented/contributed by India and which is also the basis for the binary
system which is used in computers (with a change of base from ten to two) .
Numeral systems were independently invented by every
highly developed civilization in the world: Egypt, Mesopotamia, China, Mexico
and India. Most of the other civilizations of West Asia and Mediterranean
Europe derived or developed their own numeral systems based on the Egyptian
system. The numeral system of each civilization provides an indication of the
stage of development of mathematical logic in each civilization, as we will
see, and the Indian system represents the highest stage of development: the Egyptian
system represents the first systematic stage of development, the Chinese
system represents the second systematic stage of development, and
the Indian system represents the third and final systematic stage
of development.
The very
idea of numbers contains the first seeds of any numeral system. We can imagine
different societies from the most primitive times which had numbers (at least
up to three in the simplest and most primitive system) but did not have any
method of recording numbers in the form of a written numeral system.
The first
primitive stage of
recording numbers must have started in a pictorial form. In a primitive
society, a man possessing, for example, 12 cows and 5 sheep thought of recording the
fact by drawing 12 pictures of a cow and 5 pictures of a sheep. The very
concept of representing numbers in writing (albeit pictorial) is the
characteristic of this first stage.
In the
second primitive stage, as society became larger and more complicated, the concept of numbers
must have evolved from the concrete to the abstract. Thus, finding it tiresome
to draw 12 pictures of a cow and 5 pictures of a sheep, the man in a society at
a more developed stage conceived the idea of representing each unit by an
abstract picture (most logically a simple vertical or horizontal line): thus 12
lines followed by the picture of a cow, and 5 lines followed by the picture of
a sheep. The concept of abstract numbers, as opposed to numbers as an intrinsic
aspect of some concrete material unit, is the characteristic of this second
stage.
In the
third primitive stage,
as the number of units became much larger and more cumbersome, it would be
tiresome to keep track of the number of individual pictures. Draw a series of 152
vertical lines in a row and try to count them again, to see how clumsy it would
be and how susceptible to counting errors! This must have led to the evolution
of numbers from the individual unit to the collective unit. This can be seen
even today in a system of keeping scores which is still quite commonly used: after
four vertical strokes to indicate four scores, the fifth stroke is a horizontal
stroke drawn across the earlier four strokes, indicating five or a full hand.
After that the sixth score is recorded by another vertical stroke at a little
distance from the first hand. The concept of an abstract unit consisting
of a collection of a certain fixed number of individual abstract units is the
characteristic of this third stage.
[This fixed
number was different in different primitive societies: the most common, natural
and logical number was ten in most societies since human beings have ten
fingers on the hands for counting, but it could also be (and was so in some
societies) five (one full hand) or twenty (the total number of
fingers on both hands and feet). If human beings had had twelve fingers
instead of ten, the natural numeral system would have been
mathematically even more effective, since twelve is divisible by two, three, four
and six, while ten is divisible only by two and five. And it would also have
fit in with some other aspects of nature, such as the twelve months in a
natural year, the twelve tones in a natural octave, etc.].
From this
point start the three systematic stages of development of the numeral
system:
1. The Egyptian
numeral system represents the first stage of development. This stage
involves the invention of a continuous recurring base. The base (as in
most cultures) is ten. The main problem in any numeral system that was
solved by the Egyptian system was the repetition of symbols beyond nine
times. The Egyptian system had one symbol for one, another for ten,
another for hundred, and so on, for subsequent multiples of ten (see
chart). Each symbol could be repeated as many as nine times to represent the next
number in the series. Thus to write 4596, first the symbol for thousand
was repeated four times, then the symbol for hundred five
times, then the symbol for ten nine times, and finally the symbol
for one six times:
The symbols
for 1 (10^{0}), 10 (10^{1}), 100 (10^{2}), 1,000 (10^{3}), 10,000 (10^{4}), 100,000 (10^{5}), and 1,000,000 (10^{6}), respectively are as follows:
1 (10^{0})
10 (10^{1})
100 (10^{2})
1,000 (10^{3})
10,000 (10^{4})
100,000 (10^{5})
1,000,000 (10^{6})
4596:
4096:
4006:
2. The Chinese
numeral system represents the second stage of development. Like the
Egyptian system, it has symbols to represent the numbers one, ten
and multiples of ten. But it eliminated the need to repeat these symbols
from two times to nine times to represent multiples of the
symbols. The logic used was the same as the logic involved in replacing the
twelve pictures of a cow (in the primitive stage explained earlier) with twelve
abstract symbols for one (usually a vertical line) followed by the
picture of a cow. Here the repetitions of the symbol were replaced by new
symbols representing the number of repetitions. That is, any symbol (one,
ten, hundred) required to be repeated only in eight ways:
twice, three times, four times, five times, six times, seven times, eight times
or nine times. The Chinese system therefore also invented eight
new symbols to represent the abstract numbers two to nine, and merely
placed the new symbols before the original symbols (ten, hundred,
etc.) as required in representing any number. Thus to write 4596, the
Chinese would place the following symbols in the following order: four, thousand,
five, hundred, nine, ten, six. The following
chart shows some of the Chinese numerals (a sixth century book gives these symbols
from 10^{2} to 10^{14}, see below, but in practice, the Chinese followed,
and still follow, in cases where the traditional numbers are still used,
different systems of combinations of symbols to express large numbers. In this,
many of the symbols given below have much larger values in modern usage):
1-9: 一 二 三 四 五 六 七 八 九
10^{1}: 十
10^{2}: 百
10^{3}: 千
10^{4}:
萬
10^{5}: 億
10^{6}: 兆
10^{7}:
京
10^{8}: 垓
10^{9}: 秭
10^{10}: 穰
10^{11}: 溝
10^{12}: 澗
10^{13}: 正
10^{14}: 載
Thus:
4596: 四 千 五 百 九 十 六
4096: 四 千 九 十 六
4006: 四 千 六
3.
The Indian numeral system represents the third and final stage
of development. The Chinese system had eliminated the need for repeating
symbols from two to nine times to represent the next number in any
series, but the system still required a fresh symbol to represent each next
multiple of ten (i.e. 10^{2}, 10^{3}, 10^{4}…).
The Indian system, by using a fixed positional system and a symbol for zero, eliminated
this need to invent an endless number of symbols and made it possible to
represent any finite number without any limit by a simple system of ten symbols
(1-9 and 0).
1.
१
2. २
3. ३
4. ४
5. ५
6. ६
7. ७
8. ८
9. ९
0. 0
The shapes of the actual symbols used do not matter: the
numeral symbols are different in different Indian languages, and even the
"Devanagari" numeral symbols in Hindi and Marathi, for example, have
noticeably different shapes. The Indian numeral system was borrowed by the
Arabs, who gave the symbols different shapes again, and later by the Europeans
from the Arabs with other similar changes in the shapes. It may be noted,
moreover, that some of the Devanagari (Sanskrit) numerals, which were the
ultimate basis for the shapes of the symbols in all the other systems, clearly
bear some resemblance to the initial letters of the respective Sanskrit number
words: १
(ए), ३
(त्र),
५ (प), ६ (ष).
The binary system used in computers is
a direct derivative of the Indian decimal system, with a change of base from ten
to two: so, while the Indian decimal system has ten symbols (nine
number symbols and a zero), the binary system has two symbols (one
and zero), and the place values from the right to the left are not 1,
10, 100, 1000…. as in the decimal system, but 1, 2, 4, 8, 16…. .
Thus, in the binary system:
4596:
1,000,111,110,100.
4096:
1,000,000,000,000.
4006:
111,110,100,110.
Clearly, while the binary system is
useful in the world of computers, the decimal system is more practical for the
daily use of human beings.
Now, if the Egyptian, Chinese
and Indian systems represented the three logical stages in the
development of a logical and practical numeral system, what did the numeral
systems of the other civilizations represent? They represented deviations
from the logical line of thinking, which is why their systems ultimately
failed to acquire the universality of the Indian system.
1. The Babylonian numeral system:
The Babylonian (Mesopotamian/Cuneiform)
numeral system, to begin with, had symbols for one and ten, and
derived the numbers in between accordingly by repetitions:
The numbers for 1-10 are as follows:
The symbols for the tens
numbers were also formed by repeating the symbol for ten.
The numbers for 20, 30, 40, 50 and 60:
And here was the catch: although the
Babylonians had symbols for one and ten, their numeral system was not a decimal
system (i.e. with a base of ten): it was a unique sexagesimal system
(i.e. with a base of sixty)! Therefore their place values from the right
to the left were not 1, 10, 100, 1000…. as in the decimal system, but 1, 60,
3600, 72000…. . Therefore, the symbol for one also served as the symbol
for sixty, three thousand six hundred, seventy-two thousand,
etc., depending on its position from the right in a composite numeral. The
Babylonian system had three main faults:
1. Just as the binary system
(howsoever vital to computers and cyber technology) is too small for normal
human usage, a sexagesimal system was too large and unwieldy for human
usage and computation.
2.. To be effective even as a sexagesimal
system, it should have had sixty symbols (for the numbers from one
to fifty-nine, and one for zero), but it only had symbols for one
and ten. Of course, the symbols, as we can see above, were joined
together, but that did not really improve matters. And, even if there had been
sixty different symbols, it would still have been too large and unwieldy for
common human use.
3. It did not have a symbol for zero.
Therefore, it was not clear whether the symbol for one, all by itself
and without being a part of a larger composite numeral, represented one
or sixty or three thousand six hundred or seventy-two thousand
or something bigger. In the Indian system, you can distinguish not only between
1, 10, 100, 1000, etc. because of the zeroes, but also between 40006,
40060, 40600, 46000, 4006, 4060, 4600, 406, 460 and 46. In the Egyptian and
Chinese systems, even without the zero, all these numbers could be
distinguished because the "position" of each individual number in the
composite numeral was distinguished by a different symbol (for ten, hundred,
thousand, etc.). The Babylonian system, although it was effectively used
by the Babylonians for their different purposes, was a very faulty system in
which, for example, not only could the same symbol represent 1, 60, 3600,
72000, etc., but the same combination of symbols could represent, to take the
simplest example, 3601, 3660 and 61.
[Later in time, a zero symbol was
invented, but it was not really properly understood, and was used only at the
end of a composite numeral].
To continue the same examples of the
numbers already seen in the other systems, the Babylonian system would write
them as follows:
4596: (1 x 3600, 16 x 60,
36 x 1):
4096: (1 x 3600, 8 x 60,
16 x 1):
4006: (1 x 3600, 6 x 60,
46 x 1):
2. The Mayan numeral system:
Like the Babylonian numeral system,
the Mayan (Mexican) numeral system also was not a decimal system (i.e.
with a base of ten): it was a vigesimal system (i.e. with a base of twenty).
Basically it had only three symbols, for one, five and zero,
and the other numbers between one and twenty were written by
repetitions of symbols. The Mayans also,
thus, had discovered the principle of using a zero symbol. The place
values in this system (not from the right to the left as in other
systems, but from the bottom to the top), were not 1, 10, 100, 1000…. as in the
decimal system, but 1, 20, 400, 8000…. (at least we must assume this theoretically
here for the moment for our study of the numeral system, but this was not
strictly accurate as we will see presently). The symbols from one to nineteen
were as follows:
The numbers 1-10:
The numbers 11-19:
The number 20:
The Mayan system was basically a
marvelous one: it had a strict positional system as well as a fully-developed
zero concept and symbol; but it suffered from certain faults:
1. To be fully effective as a vigesimal
system, it should have had twenty symbols (for the numbers from one
to nineteen, and one for zero), but it only had symbols for one,
five and zero. The numbers in between one and twenty
were written by repetitions of the symbols for one and five.
2. For religious reasons, to fit in
with the (roughly) 360 days in the calendar, the Mayans tweaked the base
of the vigesimal system, so that instead of the place values in this
system (from the bottom to the top) being 1, 20, 400, 8000, 160000…. as
in a regular vigesimal system, they were 1, 20, 360, 7200, 144000…. . In
short, there was a break in the regularity of the recurring base at the very
second multiple, so that the third place from the bottom represented 360
instead of 400, and after that all the subsequent bases continued at
multiples of twenty: The numbers for 1, 20, 360, 7200, 144,000, and
2,880,000 are as follows:
1.
20.
360:
7,200:
144,000:
2,880,000:
We have already seen certain numbers
written in all the numeral systems discussed so far. The following are their
forms in the Mayan numeral system:
4956: (13 x 360) + (13 x 20) + (16 x
1):
4906: (13 x 360) + (11 x 20) + (6 x1):
4006: (11 x 360) + (2 x 20) + (6 x 1):
3. The Egyptian-derived
Mediterranean and West Asian Numeral Systems:
The Egyptian numeral system
that we have already examined (called the Hieroglyphic numeral system)
was adopted by the Greeks, and from the Greeks by the Romans,
with modifications. The Egyptian Hieroglyphic numeral system, as
we have seen, was at the first stage of
development of a logical and complete system of numerals. But
unfortunately, instead of developing it in the right direction and reaching at
least the second stage of development, as for example represented in the
Chinese numeral system, the Greeks and the Romans went off
at a tangent from the logical line of development in trying to simplify and "develop"
the Hieroglyphic numeral system.
At the same
time, the Egyptians themselves "developed" another system of
numerals, distinct from the earlier system, called the Hieratic numeral
system. This system was adopted by the Greeks (and called the Greek Ionian
numeral system in opposition to the earlier Greek Attic numeral system
derived from the Egyptian Hieroglyphic numeral system) and by all the
other prominent civilizations and cultures of the Mediterranean area and West
Asia (including the Israelites and the Arabs) except the Romans.
This represented another "development" at a tangent from the logical
line of development:
a. The
Attic Greek numeral system:
The Greeks adopted the Egyptian Hieroglyphic numeral system, replacing the
Hieroglyphic symbols with Greek letters (being the first letters of the
respective Greek names of the numbers), as follows:
The numbers 1, 10, 100, 1000, 10000:
Ι Δ
Η Χ Μ
The first
ten numbers 1-10 should naturally have been written as follows:
Ι Ι Ι Ι Ι Ι
Ι Ι Ι Ι Ι Ι Ι Ι Ι Ι Ι Ι
Ι Ι Ι Ι Ι Ι Ι Ι Ι Ι Ι Ι Ι Ι Ι Ι Ι Ι Ι Ι Ι Ι Ι Ι Ι Ι Ι Δ
However, the
Greeks decided to simplify or "develop" the numeral system to reduce
the number of repetitions of a symbol within a compound numeral. Their solution
was to invent mid-way symbols between 1, 10, 100, 1000, 10000, etc., as follows:
The numbers
5, 50, 500, 5000, 50000:
Therefore, the Greek symbols for the
first ten numbers 1-10 were as follows:
Ι Ι Ι Ι Ι Ι
Ι Ι Ι Ι
Ι
Ι Ι
Ι Ι Ι
Ι Ι Ι Ι Δ
The three
numbers that we saw in the different systems already described would appear as
follows in the Greek system:
4596: Χ
Χ Χ Χ
Δ Δ Δ Δ
Ι
4096: Χ
Χ Χ Χ
Δ Δ Δ Δ
Ι
4006: Χ
Χ Χ Χ Ι
b. The
Roman numeral system: The Romans adopted the Attic Greek numeral
system, providing their own symbols for the Greek ones:
1, 5, 10, 50,
100, 500, 1000, 5000, 10000, 50000, 100,000:
I V
X L C
D M V
X L C
[The numbers 5,000 onwards have a
horizontal line above the symbol, but due to lack of such a font, the symbols
here have been underlined]
However, the Romans decided to
"develop" the system further. They found even four repetitions of a
symbol within a compound number (as in IIII for four and VIIII
for nine) too
much, and decided to reduce the fourth repetition by introducing a
minus-principle: instead of having the bigger number followed by the smaller
number four times, they decided to place one symbol of the concerned smaller
number before the bigger number to indicate "minus one". Thus:
1-10:
I II
III IV V
VI VII
VIII IX
X
Tens 10-100:
X XX
XXX LX L
LX LXX LXXX XC
C
Hundreds 100-1000:
C CC
CCC CD D
DC DCC DCCC
CM M
1000:
M
And so on.
The three numbers already shown in the other systems would appear as follows in
the Roman numeral system:
4596: MVDXCVI
4096: MVXCVI
4006: MVVI
c. The Hieratic
numeral system: The Egyptians themselves invented another new numeral
system, a sort of shorthand numeral system, where they had nine symbols for the
numbers 1-9, nine symbols for the numbers 10-90, nine symbols for
the numbers 100-900, and so on, based on the letters of the Hieratic
script. This numeral system was then adopted by the Ionian Greeks,
using the symbols of their alphabet to represent the numbers. The Hieratic
numerals and the Ionian Greek numerals are shown in the charts below:
The same system was also
adopted by almost all the cultures and civilizations of the Mediterranean area
and West Asia (except the Romans), including the Arabs and the Israelites,
using the symbols of their respective alphabets.
This exposition of the
numeral systems of the world makes it clear why the Indian numeral system was
universally adopted all over the world, and all the other numeral systems fell
into disuse (although still used as secondary symbols in scholarly works
or for other particular and restricted purposes, as for example the Roman
numeral system in western academic and religious works or a much-modified
Chinese numeral system in China).
C. INDO-ARYAN NUMBERS
One aspect
of Indian numbers which is not generally recognized is that the numbers in the
Indo-Aryan languages of North India have one feature (though not exactly a
positive feature) which makes them unique among all the languages of the world:
they are probably the only languages in the world where anyone learning the
language (any North Indian Indo-Aryan language) necessarily finds that he has
to individually learn or memorize every single number from one to hundred.
To
understand this fully, one must first understand the methods by which the
different world languages form their numbers 1-100. We will examine the
subject under the following heads:
C-I. Sexagesimal
systems (with a base of 60).
C-II.
Quindecimal systems (with a base of 15).
C-III. Vigesimal
systems (with a base of 20).
C-IV.
Decimal systems (with a base of 10) with words for 1-10 and 100.
C-V. Decimal
systems (with a base of 10) with words for units 1-9 and tens 10-100.
C-VI.
Decimal systems (with a base of 10) with words for numbers 1-19 and tens 20-100.
C-VII. Decimal
systems (with a base of 10) with words for numbers 1-100.
C-VIII.
Historical Implications of the Indo-Aryan number system.
C-I. SEXAGESIMAL
SYSTEMS (WITH A BASE OF 60):
The sexagesimal
system (with a base of 60, although with a subset of 10) is very
rare, and we will look at it before moving on to the two main systems. I
can personally think of only one language today with such a system (probably
also found in some related neighboring languages), though the ancient
Mesopotamians (Sumerians/Akkadians/Assyrians, etc., who were the only ones to
use a sexagesimal numeral system) may have had sexagesimals in
the spoken number system as well. This rare language is the Masai
language, belonging to the NiloSaharan/Sudanic language
family, and spoken in southern Kenya and northern Tanzania in east Africa. The
numbers are as follows:
1-9: nabu,
ari, üni, ungwun, miet, elle, nabishäna,
issiet, nawdu
10, 20, 30,
40, 50, 60: tomon, tigitum, ossom, arrtam, orrnom,
ïp
70, 80, 90,
100, 110: ïp-tomon, ïp-tigitum, ïp-ossom, ïp-arrtam, ïp-orrnom
Other
numbers in between 10-60 are formed by the tens word
followed by the following secondary forms of 1-9: obbo, are,
ogüni, ungwun, oimiet, oīille, nabishäna,
oissiet, nawdo
sexagesimals 60, 120, 180, 240, etc: ïp,
ari-ïp, üni-ïp, ungwun-ïp, etc. (60, 2x60, 3x60, 4x60, etc.)
Other
numbers above 60: sexagesimal (60, 120, etc) followed by 1-59.
Thus:
11 is tomon-obbo
(10+1), 99 is ïp ossom-nawdo (60+30+9), 179 is ari-ïp orrnom-nawdo (60x2+50+9).
C-II. QUINDECIMAL
SYSTEMS (WITH A BASE OF 15):
Unlikely
though it seems, there is even a language with a quindecimal system,
i.e. with a base of 15 (and it does not even have a subset of 10)!
This is the Huli language of Papua New Guinea, belonging to the Papuan
language family. The possible origin of such a system (as also the above sexagesimal
system) is hard to pinpoint: perhaps it is based on the number of days in a
lunar fortnight.
The numbers
are as follows:
1-14: mbira,
kira, tebira, maria, duria, waragaria, karia,
halira, dira, pira, bearia, hombearia, haleria,
deria
15, 30, 45,
60, 75, 90, 105 (and so on): ngui-ra, ngui-ki, ngui-tebo,
ngui-ma, ngui-dau, ngui-waraga, ngui-ka (and so on,
i.e. 15x1, 15x2, 15x3, etc.).
16-29: nguira-ni-mbira….nguira-ni-deria
(i.e. 15+1 to 15+14)
Other
numbers between the quindecimals are counting according to the serial
position: e.g. the numbers 31-44 belong to the "third series of
15" culminating in 45, the numbers 46-59 belong to the
"fourth series of 15" culminating in 60, etc. The names of the
series (covering the numbers upto 100) are as follows:
third series
31-45: ngui-tebone-gonaga (45: ngui-tebo)
fourth
series 46-60: ngui-mane-gonaga
(60: ngui-ma)
fifth series
61-75: ngui-dauni-gonaga (75: ngui-dau)
sixth series
76-90: ngui-waragane-gonaga (90:
ngui-waraga)
seventh
series 91-105: ngui-kane-gonaga
(105: ngui-ka)
Other
numbers (between the quindecimals) 31 onwards: previous quindecimal + new
series (to which the following unit belongs) + unit number. Thus 31: ngui-ki
ngui-tebone-gonaga mbira. (i.e. 30+third-series+1)
99: ngui-waraga
ngui-kane-gonaga dira (i.e. 90+seventh-series+9).
The Huli
numbers are complicated because of two things:
1. The odd
(to everyone else in the world, except the speakers of Huli) base of 15.
2. The
illogical addition of the series name (based actually on the name of the following
quindecimal) between the previous quindecimal and the unit:
thus, 31 could well have simply been ngui-ki mbira (30+1) and 99 could
have been ngui-waraga dira (90+9).
However, the
first complication is part of this rare system, and the second one can be
eliminated as shown above, and (even if it isn't eliminated, still) we get a
very regular quindecimal system.
C-III. VIGESIMAL
SYSTEMS (WITH A BASE OF 20):
Vigesimal number systems are those which are
based on 20, although they usually have a subset of 10. To learn
the numbers, one necessarily has to memorize the numbers 1-19, the vigesmals/tens
numbers from 20-100, and the regular procedure for forming the other
in-between numbers.
The two
characteristics of these languages are:
1. The vigesimal
numbers 40, 60, 80, and sometimes 100, are based on
the word for 20.
2. The other
numbers are formed by adding the numbers 1-19 to the vigesimals.
In a few
languages, the numbers 1-19 are based on an internal subset not of ten
but of five. The most perfect example of this is the Turi language
from the Austric (Austro-Asiatic) family, spoken in the adjoining
parts of Jharkhand-W. Bengal-Orissa in India, which shows this subset of five
very clearly, with the words for 5, 10 and 15 literally
meaning "one hand", "two hands" and "three hands"
respectively. Another example is the Nahuatl/Aztec language of
Mexico:
Turi
(Austric-KolMunda):
1-5: miad,
baria, pea, punia, miadti
6-10: miadti-miad,
miadti-baria, miadti-pea, miadti-punia, baranti
11-15: baranti-miad,
baranti-baria, baranti-pea, baranti-punia, peati
16-19: peati-miad,
peati-baria, peati-pea, peati-punia
20, 40, 60,
80, 100: lekacaba, bar-lekacaba, pea-lekacaba, punia-lekacaba,
miadti-lekacaba
Other
numbers: vigesimal numbers 20, 40, 60 or 80 followed by 1-19. Thus:
21: lekacaba miad (20+1), 99: punia-lekacaba peati-punia (4x20+19).
[Khmer
(Cambodian), which also belongs to the Austric family, also originally
had this subset of five, but the language now uses numbers borrowed from
the unrelated Thai language for numbers beyond 10. The Khmer
numbers 1-10 are:
muǝy, pii, bǝy, buǝn, pram,
pram-muǝy, pram-pii, pram-bǝy, pram-buǝn, dap].
Nahuatl/Aztec
(Amerindian):
1-5: ce,
ome, yey, naui, macuilli
6-10: chica-ce,
chic-ome, chicu-ey, chic-naui, matlactli
11-15: matlactli-on-ce,
matlactli-on-ome, matlactli-on-yey, matlactli-on-naui, caxtulli
16-19: caxtulli-on-ce,
caxtulli-on-ome, caxtulli-on-yey, caxtulli-on-naui
20, 40, 60,
80, 100: cem-poualli, ome-poualli,
yey-poualli, naui-poualli, macuil-poualli
Other
numbers: vigesimal numbers followed by (the word) on and the
numbers 1-19. Thus:
21: cem-poualli
on ce (20+on+1), and 99: naui-poualli
on caxtulli-on-naui (80+on+19).
[on-ce
can be shortened to oce].
The majority
of vigesimal systems, however, have a sub-set of 10. These number
systems are found in every continent (except perhaps Australia). Some examples
from the Caucasian, Basque, Burushaski, Ainu, Niger-Congo,
Austric/Austro-Asiatic, Sino-Tibetan and the Ameridian-superfamily
language-families:
Georgian
(Caucasian):
1-10: erti,
ori, sami, otxi, xuti, ekwsi, šwidi, rwa, ҫxra, ati
11-19: tertmeti,
tormeti, ҫameti, totxmeti, txutmeti,
tekwsmeti, cwidmeti, twrameti, ҫxrameti
20, 40, 60,
80, 100: oҫi, ormoҫi, samoҫi, otxmoҫi, asi
Other
numbers: vigesimal + 1-19 with the ending oҫi of the first word becoming oҫda. Thus:
21: oҫda erti (20+1), 99: otxmoҫda ҫxrameti (80+19).
[Note: x is pronounced "kh"].
Euskara/Basque
(Basque):
1-10: bat,
biga, hirur, laur, bortz, sei, zazpi,
zortzi, bederatzi, hamar
11-19: hameka,
hamabi, hamahirur, hamalaur, hamabortz, hamasei,
hamazazpi, hamazortzi, hemeretzi
20, 40, 60,
80, 100: hogei, berrogei, hiruetanogei, lauetanogei,
ehun
Other
numbers: vigesimal + ta + 1-19. Thus:
21: hogei
ta bat (20+ta+1), 99: lauetanogei ta hemeretzi
(80+ta+19).
Burushaski (Burushaski):
1-10: hǝn,
ālto, ůsko, wālto, tsůndo, mıšīndo, tǝlo,
āltǝmbo, hůnčo, tōrůmo
11-19 tůrma
+ 1-9.
20, 40, 60,
80, 100: āltǝr, ālto-āltǝr, īski-āltǝr, wālti-āltǝr, thā
Other
numbers: vigesimal + 1-19 (but before the words tōrůmo and tůrma preceded by the word ga). Thus:
21: āltǝr hǝn (20+1), 90: wālti-āltǝr ga tōrůmo,
(80+ga+10), 99: wālti-āltǝr ga tůrma hůnčo (80+ga+19).
Ainu (Ainu):
1-10: shine, tu, re,
ine, ashikne, iwan, arwan, tupesan, shinepesan,
wan
20, 40, 60, 80, 100: hotne, tu-hotne,
re-hotne, ine-hotne, ashikne-hotne
30, 50, 70, 90: wane-tu-hotne, wane-re-hotne,
wane-ine-hotne, wane-ashikne-hotne
(literally, 30 is "ten-less-than-forty", etc).
Other numbers (including 11-19): unit
+ ishama + tens. Thus:
11: shine ishama wan (1+ ishama+10), 21: shine ishama hotne (1+ishama+20), 99: shinepesan ishama
wane-ashikne-hotne (9+ishama+90).
Mende
(NigerCongo):
1-10: yira, fere, sawa,
nani, lolu, woita, wofela, wayakpa,
tau, pu
11-19: pu-mahũ-yira (10-mahũ-1)
etc.
20, 40, 60, 80, 100: nu-yira-gboyongo,
nu-fere-gboyongo, nu-sawa-gboyongo, nu-nani-gboyongo, nu-lolu-gboyongo
Other numbers: vigesimal + 1-19.
Thus:
21: nu-yira-gboyongo mahũ yira
(20-mahũ-1), 99: nu-nani-gboyongo mahũ pu-mahũ-tau (80-mahũ-19).
Savara/Saora
(Austric-KolMunda):
1-10: bo, bagu, yagi,
uñji, molloi, tuḍru, gulji, tamji, tiñji,
galji
11: galmui, 12: miggal,
13-19: miggal-aboi (13: 12+1), etc.
20, 40, 60, 80, 100: bo-koḍi, bagu-koḍi,
yagi-koḍi, uñji-koḍi, molloi-koḍi
Other numbers: vigesimal + 1-19.
Thus:
21: bo-koḍi bo (20+1), 99: uñji-koḍi
miggal-gulji (80+12+7).
[A special word is aboi instead
of bo for 1 in the number 13]
Shompeng (Austric-Nicobarese):
1-10: heng, au, luge,
fuat, taing, lagau, aing, towe, lungi,
teya
11-19: heng-mahaukoa-teya
(1+mahaukoa+10), etc.
20, 40, 60, 80, 100: heng-inai,
au-inai, luge-inai, fuat-inai, taing-inai
Other numbers: vigesimal + 1-19.
Thus:
21: heng-inai heng (20+1), 99: fuat-inai lungi-mahaukoa-teya (80+mahaukoa+19).
Lepcha/Rōng/Sikkimese
(SinoTibetan-Tibetic):
1-10: kāt, ñat, sām,
falī, fango, tarak, kakyak, kaku, kakyōt,
katī
11-19: katī kāt-thāp
(10+1+thāp), etc.
20, 40, 60, 80, 100: khā-kāt, khā-ñat,
khā-sām, khā-falī, gyo-kāt (20x1, 20x2, 20x3, 20x4, 100x1)
Other numbers: vigesimal + sa
+ 1-19. Thus:
21: khā-kāt sa kāt-thāp (20x1+sa+1+thāp),
99: khā-falī sa kakyōt-thāp (20x4+sa+9+thāp).
[Note: The word thāp is dropped
after katī, 10. Thus 30 is khā-kāt sa katī].
Garo (SinoTibetan-Tibetic):
1-9: sa,
gini, gittam, bri, boṅga, dok, sini, cet,
sku
10, 20, 30:
ci, korgrik, koraci
Other
numbers 11-39: tens+unit. Thus 11, 21, 31, etc.: ci-sa, korgrik-sa,
koraci-sa, etc.
40, 60, 80,
100: korcaṅ-gini, korcaṅ-gittam, korcaṅ-bri, ritca-sa
Other
numbers 41-99: vigesimal + 1-19. Thus:
41: korcaṅ-gini
sa, 99: korcaṅ-bri ci-sku
Welsh
(IndoEuropean-Celtic):
1-10: un,
dau, tri, pedwar, pump, chwech, saith,
wyth, naw, deg
11-15 un-ar-ddeg,
deuddeg, tri-ar-ddeg, pedwar-ar-ddeg, pymtheg
16-19 un-ar-bymtheg,
dau-ar-bymtheg, tri-ar-bymtheg, pedwar-ar-bymtheg
20, 40, 60,
80, 100: hugain, deugain, triugain, pedwarugain, cant
The numbers
from 21-99 are regularly formed by the numbers 1-19 + ar + vigesimal (here the units come
first. Note, in Old English also, the units came first, as in the nursery rhyme
"four-and-twenty blackbirds"). Thus:
21: un ar hugain (1+ar+20)
and 99: pedwar-ar-bymtheg ar pedwarugain (19+ar+80).
Irish
(IndoEuropean-Celtic):
1-10: aon,
dō, trī, keathair, kūig, sē, seakht, okht, naoi, deikh
11-19: aon-dēag
(1+10), etc.
20, 40, 60,
80, 100: fikhe, dā-fhikhid, trī-fhikhid, kheithre-fhikhid,
kēad
Other numbers: the numbers 1-19 + is +
vigesimal (here also the units come first). Thus:
21: aon
is fikhe, 99: naoi-deag is kheithre-fhikhid (19+is+80).
[But the
language also alternatively retains the original Indo-European tens numbers:
10, 20, 30,
etc: deikh, fikhe, trīokha, daikhead, kaoga,
seaska, seakhtō, okhtō, nōkha, kēad].
French
(IndoEuropean-Italic) [but only partially]:
1-10: un,
deux, trois, quatre, cinq, six, sept,
huit, neuf, dix
11-19: onze,
douze, treize, quatorze, quinze, seize, dix-sept,
dix-huit, dix-neuf
20-100: vingt,
trente, quarante, cinquante, soixante, soixante-dix,
quatre-vingts, quatre-vingt-dix, cent
The numbers
from 21-99 are generally formed as follows, e.g. 20: vingt,
1: un, 21: vingt et un
The et
("and") only comes before un, otherwise 22 vingt-deux,
etc.
But note the
words for 70, 80 and 90 mean "60+10", "4x20"
and "4x20+10" respectively. So the numbers 71-79 are soixante
et onze, soixante-douze, (60+11, 60+12) etc., and the numbers 91-99
are quatre-vingt-onze, quatre-vingt-douze, (4x20+11, 4x20+12)
etc. (81-89 are the normal quatre-vingt-un, quatre-vingt-deux,
etc.).
It is very
likely that this sub-system of 20, found in the Indo-European family
only in French and in the Celtic languages may be due to the
influence of Basque.
Yucatec/Mayan
(Amerindian):
1-10: hun,
ca, ox, can, ho, uac, uc, uaxac,
bolon, lahun
11-19: buluc,
lahca, ox-lahun, can-lahun, ho-lahun, uac-lahun,
uuc-lahun, uaxac-lahun, bolon-lahun
20, 40, 60,
80, 100: kal/hun-kal, ca-kal, ox-kal, can-kal, ho-kal
30, 50, 70,
90: lahu-ca-kal, lahu-ox-kal, lahu-cankal, lahu-hokal
(10 less than 40, etc.).
Other
numbers:
21-39
(except 30): 1-19 + tu kal. Thus: 21
is hun tu kal (1+tu+20).
Other
numbers (after 40, except the actual non-vigesimal tens numbers 50,70,90, etc., where
the word tu is dropped): 1-19 + tu
and the following vigesimal. Thus:
41 is hun tu ox-kal (1 below 60),
99 is bolon-lahun tu ho-kal (19 below 100).
[Some
additional, but not necessary, euphonic variations in
speech are:
a) 15, ho-lahun,
is sometimes contracted to ho-lhun
b) a y
is sometimes inserted between a word ending in u and a following ox
or ho. Thus: lahu-oxkal and lahu-hokal (50 and 90) become lahu-y-oxkal
and lahu-y-hokal, and similarly hun tu ox-kal, 41, becomes hun
tu y-ox-kal]
c) l
of lahun is dropped before tu. Thus bolon-lahun tu kal, 39,
becomes bolon-lahu tu kal]
[Note: This
is important since the Mayans were the only people to invent a vigesimal
numeral system. Hence also, perhaps, the system of forming the other
numbers (21-99) is slightly less regular or more complicated (but still
explicable by certain rules]
[Note: the x
is pronounced "sh" and the c as well as k as
"k"].
Yupik (EskimoAleut):
1-10: atauciq,
malruk, pingayun, cetaman, talliman, arving-legen,
malrung-legen, pingayun-legen, qulngunritaraan, qula.
11-19: qula-atauciq,
qula-malruk, qula-pingayun, akimiarunrita'ar, akimiaq,
akimiaq-ataucik, akimiaq-malruk, akimiaq-pingayun, yuinaunrita'ar
vigesimals
20, 40, 60, 80, 100: yuinaq, yuinaak-malruk, yuinaat-pingayun,
yuinaat-cetaman, yuinaat-talliman
Other
numbers: vigesimal + 1-19. Thus:
21: yuinaq
atauciq, 99: yuinaat-cetaman
yuinaunrita'ar
C-IV.
DECIMAL SYSTEMS (WITH A BASE OF 10) WITH WORDS FOR 1-10 AND 100
Decimal number systems are those which are
based on 10. The simplest types of decimal systems are those
where, to learn the numbers, one necessarily has to memorize the numbers 1-10,
and the number 100, and the regular procedure for forming the other
in-between numbers.
Typical
examples of these numbers are found in the major languages of the Sino-Tibetan
family [The sign
after each word shows the tone: low, rising, falling, etc.]:
Chinese
Mandarin (SinoTibetan-Sinitic):
1-10: yi_ , erh↘, sān‾, szә↘, wu↗, liu,
ch'i_ , pā_ , chiu↗, shih_
tens 20-90:
erh↘ shih_ , etc.
100: bai↗
Other
numbers: tens+unit. Thus 11: shih_ yi_ , 21: erh↘ shih_ yi_ , 99: chiu↗ shih_ chiu↗
Thai/Siamese (SinoTibetan-Sinitic):
1-10: hnïng_
, sɔng↗, sām↗, sī_ , hā↘, hok_ , chet_
, bpɛt_ , kɔ↘, sip_
tens 20-90:
sɔng↗ sip_ , etc. 100: hnïng_
rɔy↘
Other
numbers: tens+unit.
Thus 11: sip_
hnïng_, 21: sɔng↗ sip_ hnïng_, 99: kɔ↘ sip_ kɔ↘
Tibetan
(SinoTibetan-Tibetic):
1-10: gchig,
gnyis, gsum, bzhi, lnga, drug, bdun, brgyad,
dgu, bchu
tens 20-90: gnyis
bchu, etc. 100: brgya
Other
numbers: tens+unit. Thus 11: bchu gchig, 21: gnyis bchu gchig, 99: dgu bchu dgu
[Note:
the initial letter in lnga is small
L, not capital i]
Burmese
(SinoTibetan-Tibetic):
1-10: tit,
hnit, sũ, le,
ngā, cowk, khuhnit, shit, kɔ, ta-cheh
tens 20-90: hnit-cheh,
etc. 100: ta-yā
Other
numbers: tens+hnin+unit.
Thus 11: ta-cheh
hnin tit, 21: hnit-cheh
hnin tit, 99: kɔ-cheh hnin
kɔ
Abor-Miri (SinoTibetan-Tibetic):
1-10: ā,
ānyī, āūm, āpī, ānga, ākheng, kīnit,
pinyī, kanāng, ēing
tens 20-90:
ēing-ānyī, etc. 100: ling
Other
numbers: tens+lāng+unit. Thus 11: ēing lāng ā, 21: ēing-ānyī lāng ā, 99: ēing-kanāng lāng kanāng
[Note:
the suffix -ko is attached at the end of every composite number.
Thus: 1: ā-ko, 10: ēing-ko, 11: ēing lāng ā-ko, 20: ēing-ānyī-ko , 21: ēing-ānyī lāng ā-ko, 99: ēing-kanāng lāng kanāng-ko]
Some languages of the Austric
family:
Santali (Austric-KolMunda):
1-10: mit',
bar, pɛ, pon, mɔrɛ, turūi, ēāe, irәl,
arɛ, gɛl
tens 20-90:
bar-gɛl, etc. 100: mit-sae
Other
numbers: tens+khān+unit.
Thus: 11: gɛl
khān mit', 21: bar-gɛl
khān mit', 99: arɛ-gɛl
khān arɛ
[Alternately,
the other numbers can be formed without inserting the word khān]
Vietnamese (Austric-MonKhmer):
1-10: mot↘_
, hai, ba, bôn↗, nǎm, sau↗, bay↘↗,
tam↗, chin↗, muoi↘
tens 20-90:
hai muoi↘, etc. 100: mot↘_
trǎm
Other
numbers: tens+unit.
Thus 11: muoi↘
mot↘_ , 21: hai muoi↘ mot↘_
, 99: chin↗ muoi↘ chin↗
Khasi (Austric-MonKhmer):
1-10: ši,
ār, lāi, sāw, sàn, hinrīw, hinniew, p'rā,
k'ündāi, ši-p'ew
tens 20-90:
ār-p'ew, etc. 100: ši-spå
Other
numbers: tens+unit: Thus 21: ār-p'ew ši, 99: k'ündāi-p'ew k'ündāi
Some languages
of the Austronesian family:
Hawaiian
(Austronesian):
1-10: akahi,
alua, akolu, aha, alima, aono, ahiku,
awalu, aiwa, umi
20: iwak-alua, 30-90: kan-akolu, etc. 100: haneli
Other
numbers: tens+kumam+unit.
Thus: 11: umi
kumam-akahi 21: iwak-alua kumam-akahi, 99: kan-aiwa
kumam-aiwa
Some
languages from African families:
Hausa (SemitoHamitic-Hamitic):
1-10: daia,
biu, uku, fudu, biar, shidda, bakoi, takos,
tara, goma
tens 20-90:
gomia-biu, etc. 100: dari
Other
numbers: 11-17, etc.: tens+sha+unit. Thus 11: goma sha daia, 21: gomia-biu sha daia
18-19: following
tens+gaira+biu/daia (i.e. following tens-minus-2/1). Thus:
18: gomia-biu
gaira biu (20-minus-2),
99: dari gaira daia (100-minus-1).
Wolof (NigerCongo):
1-10: ben,
nīar, nīat, nīanit, jiūrum, jiūrumrumben, jiūrum-nīar,
jiūrum-nīat, jiūrum-nīanit, fūk
tens 20-90:
nīar-fūk, etc. 100: tēmēr
Other
numbers: tens+a+unit. Thus 11: fūk a ben, 21: nīar-fūk a ben, 99: jiūrum-nīanit-fūk a jiūrum-nīanit
Fulani (NigerCongo):
1-10: goo,
zizi, tati, nayi, joyi, jeegom, jeezizi,
jetati, jenayi, sappo
20: noogas, tens 30-90: capanze-tati, etc. 100: temedere
Other
numbers: tens+e+unit.
Thus 11: sappo
e goo, 21: noogas e
goo, 99: capanze-jenayi e
jenayi
Namagua-Hottentot (Khoisan):
1-10: ckui,
ckam, qnona, haka, kore, qnani, hû, xkhaisi,
goisi, disi
tens
20-100: ckam-disi, etc.
[even 100: disi-disi]
Other
numbers: tens+unit+ckha.
Thus: 11: disi
ckui-ckha, 21: ckam-disi ckui-ckha, 99: goisi-disi goisi-ckha
[the four
letters c, v, q, and x represent four different
types of clicking sounds. Clicking sounds as part of the language are unique in
the whole world to the Khoisan languages, though some non-Khoisan neighboring
languages like Zulu have also borrowed this feature from them]
Some languages
from the Amerindian super-family of languages from America:
Quechua/Inca
(Amerindian):
1-10: huk,
iskay, kimsa, tawa, pisqa, suqta, qanchis,
pusaq, iskun, chunka
tens 20-90: iskay-chunka,
etc. 100: pachak
Other
numbers: tens+unit+yuq/niyuq [-yuq after vowel,-niyuq
after consonant. final y in 2 is consonant]. Thus:
11: chunka-huk-niyuq,
13: chunka kimsa-yuq, 99: iskun-chunka iskun-niyuq
Guarani (Amerindian):
1-10: peteĩ,
mokoĩ, mbohapy, irundy, po, poteĩ, pokoĩ,
poapy, porundy, pa
tens 20-90:
mokoĩ-pa, etc. 100: sa
Other
numbers: tens+unit. Thus 11: pa peteĩ, 21: mokoĩ-pa peteĩ, 99: porundy-pa porundy
Tarahumara (Amerindian):
1-10: bire,
oka, beka, nawo, mari, usani, kichao,
osanawo, kimakoi, makoi
tens 20-90:
oka-makoi, etc. 100: makoi-makoi
Other
numbers: tens+wamina+unit. Thus:
11: makoi
wamina bire, 21: oka-makoi
wamina bire, 99: kimakoi-makoi
wamina kimakoi
Tonkawa (Amerindian):
1-10: wē'isbax,
gedai, med'is, sigid, gasgwa, sikwālau, sigidyē'es,
sikwē'isxw'ēl'a, sikbax
tens 20-90:
sikbax-'āla-gedai, etc.
100: sendo-wē'isbax (borrowed from Spanish)
Other
numbers: tens+'en+unit+'en.
Thus 11: sikbax-'en wē'isbax-'en,
21: sikbax-'āla-gedai-'en
wē'isbax-'en, 99: sikbax-'āla-sikwē'isxw'ēl'a-'en
sikwē'isxw'ēl'a-'en
Zuñi (Amerindian):
1-10: t'opa,
kwili, ha'i, awiten, apte, t'opaleqä, kwilileqä,
ha'eleqä, tenaleqä, astemła
tens 20-90:
kwili-qän-astemła, etc.
100: asi-astemlä
Other
numbers: tens+unit+yäłto. Thus
11: astemła t'opa-yäłto,
21: kwili-qän-astemła t'opa- yäłto, 99: tenaleqä-qän-astemła tenaleqä-yäłto
C-V. DECIMAL
SYSTEMS (WITH A BASE OF 10) WITH WORDS FOR UNITS 1-9 AND
TENS 10-100:
These are
the decimal systems where, to learn the numbers, one necessarily has to
memorize the numbers 1-10, and the tens numbers 20-100, and the regular
procedure for forming the other in-between numbers.
Typical
examples of these numbers are found in the major languages of the Uralo-Altaic
family:
Mongolian
(UraloAltaic-Altaic):
1-10: nigen,
khoyar, gorban, dörben, tabun, jirgugan, dologan,
naiman, yisun, arban,
Tens 20-100:
khorin, gochin, döchin, tabin, jiran, dalan,
nayan, yeren, jagon
Other
numbers: tens+unit, e.g. 11 is arban nigen (10+1), etc.
Turkish
(UraloAltaic-Altaic):
1-10: bir,
iki, üҫ, dört,
beş,
altï, yedi, sekiz, dokuz, on
Tens 20-100:
yirmi, otuz, kïrk, elli, altmïş, yetmiş, seksen, doksan, yüz
Other
numbers: tens+unit, e.g. 11 is on bir (10+1), etc.
Manchu
(UraloAltaic-Altaic):
1-10: emu,
juwe, ilan, duin, sunja, ninggun, nadan,
jakūn, uyun, juwan
Tens 20-100:
orin, gusin, dehi, susai, ninju, nadanju,
jakūnju, uyunju, tanggū
Other
numbers: tens+unit, e.g. 11 is juwan emu (10+1), etc.
[The only
special form is 15, tofohun].
Korean (UraloAltaic-KoreoJapanese):
1-10: hana, tul, set,
net, tasәt, yәsәt, ilgop, yәdәlp, ahop, yәl
tens 20-100:
sïmïl, sәlïn, mahïn, sühïn, yecun, ilhïn,
yәdïn, ahïn, pɛk
Other
numbers: tens+unit. Thus 11: yәl hana, 21: sïmïl hana, 99: ahïn ahop
[usually a
-ïi is inserted
after the final word. Thus 1: hanaïi, 20: sïmïlïi, 21: sïmïl hanaïi,
etc.]
Japanese (UraloAltaic-KoreoJapanese):
1-10: hitotsu,
futatsu, mittsu, yottsu, itsutsu, muttsu, nanatsu,
yattsu, kokonotsu, tō
tens
20-100: hatachi, miso, yoso, iso, muso, nanaso,
yaso, kokonoso, momo [Note: miso, yoso, etc.
can alternately be misoji, yosoji, etc]
Other
numbers: tens+amari+unit
Thus 11: tō
amari hitotsu, 21: hatachi
amari hitotsu, 99: kokonoso
amari kokonotsu
[Modern
Japanese, however, uses numbers basically borrowed from Chinese]
Hungarian
(UraloAltaic-Uralic):
1-10: egy,
kettő, három, négy, öt, hat, hét,
nyolcz, kilencz, tíz
tens 20-100:
húsz, harmincz, negyven, ötven, hatvan, hetven, nyolczvan,
kilenczven, száz
Other
numbers: tens+unit [But here, in line
with the -n endings, 10: tizen, 20: huszon]. Thus:
11: tizen-egy, 99: kilenczven-kilencz
Also,
sometimes in some other languages in Asia and Africa:
Tengima
Naga (SinoTibetan-Tibetic):
1-10: po,
kenna, sê, dā, pangu, suru,
thenā, thethā, tekwü, kerr
tens 20-100:
kerr, mekwü, serr, lhidā, lhisuru, lhithenā, lhithethā,
lhitekwü,
krā
Other
numbers: 11-13, etc. previous tens+o+1-3 [Here, 1
has the special form pokrō],
14-19, etc.
following tens+pemo+7-9.
e.g. 11 is kerr
o pokrō (10+o+1), 21 is mekwü o pokrō, (20+o+1), 99 is krā pemo tekwü (100+pemo+9)
Amharic/Ethiopian
(SemitoHamitic-Semitic):
1-10: and,
hulat, sost, arāt, am'st, sad'st, sabāt,
sam'nt, zaṭañ, ašr
tens 20-100:
hāyā, šalāsā, arbā, amsā, salsā,
sabā , samānyā, zaṭanā, mato
Other
numbers: tens+unit, e.g. 11: ašrā and, 21: hāyā and, 99: zaṭanā zaṭañ
[The only
special form is the first tens number in combining with units: ašr
becomes ašrā].
Swahili
(NigerCongo):
1-9: mosi,
pili, tatu, 'nne, tano, sita, saba, nane,
kenda
Tens 10-100:
kumi, makumi-mawili, makumi-matatu,
makumi-ma'nne, makumi-matano, makumi-sita, makumi-saba,
makumi-manane, makumi-kenda, mia
(The word
for 100 is borrowed from Arabic)
Other
numbers: tens+na+unit 1-9 [Here,
1 and 2 have special forms: moja, mbili], e.g. 11 is kumi na
moja (10+na+1).
Languages of
this category are found in the Amerindian superfamily of America as well.
One example:
Sahaptin
(Amerindian):
1-10: naxc,
nipt, mәtad, pinipt, paxad, ptәxninc, tusxas, paxatumad, t'smәst, putәmd
tens 20-100:
nibtid, mәtabtid, pinibtid, paxabtid, ptәxninseibtid, tusxaseibtid, paxatumadeibtid,
tsmaseibtid, naxcputabdid
Other
numbers: tens+unit or tens+wiya+unit. Thus:
11: putәmd wiya naxc, 21: nibtid wiya naxc, 99: tsmaseibtid wiya t'smәst
C-VI. DECIMAL
SYSTEMS (WITH A BASE OF 10) WITH WORDS FOR NUMBERS 1-19 and
TENS 20-100:
These are
the decimal systems where, to learn the numbers, one necessarily has to
memorize the numbers 1-10 and the tens numbers 20-100 and the regular
procedure for forming the other numbers in-between 21-99, but
(due perhaps to the influence of some vigesimal number systems in the
vicinity) also the separate numbers or the regular procedure for forming the
numbers 11-19.
Many
languages form the numbers differently for 11-19 than for the
other later numbers 21-29, 31-39, etc., but by a regular
procedure rather than with different words. Thus we have the following
languages from the Uralo-Altaic family:
Finnish
(Uralo-Altaic-Finno-Ugrian):
1-10: yksi,
kaksi, kolme, neljä, viisi, kuusi, seitsemän,
kahdeksan, yhdeksän, kymmenen
11-19: yksi-toista,
etc.
tens 20-90: kaksi-kymmentä, etc. 100: sata
Other
numbers: tens+unit. Thus 21: kaksi-kymmentä
yksi, 99: yhdeksän-kymmentä yhdeksän
Estonian
(Uralo-Altaic-Finno-Ugrian):
1-10: üks,
kaks, kolm, neli, viis, kuus, seitse,
kaheksa, üheksa, kümme
11-19: üks-teist, etc.
tens 20-100:
kaks-kümmend, etc. 100: sada
Other
numbers: tens+unit. Thus 21: kaks-kümmend üks, üheksa-kümmend üheksa
Some
languages of the Austronesian family:
Malay
(Austronesian):
1-10: satu,
dua, tiga, empat, lima, enam, tujuh, lapan,
sembilan, se-puluh
11-19: se-belas,
dua-belas, etc.
tens 20-90: dua-puluh,
etc., 100: se-ratus
Other
numbers: tens+unit. Thus: 21: dua-puluh satu, 99: sembilan-puluh sembilan
Tagalog (Austronesian):
1-10: isá,
dalawá, tatló, apat, limá, anim, pitó,
waló, siyam, sang-pouó
11-19: labing-isá,
etc.
tens
20-100: dalawá-ngpouó, tatló-ngpouó, apat-napouó,
limá-ngpouó, anim-napouó, pitó-ngpouó, waló-ngpouó,
siyam-napouó [ie. -ngpouó after vowel, -napouó after
consonant]
100: sangdáan
Other
numbers: tens+'t+unit. Thus 21: dalawá-ngpouó-'t isá, 99: siyam-napouó-'t siyam
Then we have
the languages where the numbers 11-19 are formed with distinct words or
by a process of fusion and inflection, but the later in-between numbers
(21-29, 31-39, etc.) are formed in a very regular way.
Some
languages of Africa:
Kanuri
(NiloSaharan/Sudanic):
1-10: tilo,
ndi, yasgә, degә,
ugu, arasgә, tulur, wusgә, lәgar, megu
tens 20-90:
pindi, piyasgә, pidegә,
piugu, pirasgә, pitulur, pitusgu, pilәgar
11-19: lәgari,
nduri, yasgәn, deri, uri, arasgәn, tulurri,
wusgәn, lәgarri
Other
numbers: tens+unit, or tens+tata+unit [units ending in vowels add a -n,
and units ending in consonants add a -nyin in the compound words].
Thus: 21: pindi
tata tilon, 99: pilәgar
tata lәgarnyin
Some
languages from the Amerindian language super-family of America:
Cherokee (Amerindian):
1-10: sowo,
tali, tsoi, nvgi, hisgi, sudali, galiquogi,
tsunela, sonela, sgohi
11-19: sadu,
talidu, tsogadu, nigadu, hisgadu, daladu, galiquadu,
neladu, soneladu
tens
20-100: tali-sgohi, tsoi-gohi, nvg-sgohi, hisgi-sgohi,
sudali-sgohi, galiqua-sgohi, tsunela-sgohi, sonela-sgohi,
sgohitsiqua
Other
numbers: tens (minus -hi)+unit. Thus 21: tali-sgo sowo, 99: sonela-sgo sonela
Navaho (Amerindian):
1-10:
dałai, nak'i, txā, dī, ashdla, hastxá,
tsosts'ed, tsebi, naast'ai, naezná
11-19:
ładzáda, nak'idzada, txádzáda, didzáda, ashdlaáda,
xastxaáda, tsosts'edzáda, tsebidzáda, naas'aidzáda
tens
20-100: nadīn, txadīn, dísdīn, ashdládīn, hastą́dīn,
tsosts'idīn, tseebídīn, náhást'édīn, naennádīn
Other
numbers: tens+ła+unit. Thus 21: nadīn ła dałai, 99: náhást'édīn ła naezná
Some of the Semitic languages (which also have dual forms in 1-19 because of grammatical gender):
Arabic
(SemitoHamitic-Semitic):
1-10 masc.: wāḥidun,
isnāni, salasatun, 'arba'atun, khamsatun,
sittatun, sab'atun, samāniyatun, tis'atun, 'asharatun
1-10 fem.: wāḥidatun,
isnatāni, salasun, 'arba'un, khamsun,
sittun, sab'un, samānin, tis'un, 'ashrun
11-12 masc.:
'aḥada-'ashar, isnā-'ashar.
11-12 fem.: 'iḥdai-'ashrat, isnatā-'ashrat
13-19 masc.:
salasata-'ashar, etc. (-tun becomes -ta).
13-19 fem.: salasa-'ashar,
etc. (-un becomes -a).
[Note: 18 is samāniya-'ashar]
Tens 20-100:
i'shrūna, salasūna,
'arba'ūna, khamsūna,
sittūna, sab'ūna,
samānūna, tis'ūna
Other
numbers 21-99: unit (m/f) followed by (the word) wa and the tens.
Thus:
21( masc.): wāḥidun-wa-i'shrūna, 99 (masc.): tis'atun-wa-tis'ūna.
Hebrew (SemitoHamitic-Semitic):
1-10 masc.: ɛḥɔd,
shnayim, shloshɔh, arbɔ'ɔh, ḥ^{a}mishɔh,
shishɔh, shiv'ɔh, shmōnɔh, tish'ɔh, 'ɛsɔrɔh
1-10 fem.: aḥad,
shtayim, shlosh, arba, ḥɔmesh, shesh,
shɛva', shmōnɛh, tesha', 'ɛsɛr
11-12 masc.:
'aḥad-'ɔsɔr, shnem-'ɔsɔr.
11-12 fem.: 'aḥad-'ɛsreh, shtem-'ɛsreh.
13-19 masc: shloshɔh-'ɔsɔr, etc. (3+'ɔsɔr). 13-19 fem.: shlosh-'ɛsreh, etc. (3+'ɛsreh).
Tens 20-100:
'ɛsrīm, shloshīm, arbɔ'īm, ḥ^{a}mishīm, shishīm, shiv'īm,
shmōnīm, tish'īm, meɔh
Other
numbers 21-99: unit (m/f) followed by (the word) w and the tens.
Thus:
21( masc.): ɛḥɔd-w-'ɛsrīm,
99 (masc.): tish'ɔh-w-tish'īm
Maltese
(SemitoHamitic-Semitic):
1-10: wieħed,
tnejn, tlieta, erbgħa, ħamsa, sitta, sebgħa,
tmienja, disgħa, għaxra
11-19: ħdax, tnax, tlettax,
erbatax, ħmistax, sittax, sbattax, tmintax, dsatax
tens 20-100: għoxrin, tletin,
erbgħin, ħamsin, sittin, sebgħin, tmenin, disgħin
Other numbers: unit+u+tens.
Thus 21: wieħed u għoxrin,
99: disgħa u
disgħin
But the Dravidian
family of languages of India as a whole falls in this category, with
clear fusion or inflection in 11-19.
Tamil
(Dravidian):
1-10: onṛu,
iraṇḍu, mūnṛu, nāngu, aindu, āṛu, ēlu, eṭṭu, onbadu,
pattu
11-19: padinonṛu,
panniraṇḍu, padimūnṛu, padināngu, padinaindu, padināṛu,
padinēlu, padineṭṭu, pattonbadu
tens
20-100: irubadu, muppadu, nāṛbadu, aimbadu, aṛubadu,
elubadu, eṇbadu, toṇṇūṛu, nūṛu
Other
numbers: tens+unit [The final -du and -ṛu of the tens become -tt
and -ṭṛ before vowels and -ttu and -ṭṛu before
consonants]. Thus:
21: irubatt-onṛu,
23: irubattu-mūnṛu, 93: toṇṇūṭṛu-mūnṛu, 99: toṇṇūṭṛ-onbadu
[In
Dravidian languages, initial e, ē, o, ō are pronounced ye, yē,
wo, wō. In Tamil, a final u is pronounce ï]
Malayalam (Dravidian):
1-10: onn,
raṇṭ, mūnn, nāl, añc, āṛ, ēl, eṭṭ,
onpat, patt
11-19: patinonn,
panṛaṇṭ, patimmūnn, patināl, patinañc, patināṛ,
patinēl, patineṭṭ, pattonpat
tens
20-100: irupat, muppat, nālpat, anpat, aṛupat,
elupat, eṇpat, toṇṇūṛ, nūṛ
Other
numbers: tens+unit [The final -at
of the tens becomes -att before vowels and -atti before
consonants. The final ūṛ of 90 becomes ūṭṛi alternately
pronounced ūṭi, before the units]. Thus 21: irupatt-onn, 23: irupatti-mūnn, 99: toṇṇūṭṛi-onpat
Kannada (Dravidian):
1-10: ondu,
erḍu, mūru, nalku, aidu, āru, ēḷu, eṇṭu,
ombattu, hattu
11-19: hannondu,
hannerḍu, hadimūru, hadinālku, hadinaidu, hadināru,
hadinēḷu, hadineṇṭu, hattombattu
tens
20-100: ippattu, mūvattu, nālvattu, aivattu, ārvattu,
eppattu, embattu, tombattu, nūru
Other
numbers: tens+unit. [The final -ttu of the tens become -tt before
vowels].
Thus 21: ippatt-ondu, 99: tombatt-ombattu
Telugu (Dravidian):
1-10: okaṭi,
reṇḍu, mūḍu, nālugu, ayidu, āru, ēḍu,
enimidi, tommidi, padi
11-19: padakoṇḍu,
panneṇḍu, padamūḍu, padanālugu, padihēni, padahāru,
padihēḍu, paddenimidi, pandommidi
tens
20-100: iruvai, muppai, nalubhai, yābhai, aravai,
ḍebbhai, enabhai, tombhai, vandala
Other
numbers: tens+unit. Thus 21: iruvai okaṭi, 99: tombhai tommidi
And so do the
languages from all the other branches of Indo-European languages
outside India:
Persian (IndoEuropean-Iranian):
1-10: yak,
dū, si, cahār, pañj, shish, haft, hasht,
nuh, dah
11-19: yāzdah,
davāzdah, sīzdah, chahārdah, pānzdah, shānzdah,
hīvdah, hījdah, nūzdah
tens
20-100: bīst, sī, chihil, pañjāh, shast, haftād,
hashtād, navad, sad
Other
numbers: tens+u+unit. Thus 21: bīst u yak, 99: navad u nuh
Armenian (IndoEuropean-ThracoPhrygian):
1-10: mēk, erkou, erekh, chors,
hing, veçh, eòthә, outhә, inә, tas
11-19: tasnmēk, tasnerkou, tasnerekh,
tasnchors, tasnhing, tasnveçh,
tasneòthә, tasnouthә, tasninә
tens 20-100:
khsan, eresoun, kharrasoun, yisoun, vathsoun,
eòthanasoun, outhsoun, innsoun, hariur
Other
numbers: tens+unit. Thus: 21: khsan mēk, 99: innsoun inә
Ancient
Greek (IndoEuropean-Hellenic):
1-10: heîs/mía/hen
(m/f/n), dúo,
treîs, téssares, pénte, héks, heptá, oktṓ,
ennéa, déka
11-19: héndeka, dṓdeka, treîs-kaì-déka, téssares-kaì-déka, pentekaídeka,
hekkaídeka, heptakaídeka, oktokaídeka, enneakaídeka
tens 20-100: eíkosi, triákonta, tessarákonta, pentḗkonta, heksḗkonta, hebdomḗkonta, ogdoḗkonta, enenḗkonta, hekatón
Other
numbers: tens+kaì+unit or unit+kaì+tens. Either form can be used. Thus:
21: eíkosi kaì heîs
or heîs kaì eíkosi, 99: enenḗkonta kaì ennéa, or ennéa kaì enenḗkonta
[Note: Greek
vowels have a tonal accent, which is marked. A special form for neuter 4: téssara]
Modern
Greek (IndoEuropean-Hellenic):
1-10: henas,
duo, treis, tessereis, pente,
eksi, hephta, okhtō,
ennia, deka
11-12: hendeka, dōdeka, 13-19: deka-treis,
etc.
tens 20-100:
eikosi, trianta, saranta, penēnta, heksēnta,
hebdomēnta, ogdonta, enenēnta, hekato
Other numbers: tens+unit. Thus: 21: eikosi-henas,
99: enenēnta-ennia
[Modern
Greek has no tonal accent, hence accent not marked here].
Albanian (IndoEuropean-Illyrian):
1-10: një, dy, tre,
katër, pesë, gjashtë, shtatë, tetë, nënd,
dhjëte
1-18: një-mbë-dhjëte, etc.
19: nëntë-mbë-dhjëte
tens 20-100: njëzet, tridhjet,
dyzet, pesë-dhjet, gjashtë-dhjet, shtatë-dhjet, tetë-dhjet,
nënd-dhjet, një-qind
Other numbers: tens+e+unit.
Thus 21: njëzet e një,
99: nënd-dhjet e nënd
[Note: 20 and 40 seem to be formed on
a principle of 1x20, 2x20].
Polish (IndoEuropean-Slavic):
1-10: jeden,
dwa, trzy, cztery, pięć, sześć, siedem,
osiem, dziewięć, dziesięć
11-19: jeden-naście,
dwa-naście, trzy-naście, czter-naście, pięt-naście,
szes-naście, siedem-naście, osiem-naście, dziewięt-naście
tens
20-100: dwa-dzieścia, trzy-dzieści, cztery-dzieści, pięć-dzieśiąt,
sześć-dzieśiąt, siedem-dzieśiąt, osiem-dzieśiąt, dziewięć-dzieśiąt,
sto
Other
numbers: tens+unit. Thus 21: dwa-dzieścia jeden, 99: dziewięć-dzieśiąt dziewięć
Russian
(IndoEuropean-Slavic):
1-10: odin,
dva, tri, cyetyrye, pyat', shyest', syem',
vosyem', dyevyat', dyesyat'
11-19: odi-nadçat',
dvye-nadçat', tri-nadçat', cyetyr-nadçat', pyat-nadçat',
shyest-nadçat', syem-nadçat', vosyem-nadçat', dyevyatnadçat'
tens
20-100: dvadçat', tridçat', sorok, pyat'-dyesyat, shyest'-dyesyat,
syem'-dyesyat, vosyem'-dyesyat, dyevyanosto, sto
Other
numbers: tens+unit: Thus 21: dvadçat' odin, 99: dyevyanosto dyevyat'
Lithuanian
(IndoEuropean-Baltic):
1-10: vienas,
du, trys, keturi, penki, šeši, septyni, aštuoni, devyni,
dešimtis
11-19: vienuolika, dvylika,
trylika,keturiolika, penkiolika, šešiolika,
septyniolika, aštuoniolika, devyniolika
tens 20-100: dvidešimt, trisdešimt,
keturiasdešimt, penkiasdešimt, šešiasdešimt,
septyniasdešimt, aštuoniasdešimt, devyniasdešimt, šimtas
Other numbers: tens+unit. Thus 21: dvidešimt
vienas,
99: devyniasdešimt devyni
Latvian (IndoEuropean-Baltic):
1-10: viens, divi, tris,
četri, pieci, seši, septiņi, astoņi, deviņi,
desmits
11-19: vienspadsmit, divspadsmit,
trispadsmit, četrpadsmit, piecpadsmit, sešpadsmit, septiņpadsmit,
astoņpadsmit, deviņpadsmit
tens 20-100: divdesmit, trisdesmit,
četrdesmit, piecdesmit, sešdesmit, septiņdesmit, astoņdesmit,
deviņdesmit, simts
Other
numbers: tens+unit. Thus 21: divdesmit
viens, 99: deviņdesmit
deviņi
Danish
(IndoEuropean-Germanic):
1-10: en/et,
to, tre, fire, fem, seks, syv, otte,
ni, ti
11-19: elleve,
tolv, tretten, fjorten, femten, seksten, sytten,
atten, nitten
tens 20-100:
tyve, tredive, fyrre, halvtreds, tres, halvfjerds,
firs, halvfems, hundrede
Other
numbers: unit+og+tens. Thus: 21: en-og-tyve, 99: ni-og-halvfems.
Norwegian
(IndoEuropean-Germanic):
1-10: en/et,
to, tre, fire, fem, seks, sju, åtte,
ni, ti
11-19: elleve,
tolv, tretten, fjorten, femten, seksten, sytten,
atten, nitten
tens 20-100:
tjue, tretti, førti, femti, seksti, sytti, åtti,
nitti, hundre
Other
numbers: unit+og+tens. Thus: 21: en-og-tjue, 99: ni-og-nitti.
Swedish
(Indo-European-Germanic):
1-10: en/ett,
två,
tre, fyra, fem, sex, sju, åtta, nio,
tio
11-19: tio, elva, tolv,
tretton, fjorton, femton, sexton, sjutton, aderton,
nitton
tens 20-100: tjugo, trettio,
fyrtio, femtio, sextio, sjuttio, åttio, nittio,
hundra
Other
numbers: tens+unit. Thus 21: tjugo-en,
99: nittio-nio
Icelandic (IndoEuropean-Germanic):
1-10: einn, tveir, ƥrīr,
fjórir, fimm, sex, sjö, átta, níu, tíu
11-19: ellefu, tólf, ƥrettán,
fjórtán, fimmtán, sextán, seytján, átján, nítjan
tens 20-100: tuttugu, ƥrjátíu,
fjörutíu, fimmtíu, sextíu, sjötíu, áttatíu, níutíu,
hundrađ
Other numbers: tens+og+unit.
Thus 21: tuttugu og einn,
99: níutíu og níu
German (IndoEuropean-Germanic):
1-10: eins, zwei, drei,
vier, fünf, sechs, sieben, acht, neun,
zehn
11-19: elf, zwölf, dreizehn,
vierzehn, fünfzehn, sechzehn, siebzehn, achtzehn,
neunzehn
tens 20-100:
zwanzig, dreissig, vierzig, fünfzig, sechzig,
siebzig, achtzig,
neunzig, hundert
Other
numbers: unit+und+tens (as one word, but eins becomes ein).
Thus:
21: einundzwanzig, 99: neunundneunzig
Dutch (IndoEuropean-Germanic):
1-10: een,
twee, drie, vier, vijf, zes, zeven, acht,
negen, tien
11-19: elf,
twaalf, dertien, veertien, vijftien, zestien,
zeventien, achttien, negentien
tens 20-100:
twintig, dertig, veertig, vijftig, zestig, zeventig,
tachtig, negentig, honderd
Other
numbers: unit+en+tens. Thus 21: een en twintig, 99: negen en negentig
Old English (IndoEuropean-Germanic):
1-10: ān, twēgen,
ƥrīe, fēower, fīf, siex, seofon, eahta,
nigon, tīen
11-19: endleofan, twelf,
ƥrēotīene,
fēowertīene,
fīftīene, siextīene, seofontīene, eahtatīene, nigontīene
tens 20-100: twentig, ƥrītig,
fēowertig,
fīftig, siextig, hundseofontig, hundeahtatig, hundnigontig,
hundtēontig
Other numbers: unit+and+tens.
Thus 21: ān and twentig,
99: nigon and hundnigontig
[ƥ is pronounced "th"]
English
(IndoEuropean-Germanic):
1-10: one,
two, three, four, five, six, seven, eight,
nine, ten
11-19: eleven,
twelve, thirteen, fourteen, fifteen, sixteen,
seventeen, eighteen, nineteen
tens 20-100:
twenty, thirty, forty, fifty, sixty, seventy,
eighty, ninety, hundred
Other
numbers: tens+unit. Thus: 21: twenty-one, 99: ninety-nine
Latin
(IndoEuropean-Italic):
1-10: unus,
duo, tres, quattuor, quinque, sex, septem,
octo, novem, decem
11-19: undecim,
duodecim, tredecim, quattuordecim, quindecim, sedecim,
septemdecim, duode-viginti, unde-viginti
tens 20-100:
viginti, triginta, quadraginta, quinquaginta, sexaginta,
septuaginta, octoginta, ninaginta, centum
Other
numbers: tens+unit (1-7) or unit (1-7)+et+tens. Either form can be used.
Tens
(including 100)+unit (8-9): duode/unde+following-tens
(i.e. 2-less-then, 1-less-then the following tens). Thus:
21: viginti-unus
or unus et viginti, 99: undecentum
Spanish
(IndoEuropean-Italic):
1-10: uno/una,
dos, tres, cuatro, cinco, séis,
siete, ocho, nueve, diez
11-19: once,
doce, trece, catorce, quince, dieciséis,
diecisiete, dieciocho, diecinueve
tens 20-100:
veinte, treinta, cuarenta, cincuenta, sesenta,
setenta, ochenta, noventa, ciento
Other
numbers: 21-29: vienti-uno, etc. Others: tens+y+unit.
Thus:
31: treinta
y uno, 99: noventa y
nueve
Portuguese
(IndoEuropean-Italic):
1-10: um/uma,
dois, três, quatro, cinco, seis, sete,
oito, nove, dez
11-19: onze,
doze, treze, catorze, quinze, dezasseis, dezassete,
dezoito, dezanove
tens 20-100:
vinte, trinta, quarenta, sessenta, setenta, oitenta,
noventa, cento
Other
numbers: tens+e+unit. Thus 21: vinte e um, 99: noventa e nove
Romanian (IndoEuropean-Italic):
1-10: unu,
doi, trei, patru, cinci, şase,
şapte, opt, nouă, zece
11-19: unsprezece,
doisprezece, treisprezece, paisprezece, cincisprezece,
şaisprezece, şaptesprezece, optsprezece, nouăsprezece
tens 20-100:
douăzeci, treizeci, paizeci, cincizeci, şaizeci, şaptezeci, optzeci, nouăzeci, o sută
Other
numbers: tens+şi+unit. Thus 21: douăzeci şi unu, 99: nouăzeci şi nouă
Italian (IndoEuropean-Italic):
1-10: uno,
due, tré, quattro, cinque, sei,
sette, otto, nove, dieci
11-19: undici,
dodici, tredici, quattordici, quindici, sedici,
diciassette, diciotto, diciannove
tens 20-100:
venti, trenta, quaranta, cinquanta, sessanta,
settanta, ottanta, novanta, cento
Other
numbers: tens+unit [last vowel of tens dropped before vowels in uno, otto].
Thus:
21: vent-uno, 99: novanta-nove
C-VII.
DECIMAL SYSTEMS (WITH A BASE OF 10) WITH WORDS FOR NUMBERS 1-100:
Finally, we
come to the most complex decimal system of all, where there is such complete fusion
and inflection between the tens and unit numbers that it
becomes necessary to learn individually the exact form of every number from 1-100,
above the usual necessity of learning the unit words 1-9 and tens
words 10-100.
Basically,
one has to first learn the numbers from 1-10, 11-19 and the tens 20-100.
The other numbers 21-99 are naturally formed by a combination of the tens
and unit words.
But these
words are fused together in such a way that it becomes necessary to
individually learn every number from 1-100. [In addition, the words 19,
29, 39, etc. are formed on the principle "one less than the following
tens" (usually except 89 and 99)].
The only
languages in the world which have a number system of this kind are the Indo-Aryan
languages of North India. We will take the example of just three of these languages: Hindi,
Marathi and Gujarati. Compare the difference in the forms in the three languages:
Hindi:
1-9: ek,
do, tīn, cār, pāñc, chah, sāt, āṭh,
nau
11-19: gyārah,
bārah, terah, caudah, pandrah, solah, satārah,
aṭhārah, unnīs
tens 10-100:
das, bīs, tīs, cālīs, pacās, sāṭh, sattar,
assī, nabbe, sau
The other
numbers are formed by unit-form+tens-form, e.g. 21: ek+bīs
= ikk-īs.
The
different changes taking place in the tens forms as well as the units
form in the numbers 21-99 must be noted:
Tens
forms:
20 bīs:
-īs (21,22,23,25,27,28), -bīs (24,26).
30 tīs:
-tīs (29,31,32,33,34,35,36,37,38).
40 cālīs:
-tālīs (39,41,43,45,47,48), -yālīs (42, 46), -vālīs
(44).
50 pacās:
-cās (49), -van (51,52,54,57,58), -pan (53,55,56).
60 sāṭh:
-saṭh (59,61,62,63,64,65,66,67,68).
70 sattar:
-hattar (69,71,72,73,74,75,76,77,78).
80 assī:
-āsī (79,81,82,83,84,85,86,87,88,89).
90 nabbe:
-nave (91,92,93,94,95,96,97,98,99).
Unit
forms:
1 ek:
ikk- (21), ikat- (31), ik- (41,61,71), iky-
(81), ikyā- (51,91).
2 do:
bā- (22,52,62,92), bat- (32), ba- (42,72), bay-
(82).
3 tīn: te- (23), ten-
(33,43), tir- (53,63,83), ti- (73), tirā- (93).
4 cār:
cau- (24,54,74), ca- (44), caun- (34,64), caur-
(84), caurā- (94).
5 pāñc:
pacc- (25), paĩ- (35,45,65), pac- (55,75,85), pañcā-
(95).
6 che:
chab- (26), chat- (36), chi- (46,76), chap-
(56), chiyā- (66,96), chiy- (86).
7 sāt:
sattā- (27,57,97), saĩ- (37,47), saḍ- (67), sat- (77), satt-
(87).
8 āṭh:
aṭṭhā- (28,58,98), aḍ- (38,48,68), aṭh- (78,88).
9 nau:
un- (29,39,59,69,79), unan- (49), nav- (89), ninyā-
(99).
Marathi:
1-9: ek,
don, tīn, cār, pāç, sahā, sāt, āṭh,
naū
11-19: akrā,
bārā, terā, çaudā, pandhrā, soḷā, satrā,
aṭhrā, ekoṇīs
tens 10-100:
dahā, vīs, tīs, cāḷīs, pannās, sāṭh, sattar,
aĩśī, navvad, śambhar
The other
numbers are formed by unit-form+tens-form, e.g. 21: ek+vīs
= ek-vīs.
The
different changes taking place in the tens forms as well as the units
form in the numbers 21-99 must be noted:
Tens
forms:
20 vīs:
-vīs (21,22,23,24,25,26,27,28).
30 tīs:
-tīs (29,31,32,33,34,35,36,37,38).
40 cāḷīs:
-cāḷīs (39,41,42,43,44,45,46,47,48).
50 pannās:
-pannās (49), -vanna (51,52,55,57,58), -panna (53,54,56).
60 sāṭh:
- sāṭh (59), -saṣṭa (61,62,63,64,65,66,67,68).
70 sattar:
-sattar (69), -hattar (71,72,73,74,75,76,77,78).
80 aĩśī: -aĩśī
(79,81,82,83,84,85,86,87,88).
90 navvad:
-navvad (89), -ṇṇav (91,92,93,94,95,96,97,98,99).
Unit
forms:
1 ek:
ek- (21,31,61), ekke- (41), ekkyā- (81,91), ekkā-
(51,71).
2 don:
bā- (22,52,62,72), bat- (32), be- (42), byā-
(82,92).
3 tīn: te- (23), teha- (33),
tre- (43,53,63), tryā- (73,83,93).
4 cār:
co- (24), çau- (34,54,64), çavve- (44), çauryā-
(74,84,94).
5 pāç:
pañc- (25), pas- (35), pañce- (45), pañçā- (55),
pā- (65), pañcyā (75,85,95) .
6 sahā:
sav- (26), chat- (36), sehe- (46), chap-
(56), sahā- (66), śahā- (76,86,96).
7 sāt:
sattā- (27,57), sada- (37), satte- (47), sadu-
(67), sattyā- (77,87,97).
8 āṭh:
aṭṭhā- (28,58), aḍ- (38), aṭṭhe- (48), aḍu-
(68), aṭṭhyā- (78,88,98).
9 naū:
ekoṇ- (29,39,49,59,69,79,89), navvyā- (99).
Gujarati:
1-9: ek,
be, traṇ, cār, pāñc, cha, sāt, āṭh,
nav
11-19: agyār,
bār, ter, caud, pandar, soḷ, sattar, aḍhār,
ogṇis
tens 10-100:
das, vīs, trīs, cālīs, pacās, sāīṭh, sitter,
ẽsī, nevũ, so
The other
numbers are formed by unit-form+tens-form, e.g. 21: ek+vīs
= ek-vīs.
The
different changes taking place in the tens forms as well as the units
form in the numbers 21-99 must be noted:
Tens
forms:
20 vīs:
-īs (25), -vīs (21,22,23,24,26,27,28).
30 trīs:
-trīs (29,31,32,33,34,35,36,37,38).
40 cālīs:
-tālīs (41,42,43,45,46,47,48), -cālīs (39), -ālīs (44).
50 pacās:
-pacās (49), -van (51,52,55,57,58), -pan (53,54,56).
60 sāīṭh:
-sāṭh (59), saṭh (61,62,63,64,65,66,67,68).
70 sitter: sitter (69), -oter (71,72,73,74,75,76,77,78).
80 ẽsī: ẽsī (79), -āsī
(81,82,83,84,85,86,87,88,89).
90 nevũ: -ṇu (91,92,93,94,95,97,98,99), -nnu (96).
Unit
forms:
1 ek:
ek- (21,41,61,71), eka- (31), ekā- (51,91), eky-
(81).
2 be:
bā- (22,52,62,92), ba- (32), be- (42), b- (72),
by- (82).
3 traṇ: te- (23,33), tre- (43,53,63),
ty- (83), t- (73), trā- (93).
4 cār:
co- (24,34,54,64), cum- (44,74), cory- (84), corā-
(94).
5 pāñc:
pacc- (25), pāã- (35,65), pis- (45), pañc- (75,85), pañcā-
(55,95).
6 cha:
cha- (26,36.96), che- (46), chap- (56), chā-
(66), chay- (86), ch- (76).
7 sāt:
sattā- (27,57,97), saḍa- (37), suḍ- (47), saḍ-
(67), sity- (77,87).
8 āṭh:
aṭṭhā- (28,58,98), aḍ- (48,68), aḍa- (38), iṭhy-
(78,88).
9 nav:
ogaṇ- (29,39,49,59), agṇo- (69), ogṇā- (79), nevy-
(89), navvā- (99).
The same
irregularity or inflectional complexity can be seen in the formation of the
numbers between 21 and 99 in all the Indo-Aryan languages
of North India (right up to Kashmiri in the extreme north, and going so
far westwards as to influence the Pashto language in the northwest which,
although it belongs to the Iranian branch, has also been influenced by
the Indo-Aryan cerebral sounds), but is found nowhere else outside
the sphere of North India . Note that the irregularity of the fusion of
the forms in one Indo-Aryan language do not correspond to those in another
Indo-Aryan language. Thus, ek (1) has one form (ek-) in Marathi
in 21, 31 and 61, but Hindi has three different forms ikk- (in 21), ikat-
(in 31) and ik- (in 61), and Gujarati has two forms ek- (in
21,61) and eka- (in 31). Or pāñc (5) has one form (paĩ-) in Hindi in 35, 45 and 65, and Gujarati has
two forms pāã- (in 35,65) and pis- (in 45),
but Marathi pāç (5) has three different forms pas- (in 35), pañce-
(in 45) and pā- (in 65).
We have
shown the numbers 21-99 in these three Indo-Aryan languages in
classified table form, but obviously it is simpler to learn each individual
number by rote than with the help of these classification tables.
This is
in sharp contrast with all the other languages in the world other than the
Indo-Aryan languages of North India. In all the other languages, it is necessary
to learn by heart at the most the numbers from 1-10, or from 1-19,
and the tens forms (20,30,40,50,60,70,80,90).
All the numbers between 21 and 99 are formed from these numbers
by some sort of regular process which does not require
all these individual numbers to be learnt by heart. This is the case with all
other languages, including all the other non-Indo-European Indian languages
(Dravidian, Austric, Sino-Tibetan, Burushaski. The Andamanese languages, as
already pointed out, do not have numbers beyond 3 or 5) as well
as all the non-Indian Indo-European languages (spoken outside India), including
even the Indo-Aryan Sinhalese language spoken to the south of India.
This feature
of the Indo-Aryan numbers has very definite practical disadvantages:
1. The first
and most obvious disadvantage is that it makes it more difficult for the
learner to learn the exact forms of the numbers 1-100 in an Indo-Aryan
language than in any other language, even if the learner is himself a speaker
of another Indo-Aryan language (though in that case, of course, he is
likely to recognize the numbers when spoken by someone else more easily
than the learner who is a speaker of a non-Indo-Aryan language).
2. The
second disadvantage is that, like all the other many languages (including,
for example, Old English and German) which have a similar word-order for the
numbers 21-99, the word-order of the tens and unit
words is irrational and unordered since the unit word comes before the tens
word. Thus, the number 45,396 (four, five, three, nine,
six) in English, for example, would be "forty-five
thousand, three hundred and ninety-six" (in the order
four, five, three, nine, six), which is
rational and ordered, but in Hindi would be "pain-tālīs
hazār tīn-sau chiyā-nave" ( in the order five,
four, three, six, nine).
This is
somewhat like the irrational and unordered American style of writing the date
as compared to the British style: 4th January 2018 is written 1/4/2018
in the American style and 4/1/2018 in the British style. Logically, the month
should come between the day and the year, and the only reason the
irrational and unordered American style is gaining ground in modern usage is
because of the political and economic clout of the U.S.A and its monopoly over
computer technology.
The
unordered nature of the Indo-Aryan numbers 21-99, compounded with the
irregular and inflected forms, adds to the difficulty of the numbers. On a
personal note, I myself regularly fumble for the right words (although I know
them well) when suddenly called upon to say, for example, 67, when I
automatically say sain-saṭh (or even chiya-…) instead of saḍ-saṭh,
and then pause and correct myself.
But the
nature of the Indo-Aryan numbers is very important from the cultural and historical
view-point. As the Muslim saying goes, "mulla ki dor masjid tak":
I find in the nature of the Indo-Aryan number system one more clear piece of
evidence for the OIT (the Out-of-India Theory of Indo-European
origins).
C-VIII.
HISTORICAL IMPLICATIONS OF THE INDO-ARYAN NUMBER SYSTEM:
The number
systems as found in the different languages in India show a great range and
variety. We do not find the most uncommon types like the sexagesimal
(based on 60, found in the Masai language in Africa), and the quindecimal
(based on 15, found in the Huli language of Papua), but within
the more common systems, the vigesimal (based on 20), and decimal
(based on 10), we have every possible variety: see the difference above
between the number systems in the closely related Santali and Turi
languages where, after the initial four numbers 1-4, there is nothing in
common, and Santali has a purely decimal system while Turi
has a purely vigesimal system with a subset of five.
The
interesting thing is that an analysis of the development of number systems in
the world presents us with an interesting point about the origin and spread of
the Indo-European languages from their original homeland, pointing
towards the geographical location of that homeland. For the purpose of the
discussion to follow, which is about the development of the Indo-European
number system, we will leave out the language families of the New World and
some isolated language families in the Old World (i.e. the Australian, Papuan,
Amerindian, and also Andamanese, as well as the interior families
of Africa: Khoisan, Niger-Congo and Nilo-Saharan, and also
Eskimo-Aleut, which straddles the northernmost parts of both the Old and
New Worlds, from Greenland to Alaska and the easternmost tips of Siberia),
since they are not relevant to this question.
It will be
seen that the decimal system dominates in the most widely spoken and
distributed language families in the Old World (Indo-European, Semito-Hamitic,
Sino-Tibetan, Uralo-Altaic, Austronesian, Dravidian),
and the vigesimal system is found in the more isolated families (in the
three language-isolate families, Basque, Burushaski and Ainu,
and in Caucasian).
It is also
likely that the vigesimal system was the original system in the Austric
family: we have the system in Turi (in its earliest form, with a clear
subset of 5), and in Savara and Nicobarese and perhaps
originally in Khmer as well (among the languages examined by us here). The
Vietnamese language was clearly influenced by its Sino-Tibetan
family neighbours in developing a decimal system: note that it also has
a tonal-system and monosyllabled words like most of its major Sino-Tibetan
neighbour languages. Santali was also probably influenced by its Indo-Aryan
and Dravidian neighbours, and Khasi by its Sino-Tibetan
and Indo-Aryan neighbours, in developing a decimal system. It may
be noted that Turi (mead, pea, punia), Santali (mit', pɛ, pon),
Khmer (muәy, bәy,
buәn) and Vietnamese (mot↘_ , ba, bôn↗) have a close correspondence in the
numbers for 1,3 and 4, but not beyond that, and Turi has basic unit
number words only upto 4 (all of which could be pointing to an original
subset of 5). A reverse influence is seen in the originally Austric-speaking
areas of eastern India, where neighboring Sino-Tibetan languages like Sikkimese
and Garo have developed vigesimal systems. We also saw how the (Indo-European)
Celtic languages like Welsh and Irish developed vigesimal
systems in what probably was originally the ancient area of the Basque
family (although Irish also retained parallel decimal word-names
for the tens), while French was influenced enough to develop words like quatre-vingts
for 80 and soixante-onze etc. for 71 etc..
The point
here is that the Indo-European languages must certainly have developed
the feature of forming the numbers 11-19 in a different way from the
other sets of numbers (21-29, 31-39, 41-49, etc.) due to
the influence of neighboring languages with vigesimal systems: we will
call this the vigesimal-effect. This could be a clue to the location of
the Original IE Homeland in India, since the eastern half of India is riddled
with languages having vigesimal systems (from Sikkimese in the
north through Savara and Turi in the central parts to Nicobarese
in the eastern islands), and we also have Burushaski in the
north-northwest - but then of course we also have the Caucasian languages in the
area of the Caucasus mountains and Basque in western Europe, which (with
possibly related now-extinct languages spread out in the intervening areas) could
likewise have influenced proto-IE in other suggested Homeland-theories.
But the Indo-European
number system nevertheless does point towards an Indian Homeland and Out-of-India
theory. This can be
examined from two angles:
1. The
stage-wise development of Indo-European numerals.
2. The
spread of the vigesimal-affected decimal number-system.
1. The
Stage-wise Development of Indo-European numerals:
The
first stage of the
Indo-European number system is represented by the Sanskrit
numbers, which are as follows:
1-9: eka,
dvi, tri, catur, pañca, ṣaṭ, sapta, aṣṭa,
nava
tens 10-90: daśa,
viṁśati, triṁśat, catvāriṁśat, pañcāśat, ṣaṣṭi,
saptati, aśīti, navati, śatam
Other
numbers: unit-form+tens.
[The tens
do not undergo any change in combination, with the sole exception of the word
for 16, where -daśa becomes -ḍaśa in combination with ṣaḍ-. And, by the regular Sanskrit phonetic rules
of sandhi or word-combination, in the unit-form+tens
combinations for 80-, a-+-a becomes ā, and i-+-a
becomes ya, so 81: ekāśīti, 82: dvyaśīti, etc].
Units
forms:
1 eka:
ekā- (11), eka- (21,31,41,51,61,71,81,91).
2 dvi:
dvā- (11,22,32), dvi- (42,52,62,72,82,92).
3 tri: trayo- (13,23,33), tri-
(43,53,63,73,83,93).
4 catur:
catur- (14,24,84,94), catus- (34), catuś- (44) catuḥ-
(54,64,74).
5 pañca:
pañca- (15,25,35,45,55,65,75,85,95).
6 ṣaṭ:
ṣo- (16), ṣaḍ- (26,86), ṣaṭ- (36,46,56,66,76), ṣaṇ-
(96).
7 sapta:
sapta- (17,27,37,47,57,67,77,87,97).
8 aṣṭa:
aṣṭā- (18,28,38,48,58,68,78,88,98).
9 nava:
ūna- (19,29,39,49,59,69,79,89), nava- (99).
Compared to
the modern Indo-Aryan forms:
a) The Sanskrit
numbers with -5, -7, -8 and even -9 are remarkably
regular (compare with the forms already shown in Hindi, Marathi
and Gujarati, for example).
b) The variety
of forms for -4 and -6 are
fully explained (except perhaps the ṣo- in 16) by the regular phonetic
rules of Sanskrit sandhi: r- becomes ḥ- before -p (54), -ṣ
(64) and -s (74), s- before -t (34),
and ś- before -c (44). Likewise, ṭ- becomes ḍ-
before voiced consonants and vowels (26,86)
and -ṇ before nasal consonants (96). These are all variations
based only on the general phonetic rules of sandhi in Sanskrit (which
apply to all Sanskrit words).
c) So we are
left with with a few (far fewer as compared to the modern Indo-Aryan
languages) variable forms for -1, -2 and -3 (apart from
the irregular form for 16 already mentioned), and hardly any fusion and
irregular inflection beyond the rules of regular sandhi.
Certain noteworthy
features of the Sanskrit numbers, which have lingered on in modern
Indo-Aryan, are:
1. The units
come before the tens in all the numbers: this feature continues
in the modern Indo-Aryan languages, and in some of the Indo-European
languages outside India (Pashto under the influence of neighboring Indo-Aryan,
and the Germanic branch languages German-Dutch-OldEnglish-Norwegian-Danish),
but is reversed in all the other modern languages (including the Germanic
branch languages English-Swedish-Icelandic) in the numbers after 20. In Ancient Greek and
Latin, both ways were allowed after 20.
2. A minus
principle (ūna- "less-than" or alternately ekona-
"one-less-than") is used for
the -9 numbers: 19: ūna-viṁśati (or ekona-viṁśati) etc., except for 99:
nava-navati. This feature continues in the modern Indo-Aryan languages
and in Latin, which takes the step further (note the Latin
tendency to innovate with a minus-principle, as when adopting the Attic
Greek numeral system) by having duode-viginti and unde-viginti
(18 and 19) etc., and even duode-centum and unde-centum
(98 and 99).
[Note:
Dravidian has this etymology for the number 9: e.g. Tamil on-badu
("one-less than-ten"). Here the prefix on- represents the Tamil
word onṛu "one", but also resembles the Sanskrit ūna
"less" and Latin unus "one"!].
But, about
two other main significant features:
1. While all
the branches of Indo-European languages show the vigesimal-effect, where
1-19 are formed differently from subsequent sets like 21-29, etc.
(not counting the Celtic branch with its vigesimal system borrowed
from Basque), the sole exception is Sanskrit.
In Sanskrit,
11, 12, etc. (ekā-daśan, dvā-daśan, etc.) are exactly similar
formations to 21, 22, etc. (eka-viṁśati, dvā-viṁśati, etc.),
although grammatically the Sanskrit numbers 1-19 are supposed to be
adjectives, while the numbers above that are supposed to be nouns. The Sanskrit
numbers, therefore, clearly represent a frozen form of the earliest Indo-European
purely decimal number-system before the vigesimal-effect
took place.
2. Although Sanskrit
is a very highly inflectional language, and the modern Indo-Aryan
languages by and large have a very-much-diluted inflectional nature, the case
is the opposite in the case of the numbers, where all the modern Indo-Aryan
languages have a strong degree of inflection as compared to Sanskrit in
the numbers 21-99. All this shows a state of affairs which leads to the
second stage [Note: The numbers from 1-4 are highly inflected in
themselves in Sanskrit and have many forms, e.g. 2: dva-, dvau-,
dvi-, dve-, etc. and 3:
tri-, trayaḥ-, trīṇi, etc. But that is not relevant to the
discussion on hand]:
The
second stage of
development of the Indo-European number system is represented by all
the Indo-European languages outside North India, where we see the vigesimal-effect
in full force. In addition, the original order of the forms is unit+tens,
and there is inflection in the formation of the numbers 11-19:
1. The vigesimal-effect,
with the numbers 11-19 formed differently from subsequent sets like 21-29,
etc., is found in all the branches of Indo-European languages
outside India.
2. The unit+tens
order for the numbers 11-19 is retained in the Iranian, Albanian,
Germanic, Baltic, Slavic and Italic branches, and
partially in the Greek branch (fully in Ancient Greek, and
partially, only for 11-12, in Modern Greek), although, among
these, most of them reverse the order in the numbers after 20.
3. The distinct
inflection in the numbers 1-19 (but, whether having a unit+tens
order or a tens+unit order thereafter, not found in the
numbers beyond 20) is found in the Iranian, Italic, Germanic
and Greek (in Ancient Greek, and for 11-12 in Modern
Greek, as pointed out above) branches.
Strangely, "all
the Indo-European languages outside North India" includes even
the Indo-Aryan Sinhalese language to the south of India which shares these
features:
Sinhalese (IndoEuropean-IndoAryan):
1-9: eka,
deka, tuna, hatara, pasa, haya, hata,
aṭa, navaya, dahaya
1-9 unit
stems: ek-, de-, tun-, hatara-, pas-, ha-,
hat-, aṭa-, nava-
11-19: ekoḷaha,
doḷaha, teḷaha, tudaha, pahaḷoha, soḷaha, hataḷoha,
aṭaḷoha, ekun-vissa
tens
10-100: dahaya, vissa, tisa, hatalisa, panasa,
hɛṭa, hɛttɛɛva, asūva, anūva, siyaya
Other
numbers: unit-stem+tens. Thus the word-order for all the numbers is unit+tens.
[And, like Sanskrit
and Latin (and the other modern Indo-Aryan languages which retain
this feature), the number -9 is expressed by a minus-principle, where ekun-
is used with the following tens-form (except, as in Sanskrit
and most other modern Indo-Aryan languages, for 99)].
Thus: 21: ek-vissa, 89: ekun-anūva. Only 99 is nava-anūva. There is no minus-principle].
[Modern
colloquial Sinhalese has simplified the system, or can it
be that colloquial Sinhalese in fact represents an archaic remnant of
the first stage, where there was a purely decimal system without
the vigesimal-effect?
In colloquial speech the word-order for all the numbers is tens+unit. Even the numbers 11-19 are similarly formed in the form of tens-stem+unit, as daha-eka, daha-deka, etc.
In colloquial speech the word-order for all the numbers is tens+unit. Even the numbers 11-19 are similarly formed in the form of tens-stem+unit, as daha-eka, daha-deka, etc.
The tens
10-100 stems: daha-, visi-, tis-, hatalis-, panas-,
hɛṭa-, hɛttɛɛ-, asū-, anū-, siya-
Thus 21: visi-eka, 99: anū-navaya, etc.]
Thus, Sinhalese
texts provide us with evidence missing in North India itself. Sinhalese
is doubtless a treasure-house of clues to the most archaic stages of Indo-European,
often giving us clues to even older stages than Sanskrit (e.g. the word watura
for "water", as in Germanic English water and Hittite
watar). These clues are not recognized because of the blinkers of the
AIT, which treats all "Indo-Aryan" languages (i.e. Indo-European
languages native to India) as belonging to one branch which entered India in
its earliest form as the Vedic Sanskrit language. Orthodox opponents of
the AIT, who also want to accord primacy to the Vedic language, also adopt these blinkers.
In this second
stage, therefore, it is clear that there was a vigesimal-effect where
only the numbers 11-19 acquired distinctly inflected forms but not the other
in-between numbers from 21-99.
This second
stage of development of the Indo-European number system is not found
recorded in any text or document in North India because the older Sanskrit
numbers of the first stage had become frozen in form and the Prakrits
are recorded from a much later post-Buddhist period in the second half
of the 1st millennium BCE, long after the departure of the other branches of Indo-European
languages westwards from India, and after the diffusion of the Vedic
Sanskrit culture to the Dravidian South, all of which must have
taken place at a point of time when the Indo-Aryan languages of the
North still had a numeral system at the second stage of development.
The third
stage of development
of the Indo-European number system, where the number system continued
to become more and more subject to inflection and fusion between the tens-forms
and the unit-forms, and the inflection in the formation of compound numbers
spread to all the numbers from 11-99, is found in its earliest forms in
most of the Prakrits and much more so in the modern Indo-Aryan
languages of North India. In this stage, all the compound numbers
between 10 and 100 acquired distinct forms with fusion and
inflection between the tens and units words. The numbers 11-19,
which had already become distinctly inflected in the second stage,
therefore got a double dose of inflection:
1. In the
first stage, we see that there is barely any inflection, where the numbers 11-19
are formed just like the subsequent sets: thus 11: eka+daśa
= ekā-daśa, 12: dvi+daśa = dvā-daśa, etc.
Compare with 21: eka+viṁśati = eka-viṁśati, 22:
dvi+viṁśati = dvā-viṁśati, etc.
2. In the
second stage, which as we saw is unrecorded in India, there must have
clearly been greater fusion and inflection in 11-19, but not in the
later sets 21-29, etc.
3. In the
third stage, we find strong inflection in all the numbers, but:
a) In the numbers
after 20, the tens-forms and unit-forms are still
recognizable: Hindi 21: ek+bīs
= ikk-īs, 22: do+bīs = bā-īs (both do- and bā-
are recognizable as forms of an original dva-).
b) In the
numbers 11-19, there is a clear case of further fusion and
inflection: Hindi 11: ek+das = gyārah, 22: do+das
+ bārah, etc., where the tens and unit elements are even
more fused, inflected and changed as to make recognition of the original
elements more difficult: the -r- element in modern Indo-Aryan numbers
from 11-19 is difficult to recognize as a development from the word for 10.
[A similar process of further inflection seems to have taken place in
the westernmost IE branch Germanic, where 11 and maybe 12,
at least, seem to have continued to become more inflected later, making
recognition of the elements difficult: English 11: one+ten
= eleven, German eins+zehn = elf (German), etc. Note
also: Germanic languages are also the only modern languages outside
India retaining the original unit+tens order in their compound
numbers after 20].
[Note
on Sanskrit vis-à-vis Prakrits vis-à-vis modern Indo-Aryan:
The earliest
beginnings of the third stage can be seen in most of the recorded Prakrits.
But the literary Prakrits were actually highly Sanskritized or Sanskrit-imitating
approximations of the spoken forms of Indo-Aryan speech of the time, and
so they do not reflect the actual state of the spoken speech of the time. Thus,
for example:
a) For the
number 22, Pali texts alternately use both dvāvīsati
(imitating Sanskrit dvāviṁśati) and bāvīsa (similar to modern
Indo-Aryan form bāvīs, etc.).
b) The Pali
word, paññāsa/paṇṇāsa for 50, is closer in form to the modern
Indo-Aryan word pannās for 50 than to the Sanskrit
word pañcāśat for 50.
But its uniform use in that form (-paññāsa/-paṇṇāsa)
in all the compound unit+tens numbers (i.e. in 49
and 51-58) reflects imitation of the similar use of the word -pañcāśat
in Sanskrit rather than the use of multiple forms in modern
Indo-Aryan languages:
Hindi: -cās (49), -van
(51,52,54,57,58), -pan (53,55,56).
Marathi: -pannās (49), -vanna
(51,52,55,57,58), -panna (53,54,56).
Gujarati: -pacās (49), -van
(51,52,55,57,58), -pan (53,54,56).
Similiarly,
its uniform use of the form pañca- (5) in all the compound unit+tens
numbers (25,35,45, etc.) reflects imitation of the similar use of the same word pañca- in Sanskrit rather than the use of multiple forms in modern
Indo-Aryan languages:
Hindi: pacc- (25), paĩ-
(35,45,65), pac- (55,75,85), pañcā- (95),
Marathi: pañc- (25), pas- (35),
pañce- (45), pañçā- (55), pā- (65), pañcyā
(75,85,95)
Gujarati: pacc- (25), pāã- (35,65), pis- (45), pañc- (75,85), pañcā-
(55,95)].
Therefore,
the area of North India was home to the first stage of development of the Indo-European
number system (as represented by Vedic and Classical Sanskrit, and
perhaps colloquial Sinhalese?), as well as to the third stage,
both of which are found only in North India, while all
the other branches of Indo-European languages outside North India
(include literary Sinhalese) represent the second stage. This
clearly indicates that the Original Homeland of all these languages was in
North India, and they migrated from India during a period when the Indo-Aryan
languages of the North were at the second stage, and shared a similar vigesimal-affected
decimal system.
2. The
Spread of the Vigesimal-Affected Decimal Number-System:
As we saw, the
first and third stages of development of the Indo-European decimal
number system, as shown by what we have called the vigesimal-effect
(i.e. where the numbers 11-19 are formed in a distinctly different way
from the later sets like 21-29, 31-39, etc.), are found only
in North India, and the second stage is found in all the
branches outside North India (and therefore must have logically existed in North
India in an intervening period, even if not recorded), shows that the Original
Homeland of all these Indo-European languages was in North India.
And an
examination of the areas and languages which have this "vigesimal-affected
decimal number-system" leads to the same conclusion:
There are
many stray languages among the thousands of native American (Amerindian)
languages with decimal systems, which have distinctly different
formations for the numbers 11-19 on the one hand and subsequent sets
like 21-29, 31-39, etc. on the other. We saw the examples of Cherokee
and Navaho, and there must be many more. The explanation for this can be
the effect of neighboring languages with vigesimal systems, and there
are many of them in America: we saw the examples of the Nahuatl (Aztec),
Yucatec (Mayan) and Yupik languages. Likewise, we saw the example
of the Kanuri language in the interior of Africa which also clearly has
a vigesimal-affected decimal system, and, again,
there may be many more such languages in Africa. But obviously, these remote
languages of America and Africa cannot have a place in the history of the
origin and spread of the Indo-European languages or number systems.
The other
languages which have vigesimal-affected decimal systems
are: some Uralo-Altaic languages (e.g. Finnish, Estonian),
some Semito-Hamitic languages (e.g. Arabic, Hebrew, Maltese),
some Austronesian languages (e.g. Malay, Tagalog), and the
Dravidian languages of South India. However, barring the Dravidian
languages, the following points may be noted about the other languages:
1. The vigesimal-affected
decimal feature is not found in the families as a whole:
thus, the other relatives of Finnish and Estonian do not have
this vigesimal-affected decimal system (check Hungarian,
Turkish, Mongolian, etc. earlier in this article). Nor do the
other relatives of Arabic, Hebrew and Maltese (check the
ancient and modern Hamitic languages, Amharic earlier in this
article, and the available data on the ancient Semitic languages). And
nor do the other relatives of Malay and Tagalog (check Hawaiian
earlier).
2. Except
perhaps Arabic and Hebrew, the other languages are clearly or
arguably influenced by Indo-European languages. Check what the Wikipedia
entry has to say about Maltese:
"Maltese
has evolved independently of Literary Arabic and its varieties into a standardized language over the past
800 years in a gradual process of Latinisation.^{[5]}^{[6]} Maltese is therefore considered an
exceptional descendant of Arabic that has no diglossic relationship with Standard Arabic or Classical Arabic,^{[7]} and is classified separately from the Arabic macrolanguage.^{[8]} Maltese is also unique among Semitic
languages since its morphology has been deeply influenced by Romance languages, namely Italian and Sicilian".
The Malay
and Tagalog languages may have been influenced by Indian languages: S.E.
Asia was under the influence of Indian culture since almost two millennia.
Finnish (and the very closely related Estonian)
are known to have a large number of Indo-European (even specifically Indo-Aryan
and Iranian) borrowings. Also, the word for "hundred" in Finnish
is sata, and in Estonian is sada.
3. The numbers
11-19 are certainly formed differently from the later numbers in all the
above languages, but in every single one of them, the tens and unit
forms are not fused together (except in Maltese, which, as seen above is
a dialect of Arabic highly influenced in its morphology by Indo-European
languages), and so the numbers 11-19 do not require to be individually
learned since they are formed by simple juxtaposition: check the numbers in all
these languages detailed earlier in this article.
In sharp
contrast to this, in the Dravidian languages:
1. All the
languages have this vigesimal-affected decimal system.
2. The tens
and unit words in 11-19 are fused together by inflection.
So the Indo-European
languages outside North India, and the Dravidian languages in South
India, are the only families of languages in the world which share this vigesimal-affected
decimal feature as a whole and in almost the same way (inflection
found only in 11-19 but not after 20). Obviously this cannot be a
coincidence.
[The only
difference is that the Dravidian numbers 1-19 have the word-order
tens+unit. This shows two things: that Dravidian was
influenced by Indo-Aryan in this vigesimal-effect, but also that
it did not change its original more logical word-order for the less
logical Indo-Aryan one].
Generally,
we find common elements between the Indo-Aryan and the Dravidian
languages which are not found in the other Indo-European languages outside
India (e.g. the cerebral/retroflex consonants, many grammatical features and
words, etc. or even words for specifically Indian flora and fauna). These are
usually attributed (in most cases probably correctly) to Dravidian
influence on the Indo-Aryan languages. But then a conclusion sought to
be drawn from these common features is that it proves that the Indo-European
homeland cannot be in India, since in that case these features should
have been found in some Indo-European languages outside India as well,
and so this proves the AIT (Aryan Invasion Theory) and disproves the OIT
(Out-of-India Theory). But this logic is extremely faulty for two reasons:
1. The other
branches of Indo-European languages, in the OIT scenario, were situated
well to the west of the Indo-Aryan languages and away from any
influence from the Dravidian languages of South India, and, in any case,
they had started migrating out northwestwards in a very early period, around
3000 BCE. So they obviously did not participate in any common Indo-Aryan-Dravidian
linguistic innovations in the interior of India, or get affected by any Dravidian
features.
2. The
example of the purely Indo-Aryan Romany (Gypsy) language of Europe -
which undisputedly migrated from India just over a thousand years ago, but did
not take with it either the retroflex consonants, or Dravidian words, or
words for specifically Indian flora and fauna - shows the faultiness of this
logic.
But in this
case, we find the common element is between the Dravidian languages of
South India and the Indo-European languages outside India, and it is not
found in either Sanskrit or the modern Indo-Aryan languages of
North India!
[The following final section modified on 14/3/2019]:
[The following final section modified on 14/3/2019]:
This gives
us the following clear picture of a decimal number system developing in
a core area in North India, occupied by the Indo-European
languages which were spread out in a sprawling area between the Austric
languages in the east and Burushaski in the north-northwest:
1. In the
first stage, the pre-proto-Indo-European stage, which has left us no
records, the number system which developed was a purely decimal system, as in Santali
or the Sino-Tibetan languages, with words only for the numbers 1-10 and
perhaps 100.
2. In the
second stage, the number system developed into a system as in Turkish,
etc., with numbers for 1-10, and the tens words 20-100. These are
the only numbers reconstructed for proto-Indo-European. This is the
system found fossilized or frozen in Vedic (although with some
inflection due to the development of sandhi). It also seems to have been
preserved in present-day Colloquial Sinhalese, and the Sinhalese
may therefore have been the first to migrate from North India, though only
southwards into Sri Lanka.
3. In the
third stage, this system continued to evolve and was influenced in its
further evolution to a small extent by the surrounding vigesimal number
systems, and developed into a vigesimal-affected decimal system, where
the unitary nature of the numbers 1-10 was extended to the next set of
ten numbers by fusing and inflecting the unit-word and tens-word numbers
for 11-19 into single unitary
words. This system formed over the whole of India, including in the Dravidian
languages to the South (and in the literary form of the Sinhalese
language even further south, due to the influence of Pali). The other
branches were still within the Indian sphere at the time, and this stage spread
out westwards and outwards from India with the expansion and migration
of the other (the non-Pūru or "non-Indo-Aryan") Indo-European
dialects, which later spread out to Central Asia, West Asia and Europe. This
system prevailed on the ground in the core area in North India, but the
fossilized system of the second stage alone continued to be recorded in
the Vedic and Classical Sanskrit texts.
4. In the
fourth stage, after the migration of the other Indo-European
dialects and the standardization of the number system in the Dravidian
languages of the South, at some time in the late second or the early first
millennium BCE, the system on the ground in the core area of North India
continued to evolve, i.e. to become more and more unitary, with the unitary
nature of the numbers 1-20 now extended to all the numbers 1-99,
by fusing and inflecting the unit-word and tens-word numbers for 21-99 into single fused words. This system came to
be recorded in its earliest form in the Prakrit texts (which often
alternate between Sanskrit-imitation forms and colloquial forms), and is
recorded in more fully developed forms only in the last thousand years or so
after the modern Indo-Aryan languages developed into literary languages.
To sum up,
India is the only place in the world to have all four stages of the decimal
system:
The first stage: In India, in non-Indo-European languages like many Sino-Tibetan
languages, and in Santali.
The second stage: 1. Colloquial Sinhalese. 2. Sanskrit (though inflected
by rules of sandhi).
The third stage: Found in many stray languages of the world, but found universally
only in two groups: 1. Dravidian languages. 2. All the Indo-European
languages outside North India, including Literary Sinhalese to
the south of India.
The fourth stage: Found universally in the whole world only in the Modern
Indo-Aryan languages of North India.
One reason
why this fourth stage developed only in the languages of North India
could be because of the early development of a very high number system in Sanskrit
(as we already saw at the very beginning of this article) as recorded in the Lalitavistara,
where there were at least in theory number words up to 10^{35} to begin
with, and, above 1000, they represented multiples of hundred (sahasra
with three zeroes, lakṣa with five zeroes, koṭi with seven
zeroes, ayuta with nine zeroes, niyuta with eleven zeroes, etc.),
just as modern American number words represent multiples of thousand (thousand
with three zeroes, million with six zeroes, billion with nine
zeroes, trillion with twelve zeroes, etc.).
In modern American
numbers, a number such as 123,456,789 would be "one-twenty-three million,
four-fifty-six thousand, seven-eighty-nine".
As per
Indian numbers in Sanskrit, this would even now be put as 12,34,56,789 or "twelve crores,
thirty-four lakhs, fifty-six thousand, seven (hundred and) eighty-nine".
The qualifying numbers for the bigger number words (thousand, lakhs,
crores, etc.) could be any number from two to ninety-nine.
Sanskrit therefore inflectionally fused each of the qualifying number word
forms from two to ninety-nine into single words. While literary
Sanskrit continued to use the earlier word forms for two to ninety-nine,
the spoken languages show us the new fused forms. Thus, in Hindi, the above
number would be bārah karoḍ, cauntīs lākh, chappan hazār, sāt sau aṭṭhānave.
This is
somewhat like the development in the second stage of formation of the
decimal numeral system. In the first stage, as we saw, the Egyptian
numeral system had different symbols for every multiple of 10 (i.e. 1,
10, 100, 1000, 10000, etc.), and each of these symbols would require to be
repeated any number of times from two to nine (twice, thrice,
four times, five times, six times, seven times, eight times or nine times). In
the second stage, the Chinese system, there were eight different
additional symbols for the numbers 2-9, and these were used as qualifying
symbols before the symbols for the multiples of 10 (Thus 6482 would be 6,
1000, 4, 100, 8, 10, 2).
In a similar way, Sanskrit numbers developed
fused qualifying words from two to ninety-nine for the
multiples of 100 above 1000 (sahasra, lakṣa, koṭi, ayuta,
niyuta, etc.).