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

By culture / time period

Name Base Sample Approx. First Appearance
Proto-cuneiform numerals10&60c. 3500–2000 BCE
Indus numeralsc. 3500–1900 BCE
Proto-Elamite numerals10&603,100 BCE
Sumerian numerals10&603,100 BCE
Egyptian numerals10
Z1V20V1M12D50I8I7C11
3,000 BCE
Babylonian numerals10&60 2,000 BCE
Aegean numerals10𐄇 𐄈 𐄉 𐄊 𐄋 𐄌 𐄍 𐄎 𐄏  ( 1 2 3 4 5 6 7 8 9 )
𐄐 𐄑 𐄒 𐄓 𐄔 𐄕 𐄖 𐄗 𐄘  ( 10 20 30 40 50 60 70 80 90 )
𐄙 𐄚 𐄛 𐄜 𐄝 𐄞 𐄟 𐄠 𐄡  ( 100 200 300 400 500 600 700 800 900 )
𐄢 𐄣 𐄤 𐄥 𐄦 𐄧 𐄨 𐄩 𐄪  ( 1000 2000 3000 4000 5000 6000 7000 8000 9000 )
𐄫 𐄬 𐄭 𐄮 𐄯 𐄰 𐄱 𐄲 𐄳  ( 10000 20000 30000 40000 50000 60000 70000 80000 90000 )		1,500 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)
零壹貳參肆伍陸柒捌玖拾佰仟萬億 (Financial, T. Chinese)
零壹贰叁肆伍陆柒捌玖拾佰仟萬億 (Financial, S. Chinese)

 1,300 BCE
Roman numeralsI V X L C D M1,000 BCE
Hebrew numerals10א 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 numerals10

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 numerals10ō α β γ δ ε ϝ ζ η θ ι
ο Αʹ Βʹ Γʹ Δʹ Εʹ Ϛʹ Ζʹ Ηʹ Θʹ		<400 BCE
Phoenician numerals10𐤙 𐤘 𐤗 𐤛𐤛𐤛 𐤛𐤛𐤚 𐤛𐤛𐤖 𐤛𐤛 𐤛𐤚 𐤛𐤖 𐤛 𐤚 𐤖 [1]<250 BCE[2]
Chinese rod numerals10𝍠 𝍡 𝍢 𝍣 𝍤 𝍥 𝍦 𝍧 𝍨 𝍩1st Century
Coptic numerals10Ⲁ Ⲃ Ⲅ Ⲇ Ⲉ Ⲋ Ⲍ Ⲏ Ⲑ2nd Century
Ge'ez numerals10፩ 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 numerals10Ա Բ Գ Դ Ե Զ Է Ը Թ ԺEarly 5th Century
Khmer numerals10០ ១ ២ ៣ ៤ ៥ ៦ ៧ ៨ ៩Early 7th Century
Thai numerals10๐ ๑ ๒ ๓ ๔ ๕ ๖ ๗ ๘ ๙7th Century[4]
Abjad numerals10غ 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 numerals10٩ ٨ ٧ ٦ ٥ ٤ ٣ ٢ ١ ٠8th Century
Vietnamese numerals (Chữ Nôm)10𠬠 𠄩 𠀧 𦊚 𠄼 𦒹 𦉱 𠔭 𠃩<9th Century
Western Arabic numerals100 1 2 3 4 5 6 7 8 99th Century
Glagolitic numerals10Ⰰ Ⰱ Ⰲ Ⰳ Ⰴ Ⰵ Ⰶ Ⰷ Ⰸ ...9th Century
Cyrillic numerals10а в г д е ѕ з и ѳ і ...10th Century
Rumi numerals1010th Century
Burmese numerals10၀ ၁ ၂ ၃ ၄ ၅ ၆ ၇ ၈ ၉11th Century[5]
Tangut numerals10𘈩 𗍫 𘕕 𗥃 𗏁 𗤁 𗒹 𘉋 𗢭 𗰗11th Century (1036)
Cistercian numerals1013th Century
Maya numerals5&20 <15th Century
Muisca numerals20<15th Century
Korean numerals (Hangul)10영 일 이 삼 사 오 육 칠 팔 구15th Century (1443)
Aztec numerals2016th Century
Sinhala numerals10෦ 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 runes1019th Century
Cherokee numerals1019th Century (1820s)
Osmanya numerals10𐒠 𐒡 𐒢 𐒣 𐒤 𐒥 𐒦 𐒧 𐒨 𐒩20th Century (1920s)
Hmong numerals10𖭐 𖭑 𖭒 𖭓 𖭔 𖭕 𖭖 𖭗 𖭘 𖭙20th Century (1959)
Kaktovik numerals5&20𝋀 𝋁 𝋂 𝋃 𝋄 𝋅 𝋆 𝋇 𝋈 𝋉 𝋊 𝋋 𝋌 𝋍 𝋎 𝋏 𝋐 𝋑 𝋒 𝋓20th Century (1994)

By type of notation

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

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.[6] There have been some proposals for standardisation.[7]

BaseNameUsage
2BinaryDigital computing, imperial and customary volume (bushel-kenning-peck-gallon-pottle-quart-pint-cup-gill-jack-fluid ounce-tablespoon)
3TernaryCantor 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
4QuaternaryChumashan languages and Kharosthi numerals
5QuinaryGumatj, Ateso, Nunggubuyu, Kuurn Kopan Noot, and Saraveca languages; common count grouping e.g. tally marks
6Senary, seximalDiceware, Ndom, Kanum, and Proto-Uralic language (suspected)
7Septimal, septenary[8]Weeks timekeeping, Western music letter notation
8OctalCharles XII of Sweden, Unix-like permissions, Squawk codes, DEC PDP-11, Yuki, Pame, compact notation for binary numbers, Xiantian (I Ching, China)
9Nonary, nonalCompact notation for ternary
10Decimal, denaryMost widely used by contemporary societies[9][10][11]
11Undecimal, unodecimal, undenaryA base-11 number system was attributed to the Māori (New Zealand) in the 19th century[12] and the Pangwa (Tanzania) in the 20th century.[13] 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.[14][15][16] Featured in popular fiction.
12Duodecimal, dozenalLanguages 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
13Tredecimal, tridecimal[17][18]Conway base 13 function.
14Quattuordecimal, quadrodecimal[17][18]Programming for the HP 9100A/B calculator[19] and image processing applications;[20] pound and stone.
15Quindecimal, pentadecimal[21][18]Telephony routing over IP, and the Huli language.
16Hexadecimal, sexadecimal, sedecimal Compact notation for binary data; tonal system; ounce and pound.
17Septendecimal, heptadecimal[21][18]
18Octodecimal[21][18]A base in which 7n is palindromic for n = 3, 4, 6, 9.
19Undevicesimal, nonadecimal[21][18]
20VigesimalBasque, Celtic, Muisca, Inuit, Yoruba, Tlingit, and Dzongkha numerals; Santali, and Ainu languages; shilling and pound
5&20Quinary-vigesimal[22][23][24]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"[22]
21The smallest base in which all fractions 1/2 to 1/18 have periods of 4 or shorter.
23Kalam language,[25] Kobon language
24Quadravigesimal[26]24-hour clock timekeeping; Greek alphabet; Kaugel language.
25Sometimes used as compact notation for quinary.
26Hexavigesimal[26][27]Sometimes used for encryption or ciphering,[28] using all letters in the English alphabet
27SeptemvigesimalTelefol[29] and Oksapmin[30] 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,[31] to provide a concise encoding of alphabetic strings,[32] or as the basis for a form of gematria.[33] Compact notation for ternary.
28Months timekeeping.
30TrigesimalThe 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.
32DuotrigesimalFound in the Ngiti language.
33Use of letters (except I, O, Q) with digits in vehicle registration plates of Hong Kong.
34Using 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.
35Covers the ten decimal digits and all letters of the English alphabet, apart from not distinguishing 0 from O.
36Hexatrigesimal[34][35]Covers the ten decimal digits and all letters of the English alphabet.
37Covers the ten decimal digits and all letters of the Spanish alphabet.
38Covers the duodecimal digits and all letters of the English alphabet.
40QuadragesimalDEC 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.
42Largest base for which all minimal primes are known.
47Smallest base for which no generalized Wieferich primes are known.
49Compact notation for septenary.
50QuinquagesimalSQUOZE encoding used to compactly represent file names and other symbols on some IBM computers. Encoding using all Gurmukhi characters plus the Gurmukhi digits.
52Covers the digits and letters assigned to base 62 apart from the basic vowel letters;[36] similar to base 26 but distinguishing upper- and lower-case letters.
56A variant of base 58.[37]
57Covers base 62 apart from I, O, l, U, and u,[38] or I, 1, l, 0, and O.[39]
58Covers base 62 apart from 0 (zero), I (capital i), O (capital o) and l (lower case L).[40]
60SexagesimalBabylonian 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).[41]
62Can be notated with the digits 0–9 and the cased letters A–Z and a–z of the English alphabet.
64TetrasexagesimalI 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 /).
72The smallest base greater than binary such that no three-digit narcissistic number exists.
80OctogesimalUsed as a sub-base in Supyire.
85Ascii85 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.
89Largest base for which all left-truncatable primes are known.
90NonagesimalRelated to Goormaghtigh conjecture for the generalized repunit numbers (111 in base 90 = 1111111111111 in base 2).
91Number of characters needed to encode all ASCII except "-" (0x2D), "\" (0x5C), and "'" (0x27). (One variant uses "\" (0x5C) in place of """ (0x22).)
92Number of characters needed to encode all of ASCII except for "`" (0x60) and """ (0x22) due to confusability.[42]
93Number of characters needed to encode 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.[43]
94Number of characters needed to encode all ASCII printable characters.[44]
95Number of characters needed to encode all ASCII printable characters plus the space character.[45]
96Number of characters needed to encode all ASCII printable characters plus the two extra duodecimal digits.
97Smallest base which is not perfect odd power (where generalized Wagstaff numbers can be factored algebraically) for which no generalized Wagstaff primes are known.
100CentesimalAs 100=102, these are two decimal digits.
121Number expressible with two undecimal digits.
125Number expressible with three quinary digits.
128Using as 128=27.
144Number expressible with two duodecimal digits.
169Number expressible with two tridecimal digits.
185Smallest base which is not a perfect power (where generalized repunits can be factored algebraically) for which no generalized repunit primes are known.
196Number expressible with two tetradecimal digits.
210Smallest base such that all fractions 1/2 to 1/10 terminate.
225Number expressible with two pentadecimal digits.
256Number expressible with eight binary digits.
360Degrees of angle.

Non-standard positional numeral systems

Bijective numeration

BaseNameUsage
1Unary (Bijective base‑1)Tally marks, Counting
10Bijective base-10To avoid zero
26Bijective base-26Spreadsheet column numeration. Also used by John Nash as part of his obsession with numerology and the uncovering of "hidden" messages.[46]

Signed-digit representation

BaseNameUsage
2Balanced binary (Non-adjacent form)
3Balanced ternaryTernary computers
4Balanced quaternary
5Balanced quinary
6Balanced senary
7Balanced septenary
8Balanced octal
9Balanced nonary
10Balanced decimalJohn Colson
Augustin Cauchy
11Balanced undecimal
12Balanced duodecimal

Complex bases

BaseNameUsage
2iQuater-imaginary baserelated 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 ± iTwindragon baseTwindragon fractal shape, related to base −4 and base 16
1 ± iNegatwindragon baserelated to base −4 and base 16

Non-integer bases

BaseNameUsage
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
φGolden ratio baseEarly Beta encoder[47]
ρPlastic number base
ψSupergolden ratio base
Silver ratio base
eBase Lowest radix economy
πBase
eπBase
Base

n-adic number

BaseNameUsage
2Dyadic number
3Triadic number
4Tetradic numberthe same as dyadic number
5Pentadic number
6Hexadic numbernot a field
7Heptadic number
8Octadic numberthe same as dyadic number
9Enneadic numberthe same as triadic number
10Decadic numbernot a field
11Hendecadic number
12Dodecadic numbernot 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

Non-positional notation

All known numeral systems developed before the Babylonian numerals are non-positional,[48] as are many developed later, such as the Roman numerals. The French Cistercian monks created their own numeral system.

See also

References

  1. ↑ 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).
  2. ↑ Cajori, Florian (September 1928). A History Of Mathematical Notations Vol I. The Open Court Company. p. 18. Retrieved June 5, 2017.
  3. ↑ Chrisomalis, Stephen (January 18, 2010). Numerical Notation: A Comparative History. Cambridge University Press. pp. 135–136. ISBN 978-0-521-87818-0.
  4. ↑ Chrisomalis 2010, p. 200.
  5. ↑ "Burmese/Myanmar script and pronunciation". Omniglot. Retrieved June 5, 2017.
  6. ↑ For the mixed roots of the word "hexadecimal", see Epp, Susanna (2010), Discrete Mathematics with Applications (4th ed.), Cengage Learning, p. 91, ISBN 9781133168669.
  7. ↑ Multiplication Tables of Various Bases, p. 45, Michael Thomas de Vlieger, Dozenal Society of America
  8. ↑ "Definition of SEPTENARY". www.merriam-webster.com. Retrieved November 21, 2023.
  9. ↑ The History of Arithmetic, Louis Charles Karpinski, 200pp, Rand McNally & Company, 1925.
  10. ↑ Histoire universelle des chiffres, Georges Ifrah, Robert Laffont, 1994.
  11. ↑ 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
  12. ↑ 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.
  13. ↑ Thomas, N.W (1920). "Duodecimal base of numeration". Man. 20 (1): 56–60. doi:10.2307/2840036. JSTOR 2840036. Retrieved July 25, 2020.
  14. ↑ 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.
  15. ↑ 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. London: Sysbase Solution Ltd. 3 (1): 13.
  16. ↑ 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...
  17. 1 2 Das & Lanjewar 2012, p. 13.
  18. 1 2 3 4 5 6 Rawat & Sah 2013.
  19. ↑ HP 9100A/B programming, HP Museum
  20. ↑ Free Patents Online
  21. 1 2 3 4 Das & Lanjewar 2012, p. 14.
  22. 1 2 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.
  23. ↑ 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.' ...
  24. ↑ 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.".
  25. ↑ 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.
  26. 1 2 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.
  27. ↑ Young, Brian; Faris, Tom; Armogida, Luigi (2019). "A nomenclature for sequence-based forensic DNA analysis". Genetics. Forensic Science International. 42: 14–20. […] 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 […]
  28. ↑ "Base 26 Cipher (Number ⬌ Words) - Online Decoder, Encoder".
  29. ↑ 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.
  30. ↑ 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..
  31. ↑ 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.
  32. ↑ Stephens, Kenneth Rod (1996), Visual Basic Algorithms: A Developer's Sourcebook of Ready-to-run Code, Wiley, p. 215, ISBN 9780471134183.
  33. ↑ Sallows, Lee (1993), "Base 27: the key to a new gematria", Word Ways, 26 (2): 67–77.
  34. ↑ 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...
  35. ↑ Balagadde1, 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.{{cite journal}}: CS1 maint: numeric names: authors list (link)
  36. ↑ "Base52". GitHub. Retrieved January 3, 2016.
  37. ↑ "Base56". Retrieved January 3, 2016.
  38. ↑ "Base57". GitHub. Retrieved January 3, 2016.
  39. ↑ "Base57". GitHub. Retrieved January 22, 2019.
  40. ↑ "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
  41. ↑ "NewBase60". Retrieved January 3, 2016.
  42. ↑ "Base92". GitHub. Retrieved January 3, 2016.
  43. ↑ "Base93". September 26, 2013. Retrieved February 13, 2017.
  44. ↑ "Base94". Retrieved January 3, 2016.
  45. ↑ "base95 Numeric System". Archived from the original on February 7, 2016. Retrieved January 3, 2016.
  46. ↑ Nasar, Sylvia (2001). A Beautiful Mind. Simon and Schuster. pp. 333–6. ISBN 0-7432-2457-4.
  47. ↑ 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
  48. ↑ Chrisomalis 2010, p. 254: Chrisomalis calls the Babylonian system "the first positional system ever".
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