2 20 As A Decimal

Numeral system with ten every bit its base of operations

The decimal numeral system (also called the base of operations-ten positional numeral system and denary ^[ane] or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to not-integer numbers of the Hindu–Standard arabic numeral organisation.^[2] The way of denoting numbers in the decimal organisation is frequently referred to as decimal notation.^[iii]

A decimal numeral (also oftentimes just decimal or, less correctly, decimal number), refers more often than not to the notation of a number in the decimal numeral system. Decimals may sometimes be identified by a decimal separator (usually "." or "," as in 25.9703 or 3,1415).^[4] Decimal may too refer specifically to the digits after the decimal separator, such equally in "3.xiv is the approximation of π to two decimals". Zero-digits after a decimal separator serve the purpose of signifying the precision of a value.

The numbers that may be represented in the decimal system are the decimal fractions. That is, fractions of the form a/10 ^north , where a is an integer, and n is a not-negative integer.

The decimal system has been extended to infinite decimals for representing whatever existent number, past using an infinite sequence of digits after the decimal separator (see decimal representation). In this context, the decimal numerals with a finite number of non-nil digits after the decimal separator are sometimes chosen terminating decimals. A repeating decimal is an space decimal that, afterwards some place, repeats indefinitely the same sequence of digits (e.yard., 5.123144144144144... = 5.123144 ).^[5] An space decimal represents a rational number, the quotient of two integers, if and just if it is a repeating decimal or has a finite number of non-naught digits.

Origin [edit]

Ten digits on two easily, the possible origin of decimal counting

Many numeral systems of ancient civilizations use x and its powers for representing numbers, mayhap because there are x fingers on two hands and people started counting by using their fingers. Examples are firstly the Egyptian numerals, and then the Brahmi numerals, Greek numerals, Hebrew numerals, Roman numerals, and Chinese numerals. Very big numbers were difficult to stand for in these quondam numeral systems, and only the best mathematicians were able to multiply or divide large numbers. These difficulties were completely solved with the introduction of the Hindu–Arabic numeral system for representing integers. This arrangement has been extended to correspond some non-integer numbers, called decimal fractions or decimal numbers, for forming the decimal numeral system.

Decimal notation [edit]

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, iii, 4, 5, half dozen, vii, viii, 9;^[6] the decimal separator is the dot "." in many countries (mostly English-speaking),^[7] and a comma "," in other countries.^[iv]

For representing a non-negative number, a decimal numeral consists of

• either a (finite) sequence of digits (such as "2017"), where the entire sequence represents an integer,

  ${\displaystyle a_{grand}a_{thousand-1}\ldots a_{0}}$

• or a decimal mark separating two sequences of digits (such every bit "20.70828")

${\displaystyle a_{yard}a_{m-1}\ldots a_{0}.b_{1}b_{2}\ldots b_{n}}$ .

If m > 0, that is, if the first sequence contains at least ii digits, information technology is generally assumed that the first digit a _m is not zero. In some circumstances it may be useful to have one or more than 0's on the left; this does not change the value represented by the decimal: for example, 3.14 = 03.14 = 003.14. Similarly, if the final digit on the right of the decimal mark is zero—that is, if b _northward = 0—it may be removed; conversely, trailing zeros may be added after the decimal mark without irresolute the represented number; ^{[notation one]} for case, 15 = 15.0 = xv.00 and v.2 = v.20 = 5.200.

For representing a negative number, a minus sign is placed before a _m .

The numeral ${\displaystyle a_{one thousand}a_{m-i}\ldots a_{0}.b_{ane}b_{2}\ldots b_{n}}$ represents the number

${\displaystyle a_{one thousand}x^{thou}+a_{m-1}ten^{grand-ane}+\cdots +a_{0}x^{0}+{\frac {b_{ane}}{10^{1}}}+{\frac {b_{2}}{10^{2}}}+\cdots +{\frac {b_{n}}{10^{due north}}}}$ .

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

When the integral role 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 by and large avoided, because of the hazard of confusion between the decimal marking and other punctuation.

In cursory, the contribution of each digit to the value of a number depends on its position in the numeral. That is, the decimal organisation is a positional numeral organization.

Decimal fractions [edit]

Decimal fractions (sometimes called decimal numbers, specially in contexts involving explicit fractions) are the rational numbers that may exist expressed as a fraction whose denominator is a power of x.^[8] For case, the decimals ${\displaystyle 0.viii,14.89,0.00024,one.618,iii.14159}$ represent the fractions 8 / 10 , 1489 / 100 , 24 / 100000 , + 1618 / 1000 and + 314159 / 100000 , and are therefore decimal numbers.

More generally, a decimal with n digits after the separator (a bespeak or comma) represents the fraction with denominator 10 ⁿ , whose numerator is the integer obtained by removing the separator.

It follows that a number is a decimal fraction if and but if it has a finite decimal representation.

Expressed every bit a fully reduced fraction, the decimal numbers are those whose denominator is a production of a power of 2 and a power of 5. Thus the smallest denominators of decimal numbers are

${\displaystyle 1=2^{0}\cdot 5^{0},2=two^{1}\cdot 5^{0},4=2^{two}\cdot five^{0},5=2^{0}\cdot 5^{1},viii=ii^{3}\cdot 5^{0},x=2^{1}\cdot 5^{one},16=ii^{4}\cdot 5^{0},20=2^{2}\cdot 5^{1},25=2^{0}\cdot 5^{2},\ldots }$

Real number approximation [edit]

Decimal numerals do not let an verbal representation for all existent numbers, east.chiliad. for the real number π. Even so, they let approximating every real number with any desired accuracy, e.thou., the decimal 3.14159 approximates the existent π, being less than 10⁻⁵ off; so decimals are widely used in scientific discipline, engineering and everyday life.

More precisely, for every real number x and every positive integer due north, there are two decimals L and u with at most n digits afterward the decimal marker such that L ≤ ten ≤ u and (u − 50) = 10⁻ⁿ .

Numbers are very frequently obtained every bit the effect of measurement. As measurements are subject to measurement doubtfulness with a known upper bound, the issue of a measurement is well-represented past a decimal with n digits afterward the decimal marking, equally soon as the absolute measurement error is bounded from above by 10⁻ⁿ . In practice, measurement results are often given with a certain number of digits after the decimal point, which indicate the mistake bounds. For example, although 0.080 and 0.08 announce the same number, the decimal numeral 0.080 suggests a measurement with an error less than 0.001, while the numeral 0.08 indicates an absolute error bounded by 0.01. In both cases, the true value of the measured quantity could be, for example, 0.0803 or 0.0796 (encounter besides meaning figures).

Space decimal expansion [edit]

For a real number 10 and an integer due north ≥ 0, let [x] _n denote the (finite) decimal expansion of the greatest number that is not greater than x that has exactly n digits subsequently the decimal mark. Let d _i announce the concluding digit of [10] _i . It is straightforward to run into that [x] _n may be obtained past appending d _{due north} to the right of [x] _{due north−ane} . This style i has

[10] _north = [x]₀.d _ane d ₂...d _n−1 d _northward ,

and the difference of [x] _{due north−ane} and [x] _n amounts to

${\displaystyle \left\vert \left[x\correct]_{n}-\left[10\right]_{northward-1}\right\vert =d_{n}\cdot 10^{-n}<10^{-northward+1}}$ ,

which is either 0, if d _n = 0, or gets arbitrarily pocket-size as northward tends to infinity. Co-ordinate to the definition of a limit, x is the limit of [x] _n when n tends to infinity. This is written as ${\textstyle \;x=\lim _{n\rightarrow \infty }[x]_{n}\;}$ or

x = [x]₀.d _i d ₂...d _n ...,

which is called an infinite decimal expansion of x .

Conversely, for any integer [x]₀ and any sequence of digits ${\textstyle \;(d_{n})_{n=1}^{\infty }}$ the (infinite) expression [x]₀.d ₁ d ₂...d _n ... is an infinite decimal expansion of a real number x . This expansion is unique if neither all d _n are equal to 9 nor all d _n are equal to 0 for n large enough (for all n greater than some natural number N).

If all d _n for n > N equal to 9 and [x] _n = [10]₀.d _i d ₂...d _n , the limit of the sequence ${\textstyle \;([10]_{north})_{north=one}^{\infty }}$ is the decimal fraction obtained by replacing the concluding digit that is not a 9, i.e.: d _Northward , by d _N + 1, and replacing all subsequent 9s by 0s (run into 0.999...).

Any such decimal fraction, i.due east.: d _n = 0 for n > Due north , may be converted to its equivalent space decimal expansion by replacing d _N by d _N − 1 and replacing all subsequent 0s by 9s (meet 0.999...).

In summary, every existent number that is not a decimal fraction has a unique infinite decimal expansion. Each decimal fraction has exactly ii infinite decimal expansions, one containing simply 0s subsequently some place, which is obtained by the higher up definition of [x] _n , and the other containing but 9s afterwards some place, which is obtained by defining [x] _{due north} as the greatest number that is less than x, having exactly north digits after the decimal marker.

Rational numbers [edit]

Long division allows calculating the infinite decimal expansion of a rational number. If the rational number is a decimal fraction, the division stops eventually, producing a decimal numeral, which may exist prolongated into an space expansion by adding infinitely many zeros. If the rational number is non a decimal fraction, the partition may go on indefinitely. Even so, as all successive remainders are less than the divisor, there are only a finite number of possible remainders, and after some place, the same sequence of digits must be repeated indefinitely in the quotient. That is, one has a repeating decimal. For example,

ane / 81 = 0.012345679012... (with the group 012345679 indefinitely repeating).

The converse is besides true: if, at some bespeak in the decimal representation of a number, the aforementioned cord of digits starts repeating indefinitely, the number is rational.

For example, if 10 is	0.4156156156...
then 10,000ten is	4156.156156156...
and 10ten is	4.156156156...
then ten,00010 − 10x, i.e. 9,990x, is	4152.000000000...
and x is	4152 / 9990

or, dividing both numerator and denominator by half-dozen, 692 / 1665 .

Decimal computation [edit]

About modern computer hardware and software systems commonly use a binary representation internally (although many early computers, such equally the ENIAC or the IBM 650, used decimal representation internally).^[9] 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; figurer programs limited literals in decimal by default. (123.1, for instance, 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 utilise internal representations which are effectively decimal for storing decimal values and doing arithmetic. Oft this arithmetic is done on data which are encoded using some variant of binary-coded decimal,^{[10] [11]} specially in database implementations, merely at that place are other decimal representations in utilize (including decimal floating point such every bit in newer revisions of the IEEE 754 Standard for Floating-Point Arithmetic).^[12]

Decimal arithmetic is used in computers so that decimal fractional results of adding (or subtracting) values with a stock-still length of their fractional part always are computed to this same length of precision. This is specially of import for financial calculations, eastward.yard., requiring in their results integer multiples of the smallest currency unit for volume keeping purposes. This is not possible in binary, because the negative powers of ${\displaystyle 10}$ have no finite binary fractional representation; and is generally impossible for multiplication (or segmentation).^{[13] [xiv]} See Capricious-precision arithmetics for exact calculations.

History [edit]

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

Many aboriginal cultures calculated with numerals based on ten, sometimes argued due to human hands typically having 10 fingers/digits.^[15] Standardized weights used in the Indus Valley civilization (c.  3300–1300 BCE) were based on the ratios: ane/20, 1/10, 1/v, ane/ii, 1, 2, v, ten, 20, 50, 100, 200, and 500, while their standardized ruler – the Mohenjo-daro ruler – was divided into ten equal parts.^{[16] [17] [xviii]} Egyptian hieroglyphs, in evidence since around 3000 BCE, used a purely decimal system,^[nineteen] equally did the Cretan hieroglyphs (c.  1625−1500 BCE) of the Minoans whose numerals are closely based on the Egyptian model.^{[20] [21]} The decimal arrangement was handed downwardly to the consecutive Statuary Age cultures of Greece, including Linear A (c. 18th century BCE−1450 BCE) and Linear B (c. 1375−1200 BCE) – the number system of classical Greece also used powers of ten, including, Roman numerals, an intermediate base of 5.^[22] Notably, the polymath Archimedes (c. 287–212 BCE) invented a decimal positional arrangement in his Sand Calculator which was based on x^{8 [22]} and later led the German mathematician Carl Friedrich Gauss to complaining what heights science would have already reached in his days if Archimedes had fully realized the potential of his ingenious discovery.^[23] Hittite hieroglyphs (since 15th century BCE) were too strictly decimal.^[24]

Some non-mathematical ancient texts such as the Vedas, dating back to 1700–900 BCE make use of decimals and mathematical decimal fractions.^[25]

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 not-positional decimal systems, and required big numbers of symbols. For instance, Egyptian numerals used different symbols for 10, twenty to 90, 100, 200 to 900, 1000, 2000, 3000, 4000, to x,000.^[26] The globe's earliest positional decimal system was the Chinese rod calculus.^[27]

The world'south primeval positional decimal system
Upper row vertical class
Lower row horizontal class

History of decimal fractions [edit]

counting rod decimal fraction 1/vii

Decimal fractions were first developed and used by the Chinese in the end of quaternary century BCE,^[28] and and so spread to the Middle Due east and from there to Europe.^{[27] [29]} The written Chinese decimal fractions were non-positional.^[29] Yet, counting rod fractions were positional.^[27]

Qin Jiushao in his book Mathematical Treatise in Nine Sections (1247^[30]) denoted 0.96644 past

寸

, meaning

寸

096644

J. Lennart Berggren notes that positional decimal fractions appear for the first fourth dimension in a book by the Arab mathematician Abu'50-Hasan al-Uqlidisi written in the 10th century.^[31] The Jewish mathematician Immanuel Bonfils used decimal fractions around 1350, anticipating Simon Stevin, but did not develop any annotation to represent them.^[32] The Persian mathematician Jamshīd al-Kāshī claimed to accept discovered decimal fractions himself in the 15th century.^[31] Al Khwarizmi introduced fraction to Islamic countries in the early on 9th century; a Chinese author has alleged that his fraction presentation was an exact copy of traditional Chinese mathematical fraction from Sunzi Suanjing.^[27] This form of fraction with numerator on peak and denominator at bottom without a horizontal bar was as well used by al-Uqlidisi and by al-Kāshī in his piece of work "Arithmetic Key".^{[27] [33]}

A forerunner of modern European decimal annotation was introduced by Simon Stevin in the 16th century.^[34]

John Napier introduced using the menses (.) to separate the integer part of a decimal number from the partial part in his book on constructing tables of logarithms, published posthumously in 1620.^{[35] : p. 8, annal p. 32)}

Natural languages [edit]

A method of expressing every possible natural number using a set of x symbols emerged in India. Several Indian languages show a straightforward decimal system. Many Indo-Aryan and Dravidian languages have numbers between 10 and xx expressed in a regular pattern of addition to 10.^[36]

The Hungarian language also uses a straightforward decimal system. All numbers between x and xx are formed regularly (e.one thousand. xi 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 give-and-take for each gild (10 十 , 100 百 , 1000 千 , ten,000 万 ), and in which eleven is expressed as x-one and 23 equally two-ten-3, and 89,345 is expressed as 8 (x thousands) 万 9 (thousand) 千 3 (hundred) 百 4 (tens) 十 v is institute in Chinese, and in Vietnamese with a few irregularities. Japanese, Korean, and Thai accept imported the Chinese decimal organization. Many other languages with a decimal system have special words for the numbers between 10 and xx, and decades. For example, in English xi is "11" not "ten-ane" or "i-teen".

Incan languages such every bit Quechua and Aymara accept an almost straightforward decimal system, in which 11 is expressed as 10 with one and 23 as ii-x with three.

Some psychologists suggest irregularities of the English names of numerals may hinder children's counting power.^[37]

Other bases [edit]

Some cultures practice, or did, utilise other bases of numbers.

• Pre-Columbian Mesoamerican cultures such equally the Maya used a base-20 system (perhaps based on using all twenty fingers and toes).
• The Yuki linguistic communication in California and the Pamean languages^[38] in Mexico have octal (base of operations-8) systems because the speakers count using the spaces between their fingers rather than the fingers themselves.^[39]
• The existence of a non-decimal base in the earliest traces of the Germanic languages is attested past the presence of words and glosses pregnant that the count is in decimal (cognates to "10-count" or "tenty-wise"); such would exist expected if normal counting is not decimal, and unusual if information technology were.^{[40] [41]} Where this counting arrangement is known, it is based on the "long hundred" = 120, and a "long g" of 1200. The descriptions like "long" only appear later on the "small hundred" of 100 appeared with the Christians. Gordon's Introduction to Old Norse p. 293, gives number names that belong to this arrangement. An expression cognate to 'one hundred and eighty' translates to 200, and the cognate to 'ii hundred' translates to 240. Goodare details the use of the long hundred in Scotland in the Heart Ages, giving examples such every bit calculations where the carry implies i C (i.e. one hundred) equally 120, etc. That the general population were not alarmed to come across such numbers suggests common plenty employ. It is besides possible to avoid hundred-similar numbers past 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 likewise a paper by W.H. Stevenson, on 'Long Hundred and its uses in England'.^{[42] [43]}
• 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.^[44]
• Many languages^[45] utilise quinary (base-v) number systems, including Gumatj, Nunggubuyu,^[46] Kuurn Kopan Noot^[47] and Saraveca. Of these, Gumatj is the only true five–25 linguistic communication known, in which 25 is the higher group of v.
• Some Nigerians utilise duodecimal systems.^[48] Then did some small communities in India and Nepal, as indicated by their languages.^[49]
• The Huli language of Papua New Guinea is reported to accept base of operations-15 numbers.^[fifty] Ngui means 15, ngui ki means 15 × 2 = thirty, and ngui ngui means 15 × 15 = 225.
• Umbu-Ungu, also known as Kakoli, is reported to take base-24 numbers.^[51] Tokapu ways 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 of operations-4 cycles.^[45]
• The Ndom language of Papua New Republic of guinea is reported to have base-6 numerals.^[52] Mer ways 6, mer an thef means half-dozen × 2 = 12, nif ways 36, and nif thef means 36×2 = 72.

Run across besides [edit]

• Algorism
• Binary-coded decimal (BCD)
• Decimal nomenclature
• Decimal reckoner
• Decimal time
• Decimal representation
• Decimal section numbering
• Decimal separator
• Decimalisation
• Densely packed decimal (DPD)
• Duodecimal
• Octal
• Scientific notation
• Serial decimal
• SI prefix

Notes [edit]

1. ^ Sometimes, the extra zeros are used for indicating the accuracy of a measurement. For example, "fifteen.00 m" may indicate that the measurement error is less than i centimetre (0.01 k), while "15 grand" may hateful that the length is roughly fifteen metres and that the error may exceed x centimetres.

References [edit]

1. ^ "denary". Oxford English language Dictionary (Online ed.). Oxford University Press. (Subscription or participating institution membership required.)
2. ^ Cajori, Florian (Feb 1926). "The History of Arithmetic. Louis Charles Karpinski". Isis. University of Chicago Press. 8 (one): 231–232. doi:x.1086/358384. ISSN 0021-1753.
3. ^ Yong, Lam Lay; Se, Ang Tian (April 2004). Fleeting Footsteps. World Scientific. 268. doi:10.1142/5425. ISBN978-981-238-696-0 . Retrieved March 17, 2022.
4. ^ ^{a b} Weisstein, Eric West. (March 10, 2022). "Decimal Point". Wolfram MathWorld . Retrieved March 17, 2022. {{cite spider web}}: CS1 maint: url-condition (link)
5. ^ The vinculum (overline) in 5.123144 indicates that the '144' sequence repeats indefinitely, i.e. 5.123144 144 144 144....
6. ^ In some countries, such as Arab speaking ones, other glyphs are used for the digits
7. ^ Weisstein, Eric West. "Decimal". mathworld.wolfram.com . Retrieved 2020-08-22 .
8. ^ "Decimal Fraction". Encyclopedia of Mathematics . Retrieved 2013-06-18 .
9. ^ "Fingers or Fists? (The Option of Decimal or Binary Representation)", Werner Buchholz, Communications of the ACM, Vol. two #12, pp. three–11, ACM Press, December 1959.
10. ^ Schmid, Hermann (1983) [1974]. Decimal Computation (1 (reprint) ed.). Malabar, Florida: Robert E. Krieger Publishing Company. ISBN0-89874-318-4.
11. ^ Schmid, Hermann (1974). Decimal Computation (1st ed.). Binghamton, New York: John Wiley & Sons. ISBN0-471-76180-X.
12. ^ Decimal Floating-Point: Algorism for Computers, Cowlishaw, Mike F., Proceedings 16th IEEE Symposium on Computer Arithmetic, ISBN 0-7695-1894-X, pp. 104–eleven, IEEE Comp. Soc., 2003
13. ^ Decimal Arithmetic – FAQ
14. ^ Decimal Floating-Betoken: Algorism for Computers, Cowlishaw, One thousand. F., Proceedings 16th IEEE Symposium on Computer Arithmetic (ARITH 16), ISBN 0-7695-1894-X, pp. 104–11, IEEE Comp. Soc., June 2003
15. ^ Dantzig, Tobias (1954), Number / The Language of Science (4th ed.), The Free Press (Macmillan Publishing Co.), p. 12, ISBN0-02-906990-four
16. ^ Sergent, Bernard (1997), Genèse de 50'Inde (in French), Paris: Payot, p. 113, ISBN two-228-89116-9
17. ^ Coppa, A.; et al. (2006). "Early Neolithic tradition of dentistry: Flintstone tips were surprisingly constructive for drilling tooth enamel in a prehistoric population". Nature. 440 (7085): 755–56. Bibcode:2006Natur.440..755C. doi:x.1038/440755a. PMID 16598247. S2CID 6787162.
18. ^ Bisht, R. S. (1982), "Excavations at Banawali: 1974–77", in Possehl, Gregory Fifty. (ed.), Harappan Civilisation: A Contemporary Perspective, New Delhi: Oxford and IBH Publishing Co., pp. 113–24
19. ^ Georges Ifrah: From I to Zero. A Universal History of Numbers, Penguin Books, 1988, ISBN 0-14-009919-0, pp. 200–13 (Egyptian Numerals)
20. ^ Graham Flegg: Numbers: their history and meaning, Courier Dover Publications, 2002, ISBN 978-0-486-42165-0, p. 50
21. ^ Georges Ifrah: From One to Zero. A Universal History of Numbers, Penguin Books, 1988, ISBN 0-fourteen-009919-0, pp. 213–18 (Cretan numerals)
22. ^ ^{a b} "Greek numbers". Retrieved 2019-07-21 .
23. ^ Menninger, Karl: Zahlwort und Ziffer. Eine Kulturgeschichte der Zahl, Vandenhoeck und Ruprecht, 3rd. ed., 1979, ISBN three-525-40725-4, pp. 150–53
24. ^ Georges Ifrah: From One to Zero. A Universal History of Numbers, Penguin Books, 1988, ISBN 0-14-009919-0, pp. 218f. (The Hittite hieroglyphic arrangement)
25. ^ (Atharva Veda v.15, i–11)
26. ^ Lam Lay Yong et al. The Fleeting Footsteps pp. 137–39
27. ^ ^{a b c d east} Lam Lay Yong, "The Development of Hindu–Standard arabic and Traditional Chinese Arithmetic", Chinese Science, 1996 p. 38, Kurt Vogel notation
28. ^ "Ancient bamboo slips for calculation enter world records volume". The Institute of Archaeology, Chinese Academy of Social Sciences . Retrieved 10 May 2017.
29. ^ ^{a b} Joseph Needham (1959). "Decimal Organization". Science and Culture in Cathay, Volume III, Mathematics and the Sciences of the Heavens and the Earth. Cambridge Academy Press.
30. ^ Jean-Claude Martzloff, A History of Chinese Mathematics, Springer 1997 ISBN 3-540-33782-2
31. ^ ^{a b} Berggren, J. Lennart (2007). "Mathematics in Medieval Islam". In Katz, Victor J. (ed.). The Mathematics of Egypt, Mesopotamia, Prc, India, and Islam: A Sourcebook. Princeton University Printing. p. 530. ISBN978-0-691-11485-ix.
32. ^ Gandz, Due south.: The invention of the decimal fractions and the application of the exponential calculus by Immanuel Bonfils of Tarascon (c. 1350), Isis 25 (1936), sixteen–45.
33. ^ Lay Yong, Lam. "A Chinese Genesis, Rewriting the history of our numeral system". Archive for History of Verbal Sciences. 38: 101–08.
34. ^ B. L. van der Waerden (1985). A History of Algebra. From Khwarizmi to Emmy Noether. Berlin: Springer-Verlag.
35. ^ Napier, John (1889) [1620]. The Structure of the Wonderful Canon of Logarithms. Translated by Macdonald, William Rae. Edinburgh: Blackwood & Sons – via Net Archive. In numbers distinguished thus by a flow in their midst, whatever is written afterwards the menstruum is a fraction, the denominator of which is unity with as many cyphers later on it as in that location are figures after the period.
36. ^ "Indian numerals". Aboriginal Indian mathematics. Archived from the original on 2007-09-29. Retrieved 2015-05-22 .
37. ^ Azar, Beth (1999). "English words may hinder math skills development". American Psychological Association Monitor. thirty (4). Archived from the original on 2007-10-21.
38. ^ Avelino, Heriberto (2006). "The typology of Pame number systems and the limits of Mesoamerica as a linguistic expanse" (PDF). Linguistic Typology. 10 (ane): 41–sixty. doi:x.1515/LINGTY.2006.002. S2CID 20412558.
39. ^ Marcia Ascher. "Ethnomathematics: A Multicultural View of Mathematical Ideas". The College Mathematics Journal. JSTOR 2686959.
40. ^ McClean, R. J. (July 1958), "Observations on the Germanic numerals", High german Life and Letters, 11 (four): 293–99, doi:x.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 .
41. ^ Voyles, Joseph (October 1987), "The fundamental numerals in pre-and proto-Germanic", The Journal of English language and Germanic Philology, 86 (4): 487–95, JSTOR 27709904 .
42. ^ Stevenson, W.H. (1890). "The Long Hundred and its uses in England". Archaeological Review. Dec 1889: 313–22.
43. ^ Poole, Reginald Lane (2006). The Exchequer in the twelfth century : the Ford lectures delivered in the University of Oxford in Michaelmas term, 1911. Clark, NJ: Lawbook Commutation. ISBN1-58477-658-7. OCLC 76960942.
44. ^ At that place is a surviving list of Ventureño language number words upwardly to 32 written down past a Spanish priest ca. 1819. "Chumashan Numerals" past Madison Due south. Beeler, in Native American Mathematics, edited past Michael P. Closs (1986), ISBN 0-292-75531-vii.
45. ^ ^{a b} Hammarström, Harald (17 May 2007). "Rarities in Numeral Systems". In Wohlgemuth, January; Cysouw, Michael (eds.). Rethinking Universals: How rarities affect linguistic theory (PDF). Empirical Approaches to Linguistic communication Typology. Vol. 45. Berlin: Mouton de Gruyter (published 2010). Archived from the original (PDF) on 19 August 2007.
46. ^ Harris, John (1982). Hargrave, Susanne (ed.). "Facts and fallacies of ancient number systems" (PDF). Work Papers of SIL-AAB Series B. viii: 153–81. Archived from the original (PDF) on 2007-08-31.
47. ^ Dawson, J. "Australian Aborigines: The Languages and Customs of Several Tribes of Aborigines in the Western District of Victoria (1881), p. xcviii.
48. ^ Matsushita, Shuji (1998). Decimal vs. Duodecimal: An interaction between two systems of numeration. 2nd Meeting of the AFLANG, Oct 1998, Tokyo. Archived from the original on 2008-ten-05. Retrieved 2011-05-29 .
49. ^ Mazaudon, Martine (2002). "Les principes de construction du nombre dans les langues tibéto-birmanes". In François, Jacques (ed.). La Pluralité (PDF). Leuven: Peeters. pp. 91–119. ISBNxc-429-1295-2.
50. ^ Cheetham, Brian (1978). "Counting and Number in Huli". Papua New Republic of guinea Periodical of Pedagogy. 14: 16–35. Archived from the original on 2007-09-28.
51. ^ Bowers, Nancy; Lepi, Pundia (1975). "Kaugel Valley systems of reckoning" (PDF). Journal of the Polynesian Society. 84 (iii): 309–24. Archived from the original (PDF) on 2011-06-04.
52. ^ Owens, Kay (2001), "The Work of Glendon Lean on the Counting Systems of Papua New Republic of guinea and Oceania", Mathematics Education Enquiry Journal, xiii (1): 47–71, Bibcode:2001MEdRJ..thirteen...47O, doi:10.1007/BF03217098, S2CID 161535519, archived from the original on 2015-09-26

2 20 As A Decimal,

Source: https://en.wikipedia.org/wiki/Decimal

Posted by: lynnantither.blogspot.com

Share this post