List of numeral systems
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Numeral systems 

List of numeral systems 
There are many different numeral systems, that is, writing systems for expressing numbers.
By culture / time period[edit]
Name  Base  Sample  Approx. First Appearance  

Protocuneiform numerals  10+60  c. 3500–2000 BCE  
Indus numerals  c. 3500–1900 BCE  
ProtoElamite numerals  10+60  3,100 BCE  
Sumerian numerals  10+60  3,100 BCE  
Egyptian numerals  10 

3,000 BCE  
Babylonian numerals  10+60  2,000 BCE  
Aegean numerals  10  𐄇 𐄈 𐄉 𐄊 𐄋 𐄌 𐄍 𐄎 𐄏 ( ) 𐄐 𐄑 𐄒 𐄓 𐄔 𐄕 𐄖 𐄗 𐄘 ( ) 𐄙 𐄚 𐄛 𐄜 𐄝 𐄞 𐄟 𐄠 𐄡 ( ) 𐄢 𐄣 𐄤 𐄥 𐄦 𐄧 𐄨 𐄩 𐄪 ( ) 𐄫 𐄬 𐄭 𐄮 𐄯 𐄰 𐄱 𐄲 𐄳 ( ) 
1,500 BCE  
Chinese numerals Japanese numerals Korean numerals (SinoKorean) Vietnamese numerals (SinoVietnamese) 
10 
零一二三四五六七八九十百千萬億 (Default, Traditional Chinese) 
1,300 BCE  
Roman numerals  I V X L C D M  1,000 BCE  
Hebrew numerals  10  א ב ג ד ה ו ז ח ט י כ ל מ נ ס ע פ צ ק ר ש ת ך ם ן ף ץ 
800 BCE  
Indian numerals  10  Tamil ௦ ௧ ௨ ௩ ௪ ௫ ௬ ௭ ௮ ௯ Malayalam ൦ ൧ ൨ ൩ ൪ ൫ ൬ ൭ ൮ ൯ Kannada ೦ ೧ ೨ ೩ ೪ ೫ ೬ ೭ ೮ ೯ Telugu ౦ ౧ ౨ ౩ ౪ ౫ ౬ ౭ ౮ ౯ Odia ୦ ୧ ୨ ୩ ୪ ୫ ୬ ୭ ୮ ୯ Bengali ০ ১ ২ ৩ ৪ ৫ ৬ ৭ ৮ ৯ Devanagari ० १ २ ३ ४ ५ ६ ७ ८ ९ Punjabi ੦ ੧ ੨ ੩ ੪ ੫ ੬ ੭ ੮ ੯ Gujarati ૦ ૧ ૨ ૩ ૪ ૫ ૬ ૭ ૮ ૯ Hindustani ۰ ۱ ۲ ۳ ۴ ۵ ۶ ۷ ۸ ۹ 
750–500 BCE  
Greek numerals  10  ō α β γ δ ε ϝ ζ η θ ι ο Αʹ Βʹ Γʹ Δʹ Εʹ Ϛʹ Ζʹ Ηʹ Θʹ 
<400 BCE  
Phoenician numerals  10  𐤙 𐤘 𐤗 𐤛𐤛𐤛 𐤛𐤛𐤚 𐤛𐤛𐤖 𐤛𐤛 𐤛𐤚 𐤛𐤖 𐤛 𐤚 𐤖 ^{[1]}  <250 BCE^{[2]}  
Chinese rod numerals  10  𝍠 𝍡 𝍢 𝍣 𝍤 𝍥 𝍦 𝍧 𝍨 𝍩  1st Century  
Coptic numerals  10  Ⲁ Ⲃ Ⲅ Ⲇ Ⲉ Ⲋ Ⲍ Ⲏ Ⲑ  2nd Century  
Ge'ez numerals  10  ፩ ፪ ፫ ፬ ፭ ፮ ፯ ፰ ፱ ፲ ፳ ፴ ፵ ፶ ፷ ፸ ፹ ፺ ፻ 
3rd–4th Century 15th Century (Modern Style)^{[3]}  
Armenian numerals  10  Ա Բ Գ Դ Ե Զ Է Ը Թ Ժ  Early 5th Century  
Khmer numerals  10  ០ ១ ២ ៣ ៤ ៥ ៦ ៧ ៨ ៩  Early 7th Century  
Thai numerals  10  ๐ ๑ ๒ ๓ ๔ ๕ ๖ ๗ ๘ ๙  7th Century^{[4]}  
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  <8th Century  
Eastern Arabic numerals  10  ٩ ٨ ٧ ٦ ٥ ٤ ٣ ٢ ١ ٠  8th Century  
Vietnamese numerals (Chữ Nôm)  10  𠬠 𠄩 𠀧 𦊚 𠄼 𦒹 𦉱 𠔭 𠃩  <9th Century  
Western Arabic numerals  10  0 1 2 3 4 5 6 7 8 9  9th Century  
Glagolitic numerals  10  Ⰰ Ⰱ Ⰲ Ⰳ Ⰴ Ⰵ Ⰶ Ⰷ Ⰸ ...  9th Century  
Cyrillic numerals  10  а в г д е ѕ з и ѳ і ...  10th Century  
Rumi numerals  10  10th Century  
Burmese numerals  10  ၀ ၁ ၂ ၃ ၄ ၅ ၆ ၇ ၈ ၉  11th Century^{[5]}  
Tangut numerals  10  𘈩 𗍫 𘕕 𗥃 𗏁 𗤁 𗒹 𘉋 𗢭 𗰗  11th Century (1036)  
Cistercian numerals  10  13th Century  
Maya numerals  5+20  <15th Century  
Muisca numerals  20  <15th Century  
Korean numerals (Hangul)  10  영 일 이 삼 사 오 육 칠 팔 구  15th Century (1443)  
Aztec numerals  20  16th Century  
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 
<18th Century  
Pentadic runes  10  19th Century  
Cherokee numerals  10  19th Century (1820s)  
Osmanya numerals  10  𐒠 𐒡 𐒢 𐒣 𐒤 𐒥 𐒦 𐒧 𐒨 𐒩  20th Century (1920s)  
Kaktovik numerals  5+20  20th Century (1994) 
By type of notation[edit]
Numeral systems are classified here as to whether they use positional notation (also known as placevalue notation), and further categorized by radix or base.
Standard positional numeral systems[edit]
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.^{[6]} There have been some proposals for standardisation.^{[7]}
Base  Name  Usage 

2  Binary  Digital computing, imperial and customary volume (bushelkenningpeckgallonpottlequartpintcupgilljackfluid ouncetablespoon) 
3  Ternary  Cantor set (all points in [0,1] that can be represented in ternary with no 1s); counting Tasbih in Islam; handfootyard and teaspoontablespoonshot 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  Diceware, Ndom, Kanum, and ProtoUralic language (suspected) 
7  Septimal  Weeks timekeeping, Western music letter notation 
8  Octal  Charles XII of Sweden, Unixlike permissions, Squawk codes, DEC PDP11, Yuki, Pame, compact notation for binary numbers, Xiantian (I Ching, China) 
9  Nonary, nonal  Base 9 encoding; compact notation for ternary 
10  Decimal (also known as denary)  Most widely used by modern civilizations^{[8]}^{[9]}^{[10]} 
11  Undecimal, unodecimal^{[11]}^{[12]}^{[13]}  A base11 number system was attributed to the Māori (New Zealand) in the 19th century^{[14]} and the Pangwa (Tanzania) in the 20th century.^{[15]} 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 10digit ISBNs. 
12  Duodecimal  Languages in the Nigerian Middle Belt Janji, GbiriNiragu, Piti, and the Nimbia dialect of Gwandara; Chepang language of Nepal, and the Mahl dialect of Maldivian; dozengrossgreat gross counting; 12hour clock and months timekeeping; years of Chinese zodiac; foot and inch; Roman fractions; penny and shilling 
13  Tredecimal, tridecimal^{[16]}^{[17]}  Base 13 encoding; Conway base 13 function. 
14  Quattuordecimal, quadrodecimal^{[16]}^{[17]}  Programming for the HP 9100A/B calculator^{[18]} and image processing applications;^{[19]} pound and stone. 
15  Quindecimal, pentadecimal^{[20]}^{[17]}  Telephony routing over IP, and the Huli language. 
16  Hexadecimal
(also known as sexadecimal and sedecimal) 
Base 16 encoding; compact notation for binary data; tonal system; ounce and pound. 
17  Septendecimal, heptadecimal^{[20]}^{[17]}  Base 17 encoding. 
18  Octodecimal^{[20]}^{[17]}  Base 18 encoding; a base such that 7^{n} is palindromic for n = 3, 4, 6, 9. 
19  Undevicesimal, nonadecimal^{[20]}^{[17]}  Base 19 encoding. 
20  Vigesimal  Basque, Celtic, Muisca, Inuit, Yoruba, Tlingit, and Dzongkha numerals; Santali, and Ainu languages; shilling and pound 
5+20  Quinaryvigesimal^{[21]}^{[22]}^{[23]}  Greenlandic, Iñupiaq, Kaktovik, Maya, Nunivak Cupʼig, and Yupʼik numerals – "widespread... in the whole territory from Alaska along the Pacific Coast to the Orinoco and the Amazon"^{[21]} 
21  Base 21 encoding; also the smallest base where all of 1/2 to 1/18 have periods of 4 or shorter.  
22  Base 22 encoding.  
23  Kalam language,^{[24]} Kobon language^{[citation needed]}  
24  Quadravigesimal^{[25]}  24hour clock timekeeping; Greek alphabet; Kaugel language. 
25  Base 25 encoding; sometimes used as compact notation for quinary.  
26  Hexavigesimal^{[25]}^{[26]}  Base 26 encoding; sometimes used for encryption or ciphering,^{[27]} using all letters in the English alphabet 
27  Septemvigesimal  Telefol^{[28]} and Oksapmin^{[29]} 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,^{[30]} to provide a concise encoding of alphabetic strings,^{[31]} or as the basis for a form of gematria.^{[32]} Compact notation for ternary. 
28  Base 28 encoding; months timekeeping.  
29  Base 29 encoding.  
30  Trigesimal  The Natural Area Code, this is the smallest base such that all of 1/2 to 1/6 terminate, a number n is a regular number if and only if 1/n terminates in base 30. 
31  Base 31 encoding.  
32  Duotrigesimal  Base 32 encoding; the Ngiti language. 
33  Use of letters (except I, O, Q) with digits in vehicle registration plates of Hong Kong.  
34  Using all numbers and all letters except I and O; the smallest base where 1/2 terminates and all of 1/2 to 1/18 have periods of 4 or shorter.  
35  Using all numbers and all letters except O.  
36  Hexatrigesimal^{[33]}^{[34]}  Base 36 encoding; use of letters A–Z with digits 0–9. 
37  Base 37 encoding; using all numbers and all letters of the Spanish alphabet.  
38  Base 38 encoding; use all duodecimal digits and all letters.  
39  Base 39 encoding.  
40  Quadragesimal  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  Base 42 encoding; largest base for which all minimal primes are known.  
45  Base 45 encoding.  
47  Smallest base for which no generalized Wieferich primes are known.  
48  Base 48 encoding.  
49  Compact notation for septenary.  
50  Quinquagesimal  Base 50 encoding; SQUOZE encoding used to compactly represent file names and other symbols on some IBM computers. Encoding using all Gurmukhi characters plus the Gurmukhi digits. 
52  Base 52 encoding, a variant of base 62 without vowels except Y and y^{[35]} or a variant of base 26 using all lower and upper case letters.  
54  Base 54 encoding.  
56  Base 56 encoding, a variant of base 58.^{[36]}  
57  Base 57 encoding, a variant of base 62 excluding I, O, l, U, and u^{[37]} or I, 1, l, 0, and O.^{[38]}  
58  Base 58 encoding, a variant of base 62 excluding 0 (zero), I (capital i), O (capital o) and l (lower case L).^{[39]}  
60  Sexagesimal  Babylonian numerals; New base 60 encoding, similar to base 62, excluding I, O, and l, but including _(underscore);^{[40]} degreesminutesseconds and hoursminutesseconds measurement systems; Ekari and Sumerian 
62  Base 62 encoding, using 0–9, A–Z, and a–z.  
64  Tetrasexagesimal  Base 64 encoding; I Ching in China. This system is conveniently coded into ASCII by using the 26 letters of the Latin alphabet in both upper and lower case (52 total) plus 10 numerals (62 total) and then adding two special characters (+ and /). 
72  Base72 encoding; the smallest base >2 such that no threedigit narcissistic number exists.  
80  Octogesimal  Base80 encoding; Supyire as a subbase. 
81  Base 81 encoding, using as 81=3^{4} is related to ternary.  
85  Ascii85 encoding. This is the minimum number of characters needed to encode a 32 bit number into 5 printable characters in a process similar to MIME64 encoding, since 85^{5} is only slightly bigger than 2^{32}. Such method is 6.7% more efficient than MIME64 which encodes a 24 bit number into 4 printable characters.  
89  Largest base for which all lefttruncatable primes are known.  
90  Nonagesimal  Related to Goormaghtigh conjecture for the generalized repunit numbers (111 in base 90 = 1111111111111 in base 2). 
91  Base 91 encoding, using all ASCII except "" (0x2D), "\" (0x5C), and "'" (0x27); one variant uses "\" (0x5C) in place of """ (0x22).  
92  Base 92 encoding, using all of ASCII except for "`" (0x60) and """ (0x22) due to confusability.^{[41]}  
93  Base 93 encoding, using all of ASCII printable characters except for "," (0x27) and "" (0x3D) as well as the Space character. "," is reserved for delimiter and "" is reserved for negation.^{[42]}  
94  Base 94 encoding, using all of ASCII printable characters.^{[43]}  
95  Base 95 encoding, a variant of base 94 with the addition of the Space character.^{[44]}  
96  Base 96 encoding, using all of ASCII printable characters as well as the two extra duodecimal digits.  
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.  
100  Centesimal  As 100=10^{2}, these are two decimal digits. 
120  Base 120 encoding.  
121  Related to base 11.  
125  Related to base 5.  
128  Using as 128=2^{7}.  
144  Two duodecimal digits.  
169  Two Tridecimal digits.  
185  Smallest base which is not perfect power (where generalized repunits can be factored algebraically) for which no generalized repunit primes are known.  
196  Two tetradecimal digits.  
200  Base 200 encoding.  
210  Smallest base such that all of 1/2 to 1/10 terminate.  
216  related to base 6.  
225  Two pentadecimal digits.  
256  Base 256 encoding, as 256=2^{8}.  
300  Base 300 encoding.  
360  Degrees for angle. 
Nonstandard positional numeral systems[edit]
Bijective numeration[edit]
Base  Name  Usage 

1  Unary (Bijective base‑1)  Tally marks, Counting 
10  Bijective base10  To avoid zero 
26  Bijective base26  Spreadsheet column numeration. Also used by John Nash as part of his obsession with numerology and the uncovering of "hidden" messages.^{[45]} 
Signeddigit representation[edit]
Base  Name  Usage 

2  Balanced binary (Nonadjacent 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 
Negative bases[edit]
The common names of the negative base numeral systems are formed using the prefix nega, giving names such as:^{[citation needed]}
Base  Name  Usage 

−2  Negabinary  
−3  Negaternary  
−4  Negaquaternary  
−5  Negaquinary  
−6  Negasenary  
−8  Negaoctal  
−10  Negadecimal  
−12  Negaduodecimal  
−16  Negahexadecimal 
Complex bases[edit]
Base  Name  Usage 

2i  Quaterimaginary base  related to base −4 and base 16 
Base  related to base −2 and base 4  
Base  related to base 2  
Base  related to base 8  
Base  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 
Noninteger bases[edit]
Base  Name  Usage 

Base  a rational noninteger base  
Base  related to duodecimal  
Base  related to decimal  
Base  related to base 2  
Base  related to base 3  
Base  
Base  
Base  usage in 12tone equal temperament musical system  
Base  
Base  a negative rational noninteger base  
Base  a negative noninteger base, related to base 2  
Base  related to decimal  
Base  related to duodecimal  
φ  Golden ratio base  Early Beta encoder^{[46]} 
ρ  Plastic number base  
ψ  Supergolden ratio base  
Silver ratio base  
e  Base  Lowest radix economy 
π  Base  
eπ  Base  
Base 
nadic number[edit]
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[edit]
 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[edit]
 Quote notation
 Redundant binary representation
 Hereditary basen notation
 Asymmetric numeral systems optimized for nonuniform probability distribution of symbols
 Combinatorial number system
Nonpositional notation[edit]
All known numeral systems developed before the Babylonian numerals are nonpositional,^{[47]} as are many developed later, such as the Roman numerals. The French Cistercian monks created their own numeral system.
See also[edit]
 History of ancient numeral systems – Symbols representing numbers
 History of the Hindu–Arabic numeral system
 List of numeral system topics
 Numeral prefix – Prefix derived from numerals or other numbers
 Radix – Number of digits of a numeral system
 Radix economy – Number of digits needed to express a number in a particular base
 Table of bases – 0 to 74 in base 2 to 36
 Timeline of numerals and arithmetic – Timeline of arithmetic
References[edit]
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... unodecimal, duodecimal, tridecimal, quadrodecimal, pentadecimal, heptadecimal, octodecimal, nona decimal, vigesimal and further are discussed...
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A student of the American Indian languages is naturally led to investigate the widespread use of the quinaryvigesimal system of counting which he meets in the whole territory from Alaska along the Pacific Coast to the Orinoco and the Amazon.
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Quinaryvigesimal. This is most frequent. The Greenland Eskimo says 'other hand two' for 7, 'first foot two' for 12, 'other foot two' for 17, and similar combinations to 20, 'man ended.' The Unalit is also quinary to twenty, which is 'man completed.' ...
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This article proposes the Unique Number Mapping as an identification scheme, that could replace the E.164 numbers, could be used both with PSTN and VoIP terminals and makes use of the elements of the ENUM technology and the hexatrigesimal number system. […] To have the shortest IDs, we should use the greatest possible number system, which is the hexatrigesimal. Here the place values correspond to powers of 36...
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Concord Numbers used in the categorisation of Luganda words encoded using either Hexatrigesimal or Duotrigesimal, standard positional numbering systems. […] We propose Hexatrigesimal system to capture numeric information exceeding 10 for adaptation purposes for other Bantu languages or other agglutinative languages.
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Thanks to Satoshi Nakamoto for inventing the Base58 encoding format
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 ^ Ward, Rachel (2008), "On Robustness Properties of Beta Encoders and Golden Ratio Encoders", IEEE Transactions on Information Theory, 54 (9): 4324–4334, arXiv:0806.1083, Bibcode:2008arXiv0806.1083W, doi:10.1109/TIT.2008.928235, S2CID 12926540
 ^ Chrisomalis 2010, p. 254: Chrisomalis calls the Babylonian system "the first positional system ever".