Decimal

The decimal numeral system (also called base-ten positional numeral system, and occasionally called denary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral system. [1] The way of denoting numbers in the decimal system is often referred to as Decimal notation.[2]

A decimal numeral, or just decimal, or, improperly decimal number, refers generally to the notation of a number in the decimal system, which contains a decimal separator (for example 10.00 or 3.14159). Sometimes these terms are used for the any numeral in the decimal system. A decimal may also refer to any digit after the decimal separator, such as in "3.14 is the approximation of π to two decimals".

The numbers that may be represented in the decimal system are the decimal fractions, that is the fractions of the form a/10n, where a is an integer, and n is a nonnegative integer.

The decimal system has been extended to infinite decimals, for representing any real number, by using an infinite sequence of digits after the decimal separator (see Decimal representation). In this context, the usual decimals are sometimes called terminating decimals. A repeating decimal, is an infinite decimal, that, after some place repeats indefinitely the same sequence of digits (for example 5.123144144144144... = 5.123144).[3] An infinite decimal represents a rational number if and only if it is a repeating decimal or is a finite decimal.

Origin

Ten fingers on two hands, the possible starting point of the decimal counting.

Many numeral systems of ancient civilisations, use ten and and its powers for representing numbers, probably because there are ten fingers on two hands and people started counting by using their fingers. Examples are Brahmi numerals, Greek numerals, Hebrew numerals, Roman numerals, and Chinese numerals. Very large numbers were difficult to represent in these old numeral systems, and, only the best mathematicians were able to multiply or divide large numbers, these difficulty were completely solved with the introduction of the Hindu–Arabic numeral system for representing integers. This system has been extended to represent some non-integer numbers, called decimal fractions or decimal numbers for forming the decimal numeral system.

Decimal notation

For writing numbers, the decimal system uses ten decimal digits, a decimal mark, and, for negative numbers, a minus sign "–". The decimal digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9;[4] the decimal separator is the dot "." in many countries, including English speaking ones, but may be a comma "," in other countries, mainly in Europe.

For representing a nonnegative number, a decimal consists of

• either a (finite) sequence of digits such that 2017, or, in full generality,
${\displaystyle a_{m}a_{m-1}\ldots a_{0}}$
(in this case, the decimal represents an integer)
• or two sequence of digits separated by a decimal mark such as 3.14159, 15.00, or in full generality
${\displaystyle a_{m}a_{n-1}\ldots a_{0}.b_{1}b_{2}\ldots b_{n}}$

It is generally assumed that, if m > 0, the first digit am is not zero, but, in some circumstances, it may be useful to have one or more 0 on the left. This does changes the value represented by the decimal, for example 3.14 = 03.14 = 003.14. Similarly, if bn =0, it may be removed, and, conversely, trailing zeros may be added without changing the represented number. For example, 15 = 15.0 = 15.00 and 5.2 = 5.20 = 5.200. sometimes, the unnecessary zeros are used for indicating the accuracy of a measure. For example, 15.00m may indicate that the measurement error is less than one centimetre, while 15m may mean that the length is roughly fifteen meters, and that the error may exceed 10cm.

For representing a negative number, a minus sign is added before am.

The numeral ${\displaystyle a_{m}a_{n-1}\ldots a_{0}.b_{1}b_{2}\ldots b_{n}}$ represents the number

${\displaystyle a_{m}10^{m}+a_{m-1}10^{m-1}+\cdots a_{1}0+a_{0}+{\frac {b_{1}}{10}}+\cdots +{\frac {b_{n}}{10^{n}}}.}$

Therefore, the contribution of each digit to the value of a number depends on its position in the numeral, that is, the decimal system is a positional numeral system

Decimal fractions

The numbers that are represented by decimal numeral are the decimal fractions (sometimes called decimal numbers), that is, the [rational number]]s that may be expressed as a fraction, the denominator of which is a power of ten.[5] For example the numerals ${\displaystyle 0.8,14.89,0.00024}$ represent the fractions 8/10, 1489/100, 24/100000. More generally, a decimal with n digits after the separator represents the fraction with denominator 10n, whose numerator is the integer obtained by removing the separator.

Expressed as a fully reduced fraction, the decimal numbers are those, whose denominator is a product of a power of 2 by a power of 5, thus the smallest denominators of decimal numbers are

${\displaystyle 1=2^{0}\cdot 5^{0},2=2^{1}\cdot 5^{0},4=2^{2}\cdot 5^{0},5=2^{0}\cdot 5^{1},8=2^{3}\cdot 5^{0},10=2^{1}\cdot 5^{1},16=2^{4}\cdot 5^{0},25=2^{0}\cdot 5^{2},\ldots }$

The integer part, or integral part of a decimal is the integer written to the left of the decimal separator (see also truncation). For a nonnegative decimal, it is the largest integer that is not greater than the decimal, the part from the decimal separator to the right is the fractional part, which equals the difference between the numeral and its integer part.

When the integral part of a numeral is zero, it may occur, typically in computing, that the integer part is not written (for example .1234, instead of 0.1234). In normal writing, this is generally avoided because of the risk of confusion between the decimal mark and other punctuation.

Trailing zeros after the decimal point are not necessary, although in science, engineering and statistics they can be retained to indicate a required precision or to show a level of confidence in the accuracy of the number: Although 0.080 and 0.08 are numerically equal, in engineering 0.080 suggests a measurement with an error of up to one part in two thousand (±0.0005), while 0.08 suggests a measurement with an error of up to one in two hundred (see significant figures).

Infinite decimal expansion

Rational numbers

Any rational number with a denominator whose only prime factors are 2 and/or 5 may be expressed as a decimal fraction and has a finite decimal expansion.[6]

1/2 = 0.5
1/20 = 0.05
1/5 = 0.2
1/50 = 0.02
1/4 = 0.25
1/40 = 0.025
1/25 = 0.04
1/8 = 0.125
1/125 = 0.008
1/10 = 0.1

If a fully reduced rational number's denominator has any prime factors other than 2 or 5, it cannot be expressed as a finite decimal fraction,[6] and has a unique eventually repeating infinite decimal expansion.

1/3 = 0.333333... (with 3 repeating)
1/9 = 0.111111... (with 1 repeating)

100 − 1 = 99 = 9 × 11:

1/11 = 0.090909...

1000 − 1 = 9 × 111 = 27 × 37:

1/27 = 0.037037037...
1/37 = 0.027027027...
1/111 = 0.009009009...

also:

1/81 = 0.012345679012... (with 012345679 repeating)

That a rational number must have a finite or recurring decimal expansion can be seen to be a consequence of the long division algorithm, in that there are at most q − 1 possible nonzero remainders on division by q, so that the recurring pattern will have a period less than q. For instance, to find 3/7 by long division:

  0.4 2 8 5 7 1 4 …
7)3.0 0 0 0 0 0 0 0
2 8                         30 ÷ 7 = 4 with a remainder of 2
2 0
1 4                       20 ÷ 7 = 2 with a remainder of 6
6 0
5 6                     60 ÷ 7 = 8 with a remainder of 4
4 0
3 5                   40 ÷ 7 = 5 with a remainder of 5
5 0
4 9                 50 ÷ 7 = 7 with a remainder of 1
1 0
7               10 ÷ 7 = 1 with a remainder of 3
3 0
2 8             30 ÷ 7 = 4 with a remainder of 2
2 0
etc.


The converse to this observation is that every recurring decimal represents a rational number p/q. This is a consequence of the fact that the recurring part of a decimal representation is, in fact, an infinite geometric series which will sum to a rational number, for instance,

${\displaystyle 0.0123123123\ldots ={\frac {123}{10000}}\sum _{k=0}^{\infty }0.001^{k}={\frac {123}{10000}}\ {\frac {1}{1-0.001}}={\frac {123}{9990}}={\frac {41}{3330}}}$

Real numbers

Every real number has a (possibly infinite) decimal representation; i.e., it can be written as

${\displaystyle x=\operatorname {sign} \sum _{i\in \mathbb {Z} }a_{i}\,10^{i}}$

where

• sign ∈ {+,−}, which is related to the sign function,
• ℤ is the set of all integers (positive, negative, and zero), and
• ai ∈ { 0,1,...,9 } for all i ∈ ℤ are its decimal digits, equal to zero for all i greater than some number (that number being the common logarithm of |x|).

Such a sum converges as more and more negative values of i are included, even if there are infinitely many non-zero ai.

Rational numbers (e.g., p/q) with prime factors in the denominator other than 2 and 5 (when reduced to simplest terms) have a unique recurring decimal representation.

Non-uniqueness of decimal representation

Consider those rational numbers which have only the factors 2 and 5 in the denominator, i.e., which can be written as p/2a5b. In this case there is a terminating decimal representation, for instance, 1/1 = 1, 1/2 = 0.5, 3/5 = 0.6, 3/25 = 0.12 and 1306/1250 = 1.0448. Such numbers are the only real numbers which do not have a unique decimal representation, as they can also be written as a representation that has a recurring 9, for instance 1 = 0.99999..., 1/2 = 0.499999..., etc. The number 0 = 0/1 is special in that it has no representation with recurring 9.

This leaves the irrational numbers, they also have unique infinite decimal representations, and can be characterised as the numbers whose decimal representations neither terminate nor recur.

So in general the decimal representation is unique, if one excludes representations that end in a recurring 9.

The same trichotomy holds for other base-n positional numeral systems:

• Terminating representation: rational where the denominator divides some nk
• Recurring representation: other rational
• Non-terminating, non-recurring representation: irrational

A version of this even holds for irrational-base numeration systems, such as golden mean base representation.

Decimal computation

Multiples of bytes
Decimal
Value Metric
1000 kB kilobyte
10002 MB megabyte
10003 GB gigabyte
10004 TB terabyte
10005 PB petabyte
10006 EB exabyte
10007 ZB zettabyte
10008 YB yottabyte
Binary
Value IEC JEDEC
1024 KiB kibibyte KB kilobyte
10242 MiB mebibyte MB megabyte
10243 GiB gibibyte GB gigabyte
10244 TiB tebibyte
10245 PiB pebibyte
10246 EiB exbibyte
10247 ZiB zebibyte
10248 YiB yobibyte
Diagram of the world's earliest decimal multiplication table (c. 305 BC) from the Warring States period

Decimal computation was carried out in ancient times in many ways, typically in rod calculus, with decimal multiplication table used in ancient China and with sand tables in India and Middle East or with a variety of abaci.

Modern computer hardware and software systems commonly use a binary representation internally (although many early computers, such as the ENIAC or the IBM 650, used decimal representation internally).[7] For external use by computer specialists, this binary representation is sometimes presented in the related octal or hexadecimal systems.

For most purposes, however, binary values are converted to or from the equivalent decimal values for presentation to or input from humans; computer programs express literals in decimal by default. (123.1, for example, is written as such in a computer program, even though many computer languages are unable to encode that number precisely.)

Both computer hardware and software also use internal representations which are effectively decimal for storing decimal values and doing arithmetic. Often this arithmetic is done on data which are encoded using some variant of binary-coded decimal,[8][9] especially in database implementations, but there are other decimal representations in use (such as in the new IEEE 754 Standard for Floating-Point Arithmetic).[10]

Decimal arithmetic is used in computers so that decimal fractional results can be computed exactly, which is not possible using a binary fractional representation, this is often important for financial and other calculations.[11][12]

History

The world's earliest decimal multiplication table was made from bamboo slips, dating from 305 BC, during the Warring States period in China.

Many ancient cultures calculated with numerals based on ten, sometimes argued due to human hands typically having ten digits.[13] Egyptian hieroglyphs, in evidence since around 3000 BC, used a purely decimal system,[14] just as the Cretan hieroglyphs (ca. 1625−1500 BC) of the Minoans whose numerals are closely based on the Egyptian model.[15][16] The decimal system was handed down to the consecutive Bronze Age cultures of Greece, including Linear A (ca. 18th century BC−1450 BC) and Linear B (ca. 1375−1200 BC) — the number system of classical Greece also used powers of ten, including, like the Roman numerals did, an intermediate base of 5.[17] Notably, the polymath Archimedes (ca. 287–212 BC) invented a decimal positional system in his Sand Reckoner which was based on 108[17] and later led the German mathematician Carl Friedrich Gauss to lament what heights science would have already reached in his days if Archimedes had fully realized the potential of his ingenious discovery.[18] The Hittites hieroglyphs (since 15th century BC), just like the Egyptian and early numerals in Greece, was strictly decimal.[19]

Some ancient texts like the Vedas dating back to 1900–1700 BCE mention decimals and mathematical decimal fractions.[citation needed]

The Egyptian hieratic numerals, the Greek alphabet numerals, the Hebrew alphabet numerals, the Roman numerals, the Chinese numerals and early Indian Brahmi numerals are all non-positional decimal systems, and required large numbers of symbols, for instance, Egyptian numerals used different symbols for 10, 20, to 90, 100, 200, to 900, 1000, 2000, 3000, 4000, to 10,000.[20] The world's earliest positional decimal system was the Chinese rod calculus[21]

The world's earliest positional decimal system
Upper row vertical form
Lower row horizontal form

History of decimal fractions

counting rod decimal fraction 1/7

Decimal fractions were first developed and used by the Chinese in the end of 4th century BC,[22] and then spread to the Middle East and from there to Europe,[21][23] the written Chinese decimal fractions were non-positional.[23] However, counting rod fractions were positional.[21]

Qin Jiushao in his book Mathematical Treatise in Nine Sections (1247) denoted 0.96644 by

, meaning
096644

[24]

J. Lennart Berggren notes that positional decimal fractions appear for the first time in a book by the Arab mathematician Abu'l-Hasan al-Uqlidisi written in the 10th century,[25] the Jewish mathematician Immanuel Bonfils used decimal fractions around 1350, anticipating Simon Stevin, but did not develop any notation to represent them.[26] The Persian mathematician Jamshīd al-Kāshī claimed to have discovered decimal fractions himself in the 15th century.[25] Al Khwarizmi introduced fraction to Islamic countries in the early 9th century, his fraction presentation was an exact copy of traditional Chinese mathematical fraction from Sunzi Suanjing.[21] This form of fraction with numerator on top and denominator at bottom without a horizontal bar was also used by 10th century Abu'l-Hasan al-Uqlidisi and 15th century Jamshīd al-Kāshī's work "Arithmetic Key".[21][27]

A forerunner of modern European decimal notation was introduced by Simon Stevin in the 16th century.[28]

Natural languages

The ingenious method of expressing every possible number using a set of ten symbols emerged in India. Several Indian languages show a straightforward decimal system. Many Indo-Aryan and Dravidian languages have numbers between 10 and 20 expressed in a regular pattern of addition to 10.[29]

The Hungarian language also uses a straightforward decimal system. All numbers between 10 and 20 are formed regularly (e.g. 11 is expressed as "tizenegy" literally "one on ten"), as with those between 20 and 100 (23 as "huszonhárom" = "three on twenty").

A straightforward decimal rank system with a word for each order (10 , 100 , 1000 , 10,000 ), and in which 11 is expressed as ten-one and 23 as two-ten-three, and 89,345 is expressed as 8 (ten thousands) 9 (thousand) 3 (hundred) 4 (tens) 5 is found in Chinese, and in Vietnamese with a few irregularities. Japanese, Korean, and Thai have imported the Chinese decimal system. Many other languages with a decimal system have special words for the numbers between 10 and 20, and decades, for example, in English 11 is "eleven" not "ten-one" or "one-teen".

Incan languages such as Quechua and Aymara have an almost straightforward decimal system, in which 11 is expressed as ten with one and 23 as two-ten with three.

Some psychologists suggest irregularities of the English names of numerals may hinder children's counting ability.[30]

Other bases

Some cultures do, or did, use other bases of numbers.

• Pre-Columbian Mesoamerican cultures such as the Maya used a base-20 system (perhaps based on using all twenty fingers and toes).
• The Yuki language in California and the Pamean languages[31] in Mexico have octal (base-8) systems because the speakers count using the spaces between their fingers rather than the fingers themselves.[32]
• The existence of a non-decimal base in the earliest traces of the Germanic languages, is attested by the presence of words and glosses meaning that the count is in decimal (cognates to ten-count or tenty-wise), such would be expected if normal counting is not decimal, and unusual if it were.[33][34] Where this counting system is known, it is based on the long hundred of 120 in number, and a long thousand of 1200 in number, the descriptions like 'long' only appear after the small hundred of 100 in number appeared with the Christians. Gordon's Introduction to Old Norse p 293, gives number names that belong to this system. An expression cognate to 'one hundred and eighty' is translated to 200, and the cognate to 'two hundred' is translated at 240. Goodare details the use of the long hundred in Scotland in the Middle Ages, giving examples, calculations where the carry implies i C (i.e. one hundred) as 120, etc. That the general population were not alarmed to encounter such numbers suggests common enough use, it is also possible to avoid hundred-like numbers by using intermediate units, such as stones and pounds, rather than a long count of pounds. Goodare gives examples of numbers like vii score, where one avoids the hundred by using extended scores. There is also a paper by W.H. Stevenson, on 'Long Hundred and its uses in England'.[citation needed]
• Many or all of the Chumashan languages originally used a base-4 counting system, in which the names for numbers were structured according to multiples of 4 and 16.[35]
• Many languages[36] use quinary (base-5) number systems, including Gumatj, Nunggubuyu,[37] Kuurn Kopan Noot[38] and Saraveca. Of these, Gumatj is the only true 5–25 language known, in which 25 is the higher group of 5.
• Some Nigerians use duodecimal systems.[39] So did some small communities in India and Nepal, as indicated by their languages.[40]
• The Huli language of Papua New Guinea is reported to have base-15 numbers.[41] Ngui means 15, ngui ki means 15 × 2 = 30, and ngui ngui means 15 × 15 = 225.
• Umbu-Ungu, also known as Kakoli, is reported to have base-24 numbers.[42] Tokapu means 24, tokapu talu means 24 × 2 = 48, and tokapu tokapu means 24 × 24 = 576.
• Ngiti is reported to have a base-32 number system with base-4 cycles.[36]
• The Ndom language of Papua New Guinea is reported to have base-6 numerals.[43] Mer means 6, mer an thef means 6 × 2 = 12, nif means 36, and nif thef means 36×2 = 72.

References

1. ^ The History of Arithmetic, Louis Charles Karpinski, 200pp, Rand McNally & Company, 1925.
2. ^ Lam Lay Yong & Ang Tian Se (2004) Fleeting Footsteps. Tracing the Conception of Arithmetic and Algebra in Ancient China, Revised Edition, World Scientific, Singapore.
3. ^ The viniculum (overline) in 5.123144 indicates that the '144' sequence repeats itself indefinitely, i.e. 5.123144144144144....
4. ^ In some countries, such as Arab speaking ones, other glyphs are used for the digits
5. ^ "Decimal Fraction". Encyclopedia of Mathematics. Retrieved 2013-06-18.
6. ^ a b Math Made Nice-n-Easy. Piscataway, N.J.: Research Education Association. 1999. p. 141. ISBN 0-87891-200-2.
7. ^ Fingers or Fists? (The Choice of Decimal or Binary Representation), Werner Buchholz, Communications of the ACM, Vol. 2 #12, pp3–11, ACM Press, December 1959.
8. ^ Schmid, Hermann (1983) [1974]. Decimal Computation (1 (reprint) ed.). Malabar, Florida, USA: Robert E. Krieger Publishing Company. ISBN 0-89874-318-4.
9. ^ Schmid, Hermann (1974). Decimal Computation (1 ed.). Binghamton, New York, USA: John Wiley & Sons. ISBN 0-471-76180-X.
10. ^ Decimal Floating-Point: Algorism for Computers, Cowlishaw, M. F., Proceedings 16th IEEE Symposium on Computer Arithmetic, ISBN 0-7695-1894-X, pp104-111, IEEE Comp. Soc., June 2003
11. ^ Decimal Arithmetic – FAQ
12. ^ Decimal Floating-Point: Algorism for Computers, Cowlishaw, M. F., Proceedings 16th IEEE Symposium on Computer Arithmetic (ARITH 16), ISBN 0-7695-1894-X, pp. 104–111, IEEE Comp. Soc., June 2003
13. ^ Dantzig, Tobias (1954), Number / The Language of Science (4th ed.), The Free Press (Macmillan Publishing Co.), p. 12, ISBN 0-02-906990-4
14. ^ Georges Ifrah: From One to Zero. A Universal History of Numbers, Penguin Books, 1988, ISBN 0-14-009919-0, pp. 200–213 (Egyptian Numerals)
15. ^ Graham Flegg: Numbers: their history and meaning, Courier Dover Publications, 2002, ISBN 978-0-486-42165-0, p. 50
16. ^ Georges Ifrah: From One to Zero. A Universal History of Numbers, Penguin Books, 1988, ISBN 0-14-009919-0, pp. 213–218 (Cretan numerals)
17. ^ a b Greek numerals
18. ^ Menninger, Karl: Zahlwort und Ziffer. Eine Kulturgeschichte der Zahl, Vandenhoeck und Ruprecht, 3rd. ed., 1979, ISBN 3-525-40725-4, pp. 150–153
19. ^ Georges Ifrah: From One to Zero. A Universal History of Numbers, Penguin Books, 1988, ISBN 0-14-009919-0, pp. 218f. (The Hittite hieroglyphic system)
20. ^ Lam Lay Yong et al The Fleeting Footsteps pp 137–139
21. Lam Lay Yong, "The Development of Hindu–Arabic and Traditional Chinese Arithmetic", Chinese Science, 1996 p38, Kurt Vogel notation
22. ^ "Ancient bamboo slips for calculation enter world records boo". The Institute of Archaeology, Chinese Academy of Social Sciences. Retrieved 10 May 2017.
23. ^ a b Joseph Needham (1959). "Decimal System". Science and Civilisation in China, Volume III, Mathematics and the Sciences of the Heavens and the Earth. Cambridge University Press.
24. ^ Jean-Claude Martzloff, A History of Chinese Mathematics, Springer 1997 ISBN 3-540-33782-2
25. ^ a b Berggren, J. Lennart (2007). "Mathematics in Medieval Islam". In Katz, Victor J. The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press. p. 530. ISBN 978-0-691-11485-9.
26. ^ Gandz, S.: The invention of the decimal fractions and the application of the exponential calculus by Immanuel Bonﬁls of Tarascon (c. 1350), Isis 25 (1936), 16–45.
27. ^ Lam Lay Yong, "A Chinese Genesis, Rewriting the history of our numeral system", Archive for History of Exact Science 38: 101–108.
28. ^ B. L. van der Waerden (1985). A History of Algebra. From Khwarizmi to Emmy Noether. Berlin: Springer-Verlag.
29. ^ "Indian numerals". Ancient Indian mathematics. Retrieved 2015-05-22.
30. ^ Azar, Beth (1999). "English words may hinder math skills development". American Psychology Association Monitor. 30 (4). Archived from the original on 2007-10-21.
31. ^ Avelino, Heriberto (2006). "The typology of Pame number systems and the limits of Mesoamerica as a linguistic area" (PDF). Linguistic Typology. 10 (1): 41–60. doi:10.1515/LINGTY.2006.002.
32. ^ Marcia Ascher. "Ethnomathematics: A Multicultural View of Mathematical Ideas". The College Mathematics Journal. Retrieved 2007-04-13.
33. ^ McClean, R. J. (July 1958), "Observations on the Germanic numerals", German Life and Letters, 11 (4): 293–299, doi:10.1111/j.1468-0483.1958.tb00018.x, Some of the Germanic languages appear to show traces of an ancient blending of the decimal with the vigesimal system.
34. ^ Voyles, Joseph (October 1987), "The cardinal numerals in pre-and proto-Germanic", The Journal of English and Germanic Philology, 86 (4): 487–495, JSTOR 27709904.
35. ^ There is a surviving list of Ventureño language number words up to 32 written down by a Spanish priest ca. 1819. "Chumashan Numerals" by Madison S. Beeler, in Native American Mathematics, edited by Michael P. Closs (1986), ISBN 0-292-75531-7.
36. ^ a b Hammarström, Harald (17 May 2007). "Rarities in Numeral Systems". In Wohlgemuth, Jan; Cysouw, Michael. Rethinking Universals: How rarities affect linguistic theory (PDF). Empirical Approaches to Language Typology. 45. Berlin: Mouton de Gruyter (published 2010). Archived from the original (PDF) on 19 August 2007.
37. ^ Harris, John (1982). Hargrave, Susanne, ed. Facts and fallacies of aboriginal number systems (PDF). Work Papers of SIL-AAB Series B. 8. pp. 153–181.
38. ^ Dawson, J. "Australian Aborigines: The Languages and Customs of Several Tribes of Aborigines in the Western District of Victoria (1881), p. xcviii.
39. ^ Matsushita, Shuji (1998). Decimal vs. Duodecimal: An interaction between two systems of numeration. 2nd Meeting of the AFLANG, October 1998, Tokyo. Archived from the original on 2008-10-05. Retrieved 2011-05-29.
40. ^ Mazaudon, Martine (2002). "Les principes de construction du nombre dans les langues tibéto-birmanes". In François, Jacques. La Pluralité (PDF). Leuven: Peeters. pp. 91–119. ISBN 90-429-1295-2
41. ^ Cheetham, Brian (1978). "Counting and Number in Huli". Papua New Guinea Journal of Education. 14: 16–35. Archived from the original on 2007-09-28.
42. ^ Bowers, Nancy; Lepi, Pundia (1975). "Kaugel Valley systems of reckoning" (PDF). Journal of the Polynesian Society. 84 (3): 309–324.
43. ^ Owens, Kay (2001), "The Work of Glendon Lean on the Counting Systems of Papua New Guinea and Oceania", Mathematics Education Research Journal, 13 (1): 47–71, doi:10.1007/BF03217098