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List of numeral systems

(Redirected from Octogesimal)

For different types of numbers, such as rational numbers, real numbers, complex numbers, etc, see List of types of numbers.

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

By culture / time period

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"A base is a natural number B whose powers (B multiplied by itself some number of times) are specially designated within a numerical system."[1]: 38  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.[1] 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 10&60 c. 3500–2000 BCE
Indus numerals unknown[2] c. 3500–1900 BCE[2]
Proto-Elamite numerals 10&60 3100 BCE
Sumerian numerals 10&60 3100 BCE
Egyptian numerals 10
Z1V20V1M12D50I8I7C11
 3000 BCE
Babylonian numerals 10&60                     2000 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  (                   )		 1500 BCE
Chinese numerals
Japanese numerals
Korean numerals (Sino-Korean)
Vietnamese numerals (Sino-Vietnamese)		 10

零一二三四五六七八九十百千萬億 (Default, Traditional Chinese)
〇一二三四五六七八九十百千万亿 (Default, Simplified Chinese)

1300 BCE
Roman numerals 5&10 I V X L C D M 1000 BCE[1]
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

Bengali ০ ১ ২ ৩ ৪ ৫ ৬ ৭ ৮ ৯

Devanagari ० १ २ ३ ४ ५ ६ ७ ८ ९

Gujarati ૦ ૧ ૨ ૩ ૪ ૫ ૬ ૭ ૮ ૯

Kannada ೦ ೧ ೨ ೩ ೪ ೫ ೬ ೭ ೮ ೯

Malayalam ൦ ൧ ൨ ൩ ൪ ൫ ൬ ൭ ൮ ൯

Odia ୦ ୧ ୨ ୩ ୪ ୫ ୬ ୭ ୮ ୯

Punjabi ੦ ੧ ੨ ੩ ੪ ੫ ੬ ੭ ੮ ੯

Tamil ௦ ௧ ௨ ௩ ௪ ௫ ௬ ௭ ௮ ௯

Telugu ౦ ౧ ౨ ౩ ౪ ౫ ౬ ౭ ౮ ౯

Tibetan ༠ ༡ ༢ ༣ ༤ ༥ ༦ ༧ ༨ ༩

Urdu 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
Kharosthi numerals 4&10 𐩇 𐩆 𐩅 𐩄 𐩃 𐩂 𐩁 𐩀 <400–250 BCE[3]
Phoenician numerals 10 𐤙 𐤘 𐤗 𐤛𐤛𐤛 𐤛𐤛𐤚 𐤛𐤛𐤖 𐤛𐤛 𐤛𐤚 𐤛𐤖 𐤛 𐤚 𐤖 [4] <250 BCE[5]
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
፼ [6]		 3rd–4th Century
15th Century (Modern Style)[7]: 135–136 
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[8]
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
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)		 late 7th/early 8th Century[9]
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[10]
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)
Vai numerals 10 ꘠ ꘡ ꘢ ꘣ ꘤ ꘥ ꘦ ꘧ ꘨ ꘩ [11] 19th Century (1832)[12]
Bamum numerals 10 ꛯ ꛦ ꛧ ꛨ ꛩ ꛪ ꛫ ꛬ ꛭ ꛮ [13] 19th Century (1896)[12]
Mende Kikakui numerals 10 𞣏 𞣎 𞣍 𞣌 𞣋 𞣊 𞣉 𞣈 𞣇 [14] 20th Century (1917)[15]
Osmanya numerals 10 𐒠 𐒡 𐒢 𐒣 𐒤 𐒥 𐒦 𐒧 𐒨 𐒩 20th Century (1920s)
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 [16] 20th Century (1930s)[17]
N'Ko numerals 10 ߉ ߈ ߇ ߆ ߅ ߄ ߃ ߂ ߁ ߀ [18] 20th Century (1949)[19]
Hmong numerals 10 𖭐 𖭑 𖭒 𖭓 𖭔 𖭕 𖭖 𖭗 𖭘 𖭑𖭐 20th Century (1959)
Garay numerals 10  [20] 20th Century (1961)[21]
Adlam numerals 10 𞥙 𞥘 𞥗 𞥖 𞥕 𞥔 𞥓 𞥒 𞥑 𞥐 [22] 20th Century (1989)[23]
Kaktovik numerals 5&20                                        
𝋀 𝋁 𝋂 𝋃 𝋄 𝋅 𝋆 𝋇 𝋈 𝋉 𝋊 𝋋 𝋌 𝋍 𝋎 𝋏 𝋐 𝋑 𝋒 𝋓 [24]		 20th Century (1994)[25]
Sundanese numerals 10 ᮰ ᮱ ᮲ ᮳ ᮴ ᮵ ᮶ ᮷ ᮸ ᮹ 20th Century (1996)[26]

By type of notation

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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

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A binary clock might use 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.[27] There have been some proposals for standardisation.[28]

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[29] 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[30] Weeks timekeeping, Western music letter notation
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[31][32][33]
11 Undecimal, unodecimal, undenary A base-11 number system was attributed to the Māori (New Zealand) in the 19th century[34] and the Pangwa (Tanzania) in the 20th century.[35] 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.[36][37][38] Featured in popular fiction.
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; penny and shilling
13 Tredecimal, tridecimal[39][40] Conway base 13 function.
14 Quattuordecimal, quadrodecimal[39][40] Programming for the HP 9100A/B calculator[41] and image processing applications;[42] pound and stone.
15 Quindecimal, pentadecimal[43][40] Telephony routing over IP, and the Huli language.
16 Hexadecimal, sexadecimal, sedecimal Compact notation for binary data; tonal system; ounce and pound.
17 Septendecimal, heptadecimal[43][40]
18 Octodecimal[43][40] A base in which 7n is palindromic for n = 3, 4, 6, 9.
19 Undevicesimal, nonadecimal[43][40]
20 Vigesimal Basque, Celtic, Muisca, Inuit, Yoruba, Tlingit, and Dzongkha numerals; Santali, and Ainu languages; shilling and pound
5&20 Quinary-vigesimal[44][45][46] 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"[44]
21 The smallest base in which all fractions ⁠1/2⁠ to ⁠1/18⁠ have periods of 4 or shorter.
23 Kalam language,[47] Kobon language[citation needed]
24 Quadravigesimal[48] 24-hour clock timekeeping; Greek alphabet; Kaugel language.
25 Sometimes used as compact notation for quinary.
26 Hexavigesimal[48][49] Sometimes used for encryption or ciphering,[50] using all letters in the English alphabet
27 Septemvigesimal Telefol,[47] Oksapmin,[51] Wambon,[52] and Hewa[53] 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,[54] to provide a concise encoding of alphabetic strings,[55] or as the basis for a form of gematria.[56] Compact notation for ternary.
28 Months timekeeping.
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.
32 Duotrigesimal Found in 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 Covers the ten decimal digits and all letters of the English alphabet, apart from not distinguishing 0 from O.
36 Hexatrigesimal[57][58] Covers the ten decimal digits and all letters of the English alphabet.
37 Covers the ten decimal digits and all letters of the Spanish alphabet.
38 Covers the duodecimal digits and all letters of the English alphabet.
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 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 Quinquagesimal 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 Covers the digits and letters assigned to base 62 apart from the basic vowel letters;[59] similar to base 26 but distinguishing upper- and lower-case letters.
56 A variant of base 58.[clarification needed][60]
57 Covers base 62 apart from I, O, l, U, and u,[61] or I, 1, l, 0, and O.[62]
58 Covers base 62 apart from 0 (zero), I (capital i), O (capital o) and l (lower case L).[63]
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]
62 Can be notated with the digits 0–9 and the cased letters A–Z and a–z of the English alphabet.
64 Tetrasexagesimal 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 The smallest base greater than binary such that no three-digit narcissistic number exists.
80 Octogesimal Used as a sub-base in Supyire.
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 MIME-64 encoding, since 855 is only slightly bigger than 232. Such method is 6.7% more efficient than MIME-64 which encodes a 24 bit number into 4 printable characters.
89 Largest base for which all left-truncatable primes are known.
90 Nonagesimal Related to Goormaghtigh conjecture for the generalized repunit numbers (111 in base 90 = 1111111111111 in base 2).
95 Number of printable ASCII characters.[65]
96 Total number of character codes in the (six) ASCII sticks containing printable characters.
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=102, these are two decimal digits.
121 Number expressible with two undecimal digits.
125 Number expressible with three quinary digits.
128 Using as 128=27.[clarification needed]
144 Number expressible with two duodecimal digits.
169 Number expressible with two tridecimal digits.
185 Smallest base which is not a perfect power (where generalized repunits can be factored algebraically) for which no generalized repunit primes are known.
196 Number expressible with two tetradecimal digits.
210 Smallest base such that all fractions ⁠1/2⁠ to ⁠1/10⁠ terminate.
225 Number expressible with two pentadecimal digits.
256 Number expressible with eight binary digits.
360 Degrees of angle.

Non-standard positional numeral systems

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Bijective numeration

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Base Name Usage
1 Unary (Bijective base‑1) 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.

Some email spam filters tag messages with a number of asterisks in an e-mail header such as X-Spam-Bar or X-SPAM-LEVEL. The larger the number, the more likely the email is considered spam.

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.[66]

Signed-digit representation

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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 bases

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Base Name Usage
2i Quater-imaginary 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

Non-integer bases

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Base Name Usage
  Base   a rational non-integer base
  Base   related to duodecimal
  Base   related to decimal
  Base   related to base 2
  Base   related to base 3
  Base  
  Base  
  Base   usage in 12-tone equal temperament musical system
  Base  
  Base   a negative rational non-integer base
  Base   a negative non-integer base, related to base 2
  Base   related to decimal
  Base   related to duodecimal
φ Golden ratio base early Beta encoder[67]
ρ Plastic number base
ψ Supergolden ratio base
  Silver ratio base
e Base   best radix economy [citation needed]
π Base  
eπ Base  
  Base  

n-adic number

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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

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  • 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

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Non-positional notation

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All known numeral systems developed before the Babylonian numerals are non-positional,[68] as are many developed later, such as the Roman numerals. The French Cistercian monks created their own numeral system.

See also

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References

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  1. ^ a b c Chrisomalis, Stephen (2004). "A cognitive typology for numerical notation". Cambridge Archaeological Journal. 14 (1): 37–52. doi:10.1017/S0959774304000034.
  2. ^ a b Chrisomalis 2010, pp. 330-333.
  3. ^ Glass, Andrew; Baums, Stefan; Salomon, Richard (September 18, 2003). "Proposal to Encode Kharoṣ ṭhī in Plane 1 of ISO/IEC 10646" (PDF). Unicode.org.
  4. ^ Everson, Michael (July 25, 2007). "Proposal to add two numbers for the Phoenician script" (PDF). UTC Document Register. Unicode Consortium. L2/07-206 (WG2 N3284).
  5. ^ Cajori, Florian (September 1928). A History Of Mathematical Notations Vol I. The Open Court Company. p. 18. Retrieved June 5, 2017.
  6. ^ "Ethiopic (Unicode block)" (PDF). Unicode Character Code Charts. Unicode Consortium.
  7. ^ Chrisomalis, Stephen (2010). Numerical Notation: A Comparative History. Cambridge University Press. ISBN 978-0-521-87818-0.
  8. ^ Chrisomalis 2010, p. 200.
  9. ^ Guo, Xianghe (July 27, 2009). "武则天为反贪发明汉语大写数字——中新网" [Wu Zetian invented Chinese capital numbers to fight corruption]. 中新社 [China News Service]. Retrieved August 15, 2024.
  10. ^ "Burmese/Myanmar script and pronunciation". Omniglot. Retrieved June 5, 2017.
  11. ^ "Vai (Unicode block)" (PDF). Unicode Character Code Charts. Unicode Consortium.
  12. ^ a b Kelly, Piers. "The invention, transmission and evolution of writing: Insights from the new scripts of West Africa". Open Science Framework.
  13. ^ "Bamum (Unicode block)" (PDF). Unicode Character Code Charts. Unicode Consortium.
  14. ^ "Mende Kikakui (Unicode block)" (PDF). Unicode Character Code Charts. Unicode Consortium.
  15. ^ Everson, Michael (October 21, 2011). "Proposal for encoding the Mende script in the SMP of the UCS" (PDF). UTC Document Register. Unicode Consortium. L2/11-301R (WG2 N4133R).
  16. ^ "Medefaidrin (Unicode block)" (PDF). Unicode Character Code Charts. Unicode Consortium.
  17. ^ Rovenchak, Andrij (July 17, 2015). "Preliminary proposal for encoding the Medefaidrin (Oberi Okaime) script in the SMP of the UCS (Revised)" (PDF). UTC Document Register. Unicode Consortium. L2/L2015.
  18. ^ "NKo (Unicode block)" (PDF). Unicode Character Code Charts. Unicode Consortium.
  19. ^ Donaldson, Coleman (January 1, 2017). "Clear Language: Script, Register And The N'ko Movement Of Manding-Speaking West Africa" (PDF). repository.upenn.edu. UPenn.
  20. ^ "Consideration of the encoding of Garay with updated user feedback (revised)" (PDF). Unicode Character Code Charts. Unicode Consortium.
  21. ^ Everson, Michael (March 22, 2016). "Proposal for encoding the Garay script in the SMP of the UCS" (PDF). UTC Document Register. Unicode Consortium. L2/L16-069 (WG2 N4709).
  22. ^ "Adlam (Unicode block)" (PDF). Unicode Character Code Charts. Unicode Consortium.
  23. ^ Everson, Michael (October 28, 2014). "Revised proposal for encoding the Adlam script in the SMP of the UCS" (PDF). UTC Document Register. Unicode Consortium. L2/L14-219R (WG2 N4628R).
  24. ^ "Kaktovik Numerals (Unicode block)" (PDF). Unicode Character Code Charts. Unicode Consortium.
  25. ^ Silvia, Eduardo (February 9, 2020). "Exploratory proposal to encode the Kaktovik numerals" (PDF). UTC Document Register. Unicode Consortium. L2/20-070.
  26. ^ "Direktori Aksara Sunda untuk Unicode" (PDF) (in Indonesian). Pemerintah Provinsi Jawa Barat. 2008.[page needed]
  27. ^ For the mixed roots of the word "hexadecimal", see Epp, Susanna (2010), Discrete Mathematics with Applications (4th ed.), Cengage Learning, p. 91, ISBN 9781133168669.
  28. ^ Multiplication Tables of Various Bases, p. 45, Michael Thomas de Vlieger, Dozenal Society of America
  29. ^ Kindra, Vladimir; Rogalev, Nikolay; Osipov, Sergey; Zlyvko, Olga; Naumov, Vladimir (2022). "Research and Development of Trinary Power Cycles". Inventions. 7 (3): 56. doi:10.3390/inventions7030056. ISSN 2411-5134.
  30. ^ "Definition of SEPTENARY". www.merriam-webster.com. Retrieved November 21, 2023.
  31. ^ The History of Arithmetic, Louis Charles Karpinski, 200pp, Rand McNally & Company, 1925.
  32. ^ Histoire universelle des chiffres, Georges Ifrah, Robert Laffont, 1994.
  33. ^ 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
  34. ^ Overmann, Karenleigh A (2020). "The curious idea that Māori once counted by elevens, and the insights it still holds for cross-cultural numerical research". Journal of the Polynesian Society. 129 (1): 59–84. doi:10.15286/jps.129.1.59-84. Retrieved July 24, 2020.
  35. ^ Thomas, N.W (1920). "Duodecimal base of numeration". Man. 20 (1): 56–60. doi:10.2307/2840036. JSTOR 2840036. Retrieved July 25, 2020.
  36. ^ Ulrich, Werner (November 1957). "Non-binary error correction codes". Bell System Technical Journal. 36 (6): 1364–1365. doi:10.1002/j.1538-7305.1957.tb01514.x.
  37. ^ Das, Debasis; Lanjewar, U.A. (January 2012). "Realistic Approach of Strange Number System from Unodecimal to Vigesimal" (PDF). International Journal of Computer Science and Telecommunications. 3 (1). London: Sysbase Solution Ltd.: 13.
  38. ^ Rawat, Saurabh; Sah, Anushree (May 2013). "Subtraction in Traditional and Strange Number System by r's and r-1's Compliments". International Journal of Computer Applications. 70 (23): 13–17. Bibcode:2013IJCA...70w..13R. doi:10.5120/12206-7640. ... unodecimal, duodecimal, tridecimal, quadrodecimal, pentadecimal, heptadecimal, octodecimal, nona decimal, vigesimal and further are discussed...
  39. ^ a b Das & Lanjewar 2012, p. 13.
  40. ^ a b c d e f Rawat & Sah 2013.
  41. ^ HP 9100A/B programming, HP Museum
  42. ^ "Image processor and image processing method".
  43. ^ a b c d Das & Lanjewar 2012, p. 14.
  44. ^ a b Nykl, Alois Richard (September 1926). "The Quinary-Vigesimal System of Counting in Europe, Asia, and America". Language. 2 (3): 165–173. doi:10.2307/408742. JSTOR 408742. OCLC 50709582 – via Google Books. p. 165: 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.
  45. ^ Eells, Walter Crosby (October 14, 2004). "Number Systems of the North American Indians". In Anderson, Marlow; Katz, Victor; Wilson, Robin (eds.). Sherlock Holmes in Babylon: And Other Tales of Mathematical History. Mathematical Association of America. p. 89. ISBN 978-0-88385-546-1 – via Google Books. Quinary-vigesimal. 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.' ...
  46. ^ Chrisomalis 2010, p. 200: "The early origin of bar-and-dot numeration alongside the Middle Formative Mesoamerican scripts, the quinary-vigesimal structure of the system, and the general increase in the frequency and complexity of numeral expressions over time all point to its indigenous development.".
  47. ^ a b Laycock, Donald (1975). "Observations on Number Systems and Semantics". In Wurm, Stephen (ed.). New Guinea Area Languages and Language Study, I: Papuan Languages and the New Guinea Linguistic Scene. Pacific Linguistics C-38. Canberra: Research School of Pacific Studies, Australian National University. pp. 219–233.
  48. ^ a b Dibbell, Julian (2010). "Introduction". The Best Technology Writing 2010. Yale University Press. p. 9. ISBN 978-0-300-16565-4. There's even a hexavigesimal digital code—our own twenty-six symbol variant of the ancient Latin alphabet, which the Romans derived in turn from the quadravigesimal version used by the ancient Greeks.
  49. ^ Young, Brian; Faris, Tom; Armogida, Luigi (2019). "A nomenclature for sequence-based forensic DNA analysis". Genetics. 42. Forensic Science International: 14–20. doi:10.1016/j.fsigen.2019.06.001. PMID 31207427. […] 2) the hexadecimal output of the hash function is converted to hexavigesimal (base-26); 3) letters in the hexavigesimal number are capitalized, while all numerals are left unchanged; 4) the order of the characters is reversed so that the hexavigesimal digits appear […]
  50. ^ "Base 26 Cipher (Number ⬌ Words) - Online Decoder, Encoder".
  51. ^ Saxe, Geoffrey B.; Moylan, Thomas (1982). "The development of measurement operations among the Oksapmin of Papua New Guinea". Child Development. 53 (5): 1242–1248. doi:10.1111/j.1467-8624.1982.tb04161.x. JSTOR 1129012..
  52. ^ "Безымянный палец • Задачи".
  53. ^ Nauka i Zhizn, 1992, issue 3, p. 48.
  54. ^ Grannis, Shaun J.; Overhage, J. Marc; McDonald, Clement J. (2002), "Analysis of identifier performance using a deterministic linkage algorithm", Proceedings. AMIA Symposium: 305–309, PMC 2244404, PMID 12463836.
  55. ^ Stephens, Kenneth Rod (1996), Visual Basic Algorithms: A Developer's Sourcebook of Ready-to-run Code, Wiley, p. 215, ISBN 9780471134183.
  56. ^ Sallows, Lee (1993), "Base 27: the key to a new gematria", Word Ways, 26 (2): 67–77.
  57. ^ Gódor, Balázs (2006). "World-wide user identification in seven characters with unique number mapping". Networks 2006: 12th International Telecommunications Network Strategy and Planning Symposium. IEEE. pp. 1–5. doi:10.1109/NETWKS.2006.300409. ISBN 1-4244-0952-7. S2CID 46702639. 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...
  58. ^ Balagadde, Robert Ssali; Premchand, Parvataneni (2016). "The Structured Compact Tag-Set for Luganda". International Journal on Natural Language Computing (IJNLC). 5 (4). 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.
  59. ^ "Base52". GitHub. Retrieved January 3, 2016.
  60. ^ "Base56". Retrieved January 3, 2016.
  61. ^ "Base57". GitHub. Retrieved January 3, 2016.
  62. ^ "Base57". GitHub. Retrieved January 22, 2019.
  63. ^ "The Base58 Encoding Scheme". Internet Engineering Task Force. November 27, 2019. Archived from the original on August 12, 2020. Retrieved August 12, 2020. Thanks to Satoshi Nakamoto for inventing the Base58 encoding format
  64. ^ "NewBase60". Retrieved January 3, 2016.
  65. ^ "base95 Numeric System". Archived from the original on February 7, 2016. Retrieved January 3, 2016.
  66. ^ Nasar, Sylvia (2001). A Beautiful Mind. Simon and Schuster. pp. 333–6. ISBN 0-7432-2457-4.
  67. ^ 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
  68. ^ Chrisomalis 2010, p. 254: Chrisomalis calls the Babylonian system "the first positional system ever".
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