List of numeral systems

Short description: none

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Numeral systems

Hindu–Arabic numeral system

* Western Arabic
* Eastern Arabic

* Bengali
* Devanagari
* Gurmukhi
* Sinhala
* Tamil

* Balinese
* Burmese
* Dzongkha
* Gujarati
* Javanese
* Khmer
* Lao
* Mongolian
* Thai

East Asian

* Chinese
* Suzhou
* Hokkien
* Japanese
* Korean
* Vietnamese
* Tangut

* Counting rods

Alphabetic

* Abjad
* Armenian
* Āryabhaṭa
* Cyrillic
* Ge'ez
* Georgian
* Greek
* Hebrew
* Roman

Former

* Aegean
* Attic
* Babylonian
* Brahmi
* Chuvash
* Egyptian
* Etruscan
* Glagolitic
* Inuit
* Kharosthi
* Mayan
* Muisca
* Quipu

* Prehistoric

Positional systems by base

* 2
* 3
* 4
* 5
* 6
* 8
* 10
* 12
* 16
* 20
* 60

Non-standard positional numeral systems

* Bijective numeration (1)
* Signed-digit representation (Balanced ternary)
* factorial
* negative
* Complex-base system (2i)
* Non-integer representation (φ)
* mixed
* Asymmetric numeral systems

List of numeral systems

* v
* t
* e

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

## Contents

* 1 By culture / time period
* 2 By type of notation
* 2.1 Standard positional numeral systems
* 2.2 Non-standard positional numeral systems
* 2.2.1 Bijective numeration
* 2.2.2 Signed-digit representation
* 2.2.3 Complex bases
* 2.2.4 Non-integer bases
* 2.2.5 n-adic number
* 2.2.6 Mixed radix
* 2.2.7 Other
* 2.3 Non-positional notation
* 3 See also
* 4 References

## 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."[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).

[3] Name  | Base  | Sample  | Approx. First Appearance

|
|
|

Proto-cuneiform numerals | 10&60 |  | -3500 c. 3500–2000 BCE

Indus numerals | unknown[2] |  | -3501 c. 3500–1900 BCE

[2]

Proto-Elamite numerals | 10&60 |  | -3101 3100 BCE

Sumerian numerals | 10&60 |  | -3100 3100 BCE

Egyptian numerals | 10 | <hiero size=8>Z1 V20 V1 M12 D50 I8 I7 C11</hiero> | -3000 3000 BCE

Babylonian numerals | 10&60 | 15px 15px 15px 15px 15px 15px 15px 15px 15px 15px | -2000 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 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 1300 BCE

Roman numerals | 5&10 | I V X L C D M | -1000 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 | -800 800 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

| -750 750–500 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 <400 BCE

Kharosthi numerals | 4&10  | 𐩇 𐩆 𐩅 𐩄 𐩃 𐩂 𐩁 𐩀
Chinese rod numerals | 10 | 𝍠 𝍡 𝍢 𝍣 𝍤 𝍥 𝍦 𝍧 𝍨 𝍩 | 1 1st century

Coptic numerals | 10 | Ⲁ Ⲃ Ⲅ Ⲇ Ⲉ Ⲋ Ⲍ Ⲏ Ⲑ | 100 2nd century

Ge'ez numerals | 10 | ፩ ፪ ፫ ፬ ፭ ፮ ፯ ፰ ፱
፲ ፳ ፴ ፵ ፶ ፷ ፸ ፹ ፺

፼ [4] | 200 3rd–4th century

15th century (Modern Style)[5]: 135–136

Armenian numerals | 10 | Ա Բ Գ Դ Ե Զ Է Ը Թ Ժ | 400 Early 5th century

Khmer numerals | 10 | ០ ១ ២ ៣ ៤ ៥ ៦ ៧ ៨ ៩ | 600 Early 7th century

Thai numerals | 10 | ๐ ๑ ๒ ๓ ๔ ๕ ๖ ๗ ๘ ๙ | 601 7th century

[6]

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 | 680 <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) | 690 late 7th/early 8th century

[7]

Eastern Arabic numerals | 10 | ٩ ٨ ٧ ٦ ٥ ٤ ٣ ٢ ١ ٠ | 701 8th century

Vietnamese numerals (Chữ Nôm) | 10 | 𠬠 𠄩 𠀧 𦊚 𠄼 𦒹 𦉱 𠔭 𠃩 | 799 <9th century

Western Arabic numerals | 10 | 0 1 2 3 4 5 6 7 8 9 | 801 9th century

Glagolitic numerals | 10 | Ⰰ Ⰱ Ⰲ Ⰳ Ⰴ Ⰵ Ⰶ Ⰷ Ⰸ ... | 800 9th century

Cyrillic numerals | 10 | а в г д е ѕ з и ѳ і ... | 900 10th century

Rumi numerals | 10 |  | 900 10th century

Burmese numerals | 10 | ၀ ၁ ၂ ၃ ၄ ၅ ၆ ၇ ၈ ၉ | 1000 11th century

[8]

Tangut numerals | 10 | Template:Tangut | 1036 11th century (1036)

Cistercian numerals | 10 |  | 1200 13th century

Maya numerals | 5&20 | 15px 15px 15px 15px 15px 15px 15px 15px 15px 15px 15px 15px 15px 15px 15px 15px 15px 15px 15px 15px
𝋠 𝋡 𝋢 𝋣 𝋤 𝋥 𝋦 𝋧 𝋨 𝋩 𝋪 𝋫 𝋬 𝋭 𝋮 𝋯 𝋰 𝋱 𝋲 𝋳 | 1400 15th century

Muisca numerals | 20 |  | 1399 15th century

Korean numerals (Hangul) | 10 | 영 일 이 삼 사 오 육 칠 팔 구 | 1443 15th century (1443)

Aztec numerals | 20 | x25px x25px x25px x25px x25px x25px x30px
(1, 5, 20, 100, 400, 800, 8000) | 1500 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 | 1699 18th century

Pentadic runes | 10 |  | 1800 19th century

Cherokee numerals | 10 |  | 1820 19th century (1820s)

Vai numerals | 10 | ꘠ ꘡ ꘢ ꘣ ꘤ ꘥ ꘦ ꘧ ꘨ ꘩ [9] | 1832 19th century (1832)[10]

Bamum numerals | 10 | ꛯ ꛦ ꛧ ꛨ ꛩ ꛪ ꛫ ꛬ ꛭ ꛮ [11] | 1896 19th century (1896)[10]

Mende Kikakui numerals | 10 | 𞣏 𞣎 𞣍 𞣌 𞣋 𞣊 𞣉 𞣈 𞣇 [12] | 1896 20th century (1917)[13]

Osmanya numerals | 10 | 𐒠 𐒡 𐒢 𐒣 𐒤 𐒥 𐒦 𐒧 𐒨 𐒩 | 1921 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 [14] | 1930 20th century (1930s)[15]

N'Ko numerals | 10 | ߉ ߈ ߇ ߆ ߅ ߄ ߃ ߂ ߁ ߀ [16] | 1949 20th century (1949)[17]

Hmong numerals | 10 | 𖭐 𖭑 𖭒 𖭓 𖭔 𖭕 𖭖 𖭗 𖭘 𖭑𖭐 | 1959 20th century (1959)

Garay numerals | 10 | Garay numbers[18] | 1961 20th century (1961)[19]

Adlam numerals | 10 | 𞥙 𞥘 𞥗 𞥖 𞥕 𞥔 𞥓 𞥒 𞥑 𞥐 [20] | 1989 20th century (1989)[21]

Kaktovik numerals | 5&20 | Template:Kaktovik digit Template:Kaktovik digit Template:Kaktovik digit Template:Kaktovik digit Template:Kaktovik digit Template:Kaktovik digit Template:Kaktovik digit Template:Kaktovik digit Template:Kaktovik digit Template:Kaktovik digit Template:Kaktovik digit Template:Kaktovik digit Template:Kaktovik digit Template:Kaktovik digit Template:Kaktovik digit Template:Kaktovik digit Template:Kaktovik digit Template:Kaktovik digit Template:Kaktovik digit Template:Kaktovik digit
𝋀 𝋁 𝋂 𝋃 𝋄 𝋅 𝋆 𝋇 𝋈 𝋉 𝋊 𝋋 𝋌 𝋍 𝋎 𝋏 𝋐 𝋑 𝋒 𝋓 [22] | 1994 20th century (1994)[23]

Sundanese numerals | 10  | ᮰ ᮱ ᮲ ᮳ ᮴ ᮵ ᮶ ᮷ ᮸ ᮹  | 20th century (1996)[24]

## 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.[25] There have been some proposals for standardisation.[26]

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[27] | 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 | Aneityum (traditional),[28] Ateso, Gumatj, Kuurn Kopan Noot, and Nunggubuyu, 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[29][30][31]
11 | Undecimal, unodecimal, undenary | A base-11 number system was mistakenly attributed to the Māori (New Zealand) in the 19th century[32] and one was reported to be used by the Pangwa (Tanzania) in the 20th century,[33] but was not confirmed by later research and is believed to also be an error.[34] 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.[35][36][37]
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[38][39] | Conway's base 13 function.
14 | Quattuordecimal, quadrodecimal[38][39] | Programming for the HP 9100A/B calculator[40] and image processing applications.[41]
15 | Quindecimal, pentadecimal[42][39] | Telephony routing over IP, and the Huli language.[34]
16 | Hexadecimal, sexadecimal, sedecimal  | Compact notation for binary data; tonal system of Nystrom.
17 | Heptadecimal, septendecimal[42][39] |
18 | Octodecimal[42][39] |
19 | Undevicesimal, nonadecimal[42][39] |
20 | Vigesimal | Basque, Celtic, Muisca, Inuit, Yoruba, Tlingit, and Dzongkha numerals; Santali, and Ainu languages.
5&20 | Quinary-vigesimal[43][44][45] | 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"[43]
21 |  | The smallest base in which all fractions ⁠1/2⁠ to ⁠1/18⁠ have periods of 4 or shorter.
24 | Quadravigesimal[46] | 24-hour clock timekeeping; Greek alphabet; Kaugel language.
25 |  | Sometimes used as compact notation for quinary.
26 | Hexavigesimal[46][47] | Sometimes used for encryption or ciphering,[48] using all letters in the English alphabet. Used to encode SHA-256 hashes into uppercase letters in InChIKey (a standard indexing system of chemical structures)[49] and SID (sequence identification, an indexing system of PCR amplicons in forensics).[47]
27 |  | Telefol,[50] 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 |  | 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. Also used to encode computer (binary) data into an alphanumerical string without confusable characters (e.g. zero and "O", eight and "B") in RFC 4648, with each character standing for 5 bits.
34 |  | The smallest base where ⁠1/2⁠ terminates and all of ⁠1/2⁠ to ⁠1/18⁠ have periods of 4 or shorter.
36 | Hexatrigesimal[57][58] | Used to encode large numbers into an alphanumeric string (26 letters, 10 numbers).
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.
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).[59]
64 |  | Used to encode computer (binary) data into a relatively compact string, with each character standing for 6 bits (RFC 4648).
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 ⁠1/2⁠ to ⁠1/10⁠ terminate.

### Non-standard positional numeral systems

#### Bijective numeration

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.[60]

#### 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 bases

Base | Name | Usage

|
|

2i | Quater-imaginary base | related to base −4 and base 16
i2 | Base i2 | related to base −2 and base 4
i24 | Base i24 | related to base 2
2ω | Base 2ω | related to base 8
ω23 | Base ω23 | 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

Base | Name | Usage

|
|

32 | Base 32 | a rational non-integer base
43 | Base 43 | related to duodecimal
52 | Base 52 | related to decimal
2 | Base 2 | related to base 2
3 | Base 3 | related to base 3
23 | Base 23 |
24 | Base 24 |
212 | Base 212 | usage in 12-tone equal temperament musical system
22 | Base 22 |
−32 | Base −32 | a negative rational non-integer base
−2 | Base −2 | a negative non-integer base, related to base 2
10 | Base 10 | related to decimal
23 | Base 23 | related to duodecimal
φ | Golden ratio base | early Beta encoder[61]
ρ | Plastic number base |
ψ | Supergolden ratio base |
1+2 | Silver ratio base |
π | Base π |
eπ | Base eπ |
eπ | Base eπ |

#### 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,[62] as are many developed later, such as the Roman numerals. The French Cistercian monks created their own numeral system.

## See also

* History of ancient numeral systems
* History of the Hindu–Arabic numeral system - Aspect of history
* List of numeral system topics
* Numeral prefix - Prefix derived from numerals or other numbers
* Radix
* Radix economy - Number of digits needed to express a number in a particular base
* Timeline of numerals and arithmetic

## References

1. ↑ 1.0 1.1 1.2 Chrisomalis, Stephen (2004). "A cognitive typology for numerical notation". Cambridge Archaeological Journal 14 (1): 37–52. doi:10.1017/S0959774304000034.
2. ↑ 2.0 2.1 Chrisomalis 2010, pp. 330-333.
3. ↑ Glass, Andrew; Baums, Stefan; Salomon, Richard (2003-09-18). "Proposal to Encode Kharoṣ ṭhī in Plane 1 of ISO/IEC 10646". https://www.unicode.org/L2/L2003/03314-kharoshthi.pdf.
4. ↑ "Ethiopic (Unicode block)". Unicode Consortium. https://unicode.org/charts/PDF/U1200.pdf.
5. ↑ Chrisomalis, Stephen (2010) (in en). Numerical Notation: A Comparative History. Cambridge University Press. ISBN 978-0-521-87818-0. https://books.google.com/books?id=ux--OWgWvBQC&pg=PA135.
6. ↑ Chrisomalis 2010, p. 200.
7. ↑ Guo, Xianghe (2009-07-27). "武则天为反贪发明汉语大写数字——中新网". https://www.chinanews.com.cn/hb/news/2009/07-27/1792519.shtml.
8. ↑ "Burmese/Myanmar script and pronunciation". http://www.omniglot.com/writing/burmese.htm.
9. ↑ "Vai (Unicode block)". Unicode Consortium. https://unicode.org/charts/PDF/UA500.pdf.
10. ↑ 10.0 10.1 Kelly, Piers. "The invention, transmission and evolution of writing: Insights from the new scripts of West Africa". https://osf.io/preprints/socarxiv/253vc/download.
11. ↑ "Bamum (Unicode block)". Unicode Consortium. https://unicode.org/charts/PDF/UA6A0.pdf.
12. ↑ "Mende Kikakui (Unicode block)". Unicode Consortium. https://unicode.org/charts/PDF/U1E800.pdf.
13. ↑ Everson, Michael (2011-10-21). "Proposal for encoding the Mende script in the SMP of the UCS". Unicode Consortium. L2/11-301R (WG2 N4133R). https://www.unicode.org/L2/L2011/11301r-n4133-mende.pdf.
14. ↑ "Medefaidrin (Unicode block)". Unicode Consortium. https://unicode.org/charts/PDF/U16E40.pdf.
15. ↑ Rovenchak, Andrij (2015-07-17). "Preliminary proposal for encoding the Medefaidrin (Oberi Okaime) script in the SMP of the UCS (Revised)". Unicode Consortium. L2/L2015. https://www.unicode.org/L2/L2015/15117r2-medefaidrin.pdf.
16. ↑ "NKo (Unicode block)". Unicode Consortium. https://unicode.org/charts/PDF/U07C0.pdf.
17. ↑ Donaldson, Coleman (2017-01-01). "Clear Language: Script, Register And The N'ko Movement Of Manding-Speaking West Africa". UPenn. https://www.unicode.org/L2/L2015/15117r2-medefaidrin.pdf.
18. ↑ "Consideration of the encoding of Garay with updated user feedback (revised)". Unicode Consortium. https://www.unicode.org/L2/L2022/22048-garay-script.pdf.
19. ↑ Everson, Michael (2016-03-22). "Proposal for encoding the Garay script in the SMP of the UCS". Unicode Consortium. L2/L16-069 (WG2 N4709). https://www.unicode.org/L2/L2016/16069-n4709-garay-revision.pdf.
20. ↑ "Adlam (Unicode block)". Unicode Consortium. https://unicode.org/charts/PDF/U1E900.pdf.
21. ↑ Everson, Michael (2014-10-28). "Revised proposal for encoding the Adlam script in the SMP of the UCS". Unicode Consortium. L2/L14-219R (WG2 N4628R). https://www.unicode.org/L2/L2014/14219r-n4628-adlam.pdf.
22. ↑ "Kaktovik Numerals (Unicode block)". Unicode Consortium. https://unicode.org/charts/PDF/U1D2C0.pdf.
23. ↑ Silvia, Eduardo (2020-02-09). "Exploratory proposal to encode the Kaktovik numerals". Unicode Consortium. L2/20-070. https://www.unicode.org/L2/L2020/20070-kaktovik-numerals.pdf.
24. ↑ "Direktori Aksara Sunda untuk Unicode" (in id). Pemerintah Provinsi Jawa Barat. 2008. http://file.upi.edu/Direktori/FPBS/JUR._PEND._BHS._DAN_SASTRA_INDONESIA/197006242006041-TEDI_PERMADI/Direktori_Aksara_Sunda_untuk_Unicode.pdf.
25. ↑ For the mixed roots of the word "hexadecimal", see Epp, Susanna (2010), Discrete Mathematics with Applications (4th ed.), Cengage Learning, p. 91, ISBN 9781133168669, https://books.google.com/books?id=HUAIAAAAQBAJ&pg=PA91 .
26. ↑ Multiplication Tables of Various Bases, p. 45, Michael Thomas de Vlieger, Dozenal Society of America
27. ↑ Kindra, Vladimir; Rogalev, Nikolay; Osipov, Sergey; Zlyvko, Olga; Naumov, Vladimir (2022). "Research and Development of Trinary Power Cycles" (in en). Inventions 7 (3): 56. doi:10.3390/inventions7030056. ISSN 2411-5134.
28. ↑ The Aneityum language of Aneityum, Vanuatu traditionally had a quinary system, although this had largely been replaced by a decimal system, based on English numbers, by the early 20th century.
29. ↑ The History of Arithmetic, Louis Charles Karpinski, 200pp, Rand McNally & Company, 1925.
30. ↑ Histoire universelle des chiffres, Georges Ifrah, Robert Laffont, 1994.
31. ↑ 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
32. ↑ 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. http://www.thepolynesiansociety.org/jps/index.php/JPS/article/view/458. Retrieved 24 July 2020.
33. ↑ Thomas, N.W (1920). "Duodecimal base of numeration". Man 20 (1): 56–60. doi:10.2307/2840036. http://www.jstor.com/stable/2840036. Retrieved 25 July 2020.
34. ↑ 34.0 34.1 Hammarström, Harald (2010). "Rarities in numeral systems". Rethinking Universals. pp. 11–60. doi:10.1515/9783110220933.11. ISBN 9783110220933.
35. ↑ 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. https://archive.org/details/bstj36-6-1341/page/n23/mode/2up?q=unodecimal.
36. ↑ Das, Debasis; Lanjewar, U.A. (January 2012). "Realistic Approach of Strange Number System from Unodecimal to Vigesimal". International Journal of Computer Science and Telecommunications (London: Sysbase Solution Ltd.) 3 (1): 13. https://www.ijcst.org/Volume3/Issue1/p2_3_1.pdf.
37. ↑ 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. doi:10.5120/12206-7640. Bibcode: 2013IJCA...70w..13R. "... unodecimal, duodecimal, tridecimal, quadrodecimal, pentadecimal, heptadecimal, octodecimal, nona decimal, vigesimal and further are discussed...".
38. ↑ 38.0 38.1 Das & Lanjewar 2012, p. 13.
39. ↑ 39.0 39.1 39.2 39.3 39.4 39.5 Rawat & Sah 2013.
40. ↑ HP 9100A/B programming, HP Museum
41. ↑ "Image processor and image processing method". https://www.freepatentsonline.com/6690378.html.
42. ↑ 42.0 42.1 42.2 42.3 Das & Lanjewar 2012, p. 14.
43. ↑ 43.0 43.1 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. OCLC 50709582. https://books.google.com/books?id=1GwUAAAAIAAJ&q=Nykl&pg=RA1-PA165. "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.".
44. ↑ Eells, Walter Crosby (October 14, 2004). "Number Systems of the North American Indians". Sherlock Holmes in Babylon: And Other Tales of Mathematical History. Mathematical Association of America. p. 89. ISBN 978-0-88385-546-1. https://books.google.com/books?id=BKRE5AjRM3AC&pg=PA89. "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.' ..."
45. ↑ 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.".
46. ↑ 46.0 46.1 Dibbell, Julian (2010). "Introduction". The Best Technology Writing 2010. Yale University Press. p. 9. ISBN 978-0-300-16565-4. https://books.google.com/books?id=DKPovyrXRwkC&pg=PT9. "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."
47. ↑ 47.0 47.1 Young, Brian; Faris, Tom; Armogida, Luigi (2019). "A nomenclature for sequence-based forensic DNA analysis" (in en). Genetics (Forensic Science International) 42: 14–20. doi:10.1016/j.fsigen.2019.06.001. PMID 31207427. https://www.sciencedirect.com/science/article/pii/S1872497319300997. "[…] 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 […]".
48. ↑ "Base 26 Cipher (Number ⬌ Words) - Online Decoder, Encoder". http://www.dcode.fr/base-26-cipher.
49. ↑ "Technical FAQ - InChI Trust". http://www.inchi-trust.org/technical-faq/#13.1.
50. ↑ Cite error: Invalid `<ref>` tag; no text was provided for refs named `Laycock 1975 219–233`
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. .
52. ↑ "Безымянный палец • Задачи". https://elementy.ru/problems/264/Bezymyannyy_palets.
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, PMID 12463836 .
55. ↑ Stephens, Kenneth Rod (1996), Visual Basic Algorithms: A Developer's Sourcebook of Ready-to-run Code, Wiley, p. 215, ISBN 9780471134183, https://archive.org/details/visualbasicalgor00step/page/215 .
56. ↑ Sallows, Lee (1993), "Base 27: the key to a new gematria", Word Ways 26 (2): 67–77, http://digitalcommons.butler.edu/wordways/vol26/iss2/2/ .
57. ↑ Gódor, Balázs (2006). "World-wide user identification in seven characters with unique number mapping" (in en). 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. "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" (in en). International Journal on Natural Language Computing 5 (4): 01–21. doi:10.5121/ijnlc.2016.5401. https://www.academia.edu/28219615. "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. ↑ "NewBase60". http://tantek.pbworks.com/w/page/19402946/NewBase60.
60. ↑ Nasar, Sylvia (2001). A Beautiful Mind. Simon and Schuster. pp. 333–6. ISBN 0-7432-2457-4. https://archive.org/details/beautifulmindli00nasa.
61. ↑ Ward, Rachel (2008), "On Robustness Properties of Beta Encoders and Golden Ratio Encoders", IEEE Transactions on Information Theory 54 (9): 4324–4334, doi:10.1109/TIT.2008.928235, Bibcode: 2008arXiv0806.1083W
62. ↑ Chrisomalis 2010, p. 254: Chrisomalis calls the Babylonian system "the first positional system ever".

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