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List of numeral systems

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Summary

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There are many different numeral systems, that is, writing systems for expressing numbers.

By culture / time period

"A base is a natural number B whose powers (B multiplied by itself some number of times) are specially designated within a numerical system." The term is not equivalent to radix, as it applies to all numerical notation systems (not just positional ones with a radix) and most systems of spoken numbers. Some systems have two bases, a smaller (subbase) and a larger (base); an example is Roman numerals, which are organized by fives (V=5, L=50, D=500, the subbase) and tens (X=10, C=100, M=1,000, the base).

Name	Base	Sample	Approx. First Appearance
Proto-cuneiform numerals	1060
Indus numerals	unknown
Proto-Elamite numerals	1060
Sumerian numerals	1060
Egyptian numerals	10	Z1 V20 V1 M12 D50 I8 I7 C11
Babylonian numerals	1060	[[File:Babylonian 1.svg	15px]] [[File:Babylonian 2.svg	15px]] [[File:Babylonian 3.svg	15px]] [[File:Babylonian 4.svg	15px]] [[File:Babylonian 5.svg	15px]] [[File:Babylonian 6.svg	15px]] [[File:Babylonian 7.svg	15px]] [[File:Babylonian 8.svg	15px]] [[File:Babylonian 9.svg	15px]] [[File:Babylonian 10.svg	15px]]
Aegean numerals	10	𐄇 𐄈 𐄉 𐄊 𐄋 𐄌 𐄍 𐄎 𐄏 ( [[File:Aegean numeral 1.svg	10px	frameless	1]] [[File:Aegean numeral 2.svg	10px	frameless	2]] [[File:Aegean numeral 3.svg	10px	frameless	3]] [[File:Aegean numeral 4.svg	10px	frameless	4]] [[File:Aegean numeral 5.svg	10px	frameless	5]] [[File:Aegean numeral 6.svg	10px	frameless	6]] [[File:Aegean numeral 7.svg	10px	frameless	7]] [[File:Aegean numeral 8.svg	10px	frameless	8]] [[File:Aegean numeral 9.svg	10px	frameless	9]] )
𐄐 𐄑 𐄒 𐄓 𐄔 𐄕 𐄖 𐄗 𐄘 ( [[File:Aegean numeral 10.svg	10px	frameless	10]] [[File:Aegean numeral 20.svg	10px	frameless	20]] [[File:Aegean numeral 30.svg	10px	frameless	30]] [[File:Aegean numeral 40.svg	10px	frameless	40]] [[File:Aegean numeral 50.svg	10px	frameless	50]] [[File:Aegean numeral 60.svg	10px	frameless	60]] [[File:Aegean numeral 70.svg	10px	frameless	70]] [[File:Aegean numeral 80.svg	10px	frameless	80]] [[File:Aegean numeral 90.svg	10px	frameless	90]] )
𐄙 𐄚 𐄛 𐄜 𐄝 𐄞 𐄟 𐄠 𐄡 ( [[File:Aegean numeral 100.svg	10px	frameless	100]] [[File:Aegean numeral 200.svg	10px	frameless	200]] [[File:Aegean numeral 300.svg	10px	frameless	300]] [[File:Aegean numeral 400.svg	10px	frameless	400]] [[File:Aegean numeral 500.svg	10px	frameless	500]] [[File:Aegean numeral 600.svg	10px	frameless	600]] [[File:Aegean numeral 700.svg	10px	frameless	700]] [[File:Aegean numeral 800.svg	10px	frameless	800]] [[File:Aegean numeral 900.svg	10px	frameless	900]] )
𐄢 𐄣 𐄤 𐄥 𐄦 𐄧 𐄨 𐄩 𐄪 ( [[File:Aegean numeral 1000.svg	10px	frameless	1000]] [[File:Aegean numeral 2000.svg	10px	frameless	2000]] [[File:Aegean numeral 3000.svg	10px	frameless	3000]] [[File:Aegean numeral 4000.svg	10px	frameless	4000]] [[File:Aegean numeral 5000.svg	10px	frameless	5000]] [[File:Aegean numeral 6000.svg	10px	frameless	6000]] [[File:Aegean numeral 7000.svg	10px	frameless	7000]] [[File:Aegean numeral 8000.svg	10px	frameless	8000]] [[File:Aegean numeral 9000.svg	10px	frameless	9000]] )
𐄫 𐄬 𐄭 𐄮 𐄯 𐄰 𐄱 𐄲 𐄳 ( [[File:Aegean numeral 10000.svg	10px	frameless	10000]] [[File:Aegean numeral 20000.svg	10px	frameless	20000]] [[File:Aegean numeral 30000.svg	10px	frameless	30000]] [[File:Aegean numeral 40000.svg	10px	frameless	40000]] [[File:Aegean numeral 50000.svg	10px	frameless	50000]] [[File:Aegean numeral 60000.svg	10px	frameless	60000]] [[File:Aegean numeral 70000.svg	10px	frameless	70000]] [[File:Aegean numeral 80000.svg	10px	frameless	80000]] [[File:Aegean numeral 90000.svg	10px	frameless	90000]] )
Chinese numerals
Japanese numerals
Korean numerals (Sino-Korean)
Vietnamese numerals (Sino-Vietnamese)	10
Roman numerals	510	I V X L C D M
Hebrew numerals	10
Indian numerals	10
Greek numerals	10	ō α β γ δ ε ϝ ζ η θ ι
ο Αʹ Βʹ Γʹ Δʹ Εʹ Ϛʹ Ζʹ Ηʹ Θʹ	{{sort	-400
Kharosthi numerals	410	𐩇 𐩆 𐩅 𐩄 𐩃 𐩂 𐩁 𐩀	{{sort	-401
Phoenician numerals	10	𐤙 𐤘 𐤗 𐤛𐤛𐤛 𐤛𐤛𐤚 𐤛𐤛𐤖 𐤛𐤛 𐤛𐤚 𐤛𐤖 𐤛 𐤚 𐤖	{{sort	-250
Chinese rod numerals	10	𝍠 𝍡 𝍢 𝍣 𝍤 𝍥 𝍦 𝍧 𝍨 𝍩
Coptic numerals	10	Ⲁ Ⲃ Ⲅ Ⲇ Ⲉ Ⲋ Ⲍ Ⲏ Ⲑ
Ge'ez numerals	10	፩ ፪ ፫ ፬ ፭ ፮ ፯ ፰ ፱
፲ ፳ ፴ ፵ ፶ ፷ ፸ ፹ ፺
፻
፼
15th Century (Modern Style)
Armenian numerals	10	Ա Բ Գ Դ Ե Զ Է Ը Թ Ժ
Khmer numerals	10	០ ១ ២ ៣ ៤ ៥ ៦ ៧ ៨ ៩
Thai numerals	10	๐ ๑ ๒ ๓ ๔ ๕ ๖ ๗ ๘ ๙
Abjad numerals	10	غ ظ ض ذ خ ث ت ش ر ق ص ف ع س ن م ل ك ي ط ح ز و هـ د ج ب ا	{{sort	680
Chinese numerals (financial)	10	零壹貳參肆伍陸柒捌玖拾佰仟萬億 (T. Chinese)
零壹贰叁肆伍陆柒捌玖拾佰仟萬億 (S. Chinese)
Eastern Arabic numerals	10	٩ ٨ ٧ ٦ ٥ ٤ ٣ ٢ ١ ٠
Vietnamese numerals (Chữ Nôm)	10	𠬠𠄩𠀧𦊚𠄼𦒹𦉱𠔭𠃩	{{sort	799
Western Arabic numerals	10	0 1 2 3 4 5 6 7 8 9
Glagolitic numerals	10	Ⰰ Ⰱ Ⰲ Ⰳ Ⰴ Ⰵ Ⰶ Ⰷ Ⰸ ...
Cyrillic numerals	10	а в г д е ѕ з и ѳ і ...
Rumi numerals	10	[[File:Rumi numerals 1-9.svg	left	150px]]
Burmese numerals	10	၀ ၁ ၂ ၃ ၄ ၅ ၆ ၇ ၈ ၉
Tangut numerals	10
Cistercian numerals	10	[[File:Cistercian numerals.svg	frameless	upright]]
Maya numerals	520	[[Image:0 maia.svg	15px]] [[Image:1 maia.svg	15px]] [[Image:2 maia.svg	15px]] [[Image:3 maia.svg	15px]] [[Image:4 maia.svg	15px]] [[Image:5 maia.svg	15px]] [[Image:6 maia.svg	15px]] [[Image:7 maia.svg	15px]] [[Image:8 maia.svg	15px]] [[Image:9 maia.svg	15px]] [[Image:10 maia.svg	15px]] [[Image:11 maia.svg	15px]] [[Image:12 maia.svg	15px]] [[Image:13 maia.svg	15px]] [[Image:14 maia.svg	15px]] [[Image:15 maia.svg	15px]] [[Image:16 maia.svg	15px]] [[Image:17 maia.svg	15px]] [[Image:18 maia.svg	15px]] [[Image:19 maia.svg	15px]]
𝋠 𝋡 𝋢 𝋣 𝋤 𝋥 𝋦 𝋧 𝋨 𝋩 𝋪 𝋫 𝋬 𝋭 𝋮 𝋯 𝋰 𝋱 𝋲 𝋳	{{sort	1400
Muisca numerals	20	[[File:Muisca cyphers acc acosta humboldt zerda.svg	frameless	upright=1.5]]	{{sort	1399
Korean numerals (Hangul)	10	영 일 이 삼 사 오 육 칠 팔 구
Aztec numerals	20	[[Image:Uno Nahuatl.svg	x25px]] [[Image:Cinco Nahuatl.svg	x25px]] [[Image:Mexica 20.svg	x25px]] [[Image:Mexica 100.svg	x25px]] [[Image:Mexica 400.svg	x25px]] [[Image:Mexica 800.svg	x25px]] [[Image:Mexica 8000.svg	x30px]]
(1, 5, 20, 100, 400, 800, 8000)
Sinhala numerals	10		෦ ෧ ෨ ෩ ෪ ෫ ෬ ෭ ෮ ෯ 𑇡 𑇢 𑇣
𑇤 𑇥 𑇦 𑇧 𑇨 𑇩 𑇪 𑇫 𑇬 𑇭 𑇮 𑇯 𑇰 𑇱 𑇲 𑇳 𑇴	{{sort	1699
Pentadic runes	10	[[File:Pentimal Runes 1 through 10.svg	frameless	upright]]
Cherokee numerals	10	[[File:Cherokee Numbers – cropped (1-20).png	frameless	upright=2]]
Vai numerals	10	꘠ ꘡ ꘢ ꘣ ꘤ ꘥ ꘦ ꘧ ꘨ ꘩
Bamum numerals	10	ꛯ ꛦ ꛧ ꛨ ꛩ ꛪ ꛫ ꛬ ꛭ ꛮ
Mende Kikakui numerals	10	𞣏 𞣎 𞣍 𞣌 𞣋 𞣊 𞣉 𞣈 𞣇
Osmanya numerals	10	𐒠 𐒡 𐒢 𐒣 𐒤 𐒥 𐒦 𐒧 𐒨 𐒩
Medefaidrin numerals	20	𖺀 𖺁/𖺔 𖺂/𖺕 𖺃/𖺖 𖺄 𖺅 𖺆 𖺇 𖺈 𖺉 𖺊 𖺋 𖺌 𖺍 𖺎 𖺏 𖺐 𖺑 𖺒 𖺓
N'Ko numerals	10	߉ ߈ ߇ ߆ ߅ ߄ ߃ ߂ ߁ ߀
Hmong numerals	10
Garay numerals	10	[[File:Garay numbers.png	Garay numbers]]
Adlam numerals	10	𞥙 𞥘 𞥗 𞥖 𞥕 𞥔 𞥓 𞥒 𞥑 𞥐
Kaktovik numerals	520
𝋀 𝋁 𝋂 𝋃 𝋄 𝋅 𝋆 𝋇 𝋈 𝋉 𝋊 𝋋 𝋌 𝋍 𝋎 𝋏 𝋐 𝋑 𝋒 𝋓
Sundanese numerals	10	᮰ ᮱ ᮲ ᮳ ᮴ ᮵ ᮶ ᮷ ᮸ ᮹	20th Century (1996)

By type of notation

Positional numeral systems Numeral systems are classified here as to whether they use positional notation (also known as place-value notation), and further categorized by radix or base.

Standard positional numeral systems

LEDs]] to express binary values. In this clock, each column of LEDs shows a [[binary-coded decimal]] numeral of the traditional [[sexagesimal]] time.

The common names are derived somewhat arbitrarily from a mix of Latin and Greek, in some cases including roots from both languages within a single name. There have been some proposals for standardisation.

Base	Name	Usage
2	Binary	Digital computing, imperial and customary volume (bushel-kenning-peck-gallon-pottle-quart-pint-cup-gill-jack-fluid ounce-tablespoon)
3	Ternary, trinary	Cantor set (all points in [0,1] that can be represented in ternary with no 1s); counting Tasbih in Islam; hand-foot-yard and teaspoon-tablespoon-shot measurement systems; most economical integer base
4	Quaternary	Chumashan languages and Kharosthi numerals
5	Quinary	Gumatj, Ateso, Nunggubuyu, Kuurn Kopan Noot, and Saraveca languages; common count grouping e.g. tally marks
6	Senary, seximal	Diceware, Ndom, Kanum, and Proto-Uralic language (suspected)
7	Septimal, Septenary
8	Octal	Charles XII of Sweden, Unix-like permissions, Squawk codes, DEC PDP-11, Yuki, Pame, compact notation for binary numbers, Xiantian (I Ching, China)
9	Nonary, nonal	Compact notation for ternary
10	Decimal, denary	Most widely used by contemporary societies
11	Undecimal, unodecimal, undenary	A base-11 number system was mistakenly attributed to the Māori (New Zealand) in the 19th century and one was reported to be used by the Pangwa (Tanzania) in the 20th century, but was not confirmed by later research and is believed to also be an error. Briefly proposed during the French Revolution to settle a dispute between those proposing a shift to duodecimal and those who were content with decimal. Used as a check digit in ISBN for 10-digit ISBNs. Applications in computer science and technology.
12	Duodecimal, dozenal	Languages in the Nigerian Middle Belt Janji, Gbiri-Niragu, Piti, and the Nimbia dialect of Gwandara; Chepang language of Nepal, and the Mahl dialect of Maldivian; dozen-gross-great gross counting; 12-hour clock and months timekeeping; years of Chinese zodiac; foot and inch; Roman fractions.
13	Tredecimal, tridecimal	Conway's base 13 function.
14	Quattuordecimal, quadrodecimal	Programming for the HP 9100A/B calculator and image processing applications.
15	Quindecimal, pentadecimal	Telephony routing over IP, and the Huli language.
16	Hexadecimal, sexadecimal, sedecimal	Compact notation for binary data; tonal system of Nystrom.
17	Heptadecimal, septendecimal
18	Octodecimal
19	Undevicesimal, nonadecimal
20	Vigesimal	Basque, Celtic, Muisca, Inuit, Yoruba, Tlingit, and Dzongkha numerals; Santali, and Ainu languages.
5&20	first=Alois Richard	last=Nykl	date=September 1926	title=The Quinary-Vigesimal System of Counting in Europe, Asia, and America	pages=165–173	journal=Language	volume=2	issue=3	url=https://books.google.com/books?id=1GwUAAAAIAAJ&q=Nykl&pg=RA1-PA165	quote-page=165	quote=A student of the American Indian languages is naturally led to investigate the wide-spread use of the quinary-vigesimal system of counting which he meets in the whole territory from Alaska along the Pacific Coast to the Orinoco and the Amazon.	doi=10.2307/408742	oclc=50709582	jstor=408742	via=Google Books}}	Greenlandic, Iñupiaq, Kaktovik, Maya, Nunivak Cupʼig, and Yupʼik numerals – "wide-spread... in the whole territory from Alaska along the Pacific Coast to the Orinoco and the Amazon"
21		The smallest base in which all fractions to have periods of 4 or shorter.
23		last=Laycock	first=Donald	author-link1=Donald Laycock	date=1975	editor-last=Wurm	editor-first=Stephen	editor-link1=Stephen Wurm	series=Pacific Linguistics C-38	title=New Guinea Area Languages and Language Study, I: Papuan Languages and the New Guinea Linguistic Scene	publisher=Canberra: Research School of Pacific Studies, Australian National University	pages=219–233	chapter=Observations on Number Systems and Semantics}}
24	Quadravigesimal	24-hour clock timekeeping; Greek alphabet; Kaugel language.
25		Sometimes used as compact notation for quinary.
26	Hexavigesimal	Sometimes used for encryption or ciphering, using all letters in the English alphabet
27		Telefol, Oksapmin, Wambon, and Hewa languages. Mapping the nonzero digits to the alphabet and zero to the space is occasionally used to provide checksums for alphabetic data such as personal names, to provide a concise encoding of alphabetic strings, or as the basis for a form of gematria. Compact notation for ternary.
28		Months timekeeping.
30		The Natural Area Code, this is the smallest base such that all of to terminate, a number n is a regular number if and only if terminates in base 30.
32	Duotrigesimal	Found in the Ngiti language.
34		The smallest base where terminates and all of to have periods of 4 or shorter.
36	Hexatrigesimal
40		DEC RADIX 50/MOD40 encoding used to compactly represent file names and other symbols on Digital Equipment Corporation computers. The character set is a subset of ASCII consisting of space, upper case letters, the punctuation marks "$", ".", and "%", and the numerals.
42		Largest base for which all minimal primes are known.
47		Smallest base for which no generalized Wieferich primes are known.
49		Compact notation for septenary.
50		SQUOZE encoding used to compactly represent file names and other symbols on some IBM computers. Encoding using all Gurmukhi characters plus the Gurmukhi digits.
60	Sexagesimal	Babylonian numerals and Sumerian; degrees-minutes-seconds and hours-minutes-seconds measurement systems; Ekari; covers base 62 apart from I, O, and l, but including _(underscore).
64
72		The smallest base greater than binary such that no three-digit narcissistic number exists.
80		Used as a sub-base in Supyire.
89		Largest base for which all left-truncatable primes are known.
90		Related to Goormaghtigh conjecture for the generalized repunit numbers (111 in base 90 = 1111111111111 in base 2).
97		Smallest base which is not perfect odd power (where generalized Wagstaff numbers can be factored algebraically) for which no generalized Wagstaff primes are known.
185		Smallest base which is not a perfect power (where generalized repunits can be factored algebraically) for which no generalized repunit primes are known.
210		Smallest base such that all fractions to terminate.

[[Non-standard positional numeral systems]]

[[Bijective numeration]]

Base	Name	Usage
1	Unary(Bijectivebase1)	Tally marks, Counting. Unary numbering is used as part of some data compression algorithms such as Golomb coding. It also forms the basis for the Peano axioms for formalizing arithmetic within mathematical logic. A form of unary notation called Church encoding is used to represent numbers within lambda calculus.
10	Bijective base-10	To avoid zero
26	Bijective base-26	Spreadsheet column numeration. Also used by John Nash as part of his obsession with numerology and the uncovering of "hidden" messages.

[[Signed-digit representation]]

Base	Name	Usage
2	Balanced binary (Non-adjacent form)
3	Balanced ternary	Ternary computers
4	Balanced quaternary
5	Balanced quinary
6	Balanced senary
7	Balanced septenary
8	Balanced octal
9	Balanced nonary
10	Balanced decimal	John Colson
Augustin Cauchy
11	Balanced undecimal
12	Balanced duodecimal

[[Complex-base system|Complex bases]]

Base	Name	Usage
2i	Quater-imaginary base	related to base −4 and base 16
i\sqrt{2}	Base i\sqrt{2}	related to base −2 and base 4
i \sqrt[4]{2}	Base i \sqrt[4]{2}	related to base 2
2 \omega	Base 2 \omega	related to base 8
\omega \sqrt[3]{2}	Base \omega \sqrt[3]{2}	related to base 2
−1 ± i	Twindragon base	Twindragon fractal shape, related to base −4 and base 16
1 ± i	Negatwindragon base	related to base −4 and base 16

[[Non-integer representation|Non-integer bases]]

Base	Name	Usage
\frac{3}{2}	Base \frac{3}{2}	a rational non-integer base
\frac{4}{3}	Base \frac{4}{3}	related to duodecimal
\frac{5}{2}	Base \frac{5}{2}	related to decimal
\sqrt{2}	Base \sqrt{2}	related to base 2
\sqrt{3}	Base \sqrt{3}	related to base 3
\sqrt[3]{2}	Base \sqrt[3]{2}
\sqrt[4]{2}	Base \sqrt[4]{2}
\sqrt[12]{2}	Base \sqrt[12]{2}	usage in 12-tone equal temperament musical system
2\sqrt{2}	Base 2\sqrt{2}
-\frac{3}{2}	Base -\frac{3}{2}	a negative rational non-integer base
-\sqrt{2}	Base -\sqrt{2}	a negative non-integer base, related to base 2
\sqrt{10}	Base \sqrt{10}	related to decimal
2\sqrt{3}	Base 2\sqrt{3}	related to duodecimal
φ	Golden ratio base	early Beta encoder
ρ	Plastic number base
ψ	Supergolden ratio base
1+\sqrt{2}	Silver ratio base
e	Base e	best radix economy
π	Base \pi
e[](pi)	Base e\pi
e^\pi	Base e^\pi

[[p-adic number|''n''-adic number]]

Base	Name	Usage
2	Dyadic number
3	Triadic number
4	Tetradic number	the same as dyadic number
5	Pentadic number
6	Hexadic number	not a field
7	Heptadic number
8	Octadic number	the same as dyadic number
9	Enneadic number	the same as triadic number
10	Decadic number	not a field
11	Hendecadic number
12	Dodecadic number	not a field

[[Mixed radix]]

Factorial number system {1, 2, 3, 4, 5, 6, ...}
Even double factorial number system {2, 4, 6, 8, 10, 12, ...}
Odd double factorial number system {1, 3, 5, 7, 9, 11, ...}
Primorial number system {2, 3, 5, 7, 11, 13, ...}
Fibonorial number system {1, 2, 3, 5, 8, 13, ...}
{60, 60, 24, 7} in timekeeping
{60, 60, 24, 30 (or 31 or 28 or 29), 12, 10, 10, 10} in timekeeping
(12, 20) traditional English monetary system (£sd)
(20, 18, 13) Maya timekeeping

Other

Quote notation
Redundant binary representation
Hereditary base-n notation
Asymmetric numeral systems optimized for non-uniform probability distribution of symbols
Combinatorial number system

Non-positional notation

All known numeral systems developed before the Babylonian numerals are non-positional, as are many developed later, such as the Roman numerals. The French Cistercian monks created their own numeral system.

References

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"Ethiopic (Unicode block)". Unicode Consortium.
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Guo, Xianghe. (2009-07-27). "武则天为反贪发明汉语大写数字——中新网".
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"Vai (Unicode block)". Unicode Consortium.
"The invention, transmission and evolution of writing: Insights from the new scripts of West Africa".
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(2017-01-01). "Clear Language: Script, Register And The N'ko Movement Of Manding-Speaking West Africa". UPenn.
"Consideration of the encoding of Garay with updated user feedback (revised)". Unicode Consortium.
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"Adlam (Unicode block)". Unicode Consortium.
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(2008). "Direktori Aksara Sunda untuk Unicode". Pemerintah Provinsi Jawa Barat.
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[http://www.dozenal.org/drupal/sites_bck/default/files/MultiplicationSynopsis.pdf Multiplication Tables of Various Bases], p. 45, Michael Thomas de Vlieger, [[Dozenal Society of America]]
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''The Universal History of Numbers: From prehistory to the invention of the computer'', [[Georges Ifrah]], {{isbn. 0-471-39340-1, John Wiley and Sons Inc., New York, 2000. Translated from the French by David Bellos, E.F. Harding, Sophie Wood and Ian Monk
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