# List of numeral systems

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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
--- | --- | --- | ---
Proto-cuneiform numerals |  |  |
Proto-Elamite numerals |  |  |
Sumerian numerals | 10+60 |  | 3,100 BCE
Egyptian numerals | 10 |
| 3,000 BCE
Elamite numerals |  |  |
Indus numerals |  |  |
Babylonian numerals | 10+60 |  | 2,000 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) | 1,600 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
Bengali numerals | 10 | ০ ১ ২ ৩ ৪ ৫ ৬ ৭ ৮ ৯ | 1,400 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 ௧ ௨ ௩ ௪ ௫ ௬ ௭ ௮ ௯ ௰ Devanagari ० १ २ ३ ४ ५ ६ ७ ८ ९ Tibetan ༠ ༡ ༢ ༣ ༤ ༥ ༦ ༧ ༨ ༩ | 750 – 690 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
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 | 하나 둘 셋 넷 다섯 여섯 일곱 여덟 아홉 열 | 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
Pentimal runes | 10 |  | 19th Century
Cherokee numerals | 10 |  | 19th Century (1820s)
Kaktovik Inupiaq 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 place-value notation), and further categorized by radix or base.

### Standard positional numeral systems[edit]

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]

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 | 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 | Data transmission, DNA bases and Hilbert curves; 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 Proto-Uralic language (suspected)
7 | Septenary | Weeks timekeeping, Western music letter notation
8 | Octal | Charles XII of Sweden, Unix-like permissions, Squawk codes, DEC PDP-11, compact notation for binary numbers, Xiantian (I Ching, China)
9 | Nonary | Base9 encoding; compact notation for ternary
10 | Decimal (also known as denary) | Most widely used by modern civilizations[8][9][10]
11 | Undecimal | A base-11 number system was attributed to the Māori (New Zealand) in the 19th century[11] and the Pangwa (Tanzania) in the 20th century.[12] 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.
12 | Duodecimal | 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 | Tridecimal | Base13 encoding; Conway base 13 function.
14 | Tetradecimal | Programming for the HP 9100A/B calculator[13] and image processing applications;[14] pound and stone.
15 | Pentadecimal | Telephony routing over IP, and the Huli language.
16 | Hexadecimal (also known as sexadecimal) | Base16 encoding; compact notation for binary data; tonal system; ounce and pound.
17 | Heptadecimal | Base17 encoding.
18 | Octodecimal | Base18 encoding; a base such that 7n is palindromic for n = 3, 4, 6, 9.
19 | Enneadecimal | Base19 encoding.
20 | Vigesimal | Basque, Celtic, Maya, Muisca, Inuit, Yoruba, Tlingit, and Dzongkha numerals; Santali, and Ainu languages; shilling and pound
21 | Unvigesimal | Base21 encoding; also the smallest base where all of 1/2 to 1/18 have periods of 4 or shorter.
22 | Duovigesimal | Base22 encoding.
23 | Trivigesimal | Kalam language,[15] Kobon language[citation needed]
24 | Tetravigesimal | 24-hour clock timekeeping; Kaugel language.
25 | Pentavigesimal | Compact notation for quinary.
26 | Hexavigesimal | Base26 encoding; sometimes used for encryption or ciphering,[16] using all letters in the English alphabet
27 | Heptavigesimal Septemvigesimal | Telefol[17] and Oksapmin[18] 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,[19] to provide a concise encoding of alphabetic strings,[20] or as the basis for a form of gematria.[21] Compact notation for ternary.
28 | Octovigesimal | Base28 encoding; months timekeeping.
29 | Enneavigesimal | Base29 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 | Untrigesimal | Base31 encoding.
32 | Duotrigesimal | Base32 encoding; the Ngiti language.
33 | Tritrigesimal | Use of letters (except I, O, Q) with digits in vehicle registration plates of Hong Kong.
34 | Tetratrigesimal | 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 | Pentatrigesimal | Using all numbers and all letters except O.
36 | Hexatrigesimal | Base36 encoding; use of letters with digits.
37 | Heptatrigesimal | Base37 encoding; using all numbers and all letters of the Spanish alphabet.
38 | Octotrigesimal | Base38 encoding; use all duodecimal digits and all letters.
39 | Enneatrigesimal | Base39 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 | Duoquadragesimal | Base42 encoding; largest base for which all minimal primes are known.
45 | Pentaquadragesimal | Base45 encoding.
47 | Septaquadragesimal | Smallest base for which no generalized Wieferich primes are known.
48 | Octoquadragesimal | Base48 encoding.
49 | Enneaquadragesimal | Compact notation for septenary.
50 | Quinquagesimal | Base50 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 | Duoquinquagesimal | Base52 encoding, a variant of Base62 without vowels except Y and y[22] or a variant of Base26 using all lower and upper case letters.
54 | Tetraquinquagesimal | Base54 encoding.
56 | Hexaquinquagesimal | Base56 encoding, a variant of Base58.[23]
57 | Heptaquinquagesimal | Base57 encoding, a variant of Base62 excluding I, O, l, U, and u[24] or I, 1, l, 0, and O.[25]
58 | Octoquinquagesimal | Base58 encoding, a variant of Base62 excluding 0 (zero), I (capital i), O (capital o) and l (lower case L).[26]
60 | Sexagesimal | Babylonian numerals; NewBase60 encoding, similar to Base62, excluding I, O, and l, but including _(underscore);[27] degrees-minutes-seconds and hours-minutes-seconds measurement systems; Ekari and Sumerian languages.
62 | Duosexagesimal | Base62 encoding, using 0–9, A–Z, and a–z.
64 | Tetrasexagesimal | Base64 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 | Duoseptuagesimal | Base72 encoding; the smallest base >2 such that no three-digit narcissistic number exists.
80 | Octogesimal | Base80 encoding.
81 | Unoctogesimal | Base81 encoding, using as 81=34 is related to ternary.
85 | Pentoctogesimal | 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 | Enneaoctogesimal | 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).
91 | Unnonagesimal | Base91 encoding, using all ASCII except "-" (0x2D), "\" (0x5C), and "'" (0x27); one variant uses "\" (0x5C) in place of """ (0x22).
92 | Duononagesimal | Base92 encoding, using all of ASCII except for "`" (0x60) and """ (0x22) due to confusability.[28]
93 | Trinonagesimal | Base93 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.[29]
94 | Tetranonagesimal | Base94 encoding, using all of ASCII printable characters.[30]
95 | Pentanonagesimal | Base95 encoding, a variant of Base94 with the addition of the Space character.[31]
96 | Hexanonagesimal | Base96 encoding, using all of ASCII printable characters as well as the two extra duodecimal digits.
97 | Septanonagesimal | 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.
120 | Centevigesimal | Base120 encoding.
121 | Centeunvigesimal | Related to base 11.
125 | Centepentavigesimal | Related to base 5.
128 | Centeoctovigesimal | Using as 128=27.
144 | Centetetraquadragesimal | Two duodecimal digits.
169 | Centenovemsexagesimal | Two Tridecimal digits.
185 | Centepentoctogesimal | Smallest base which is not perfect power (where generalized repunits can be factored algebraically) for which no generalized repunit primes are known.
196 | Centehexanonagesimal | Two tetradecimal digits.
200 | Duocentesimal | Base200 encoding.
210 | Duocentedecimal | Smallest base such that all of 1/2 to 1/10 terminate.
216 | Duocentehexidecimal | related to base 6.
225 | Duocentepentavigesimal | Two pentadecimal digits.
256 | Duocentehexaquinquagesimal | Base256 encoding, as 256=28.
300 | Trecentesimal | Base300 encoding.
360 | Trecentosexagesimal | Degrees for angle.

### Non-standard positional numeral systems[edit]

#### Bijective numeration[edit]

Base | Name | Usage
--- | --- | ---
1 | Unary (Bijective base‑1) | Tally marks, Counting
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.[32]

#### Signed-digit representation[edit]

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 |

#### Negative bases[edit]

The common names of the negative base numeral systems are formed using the prefix nega-, giving names such as:

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

#### Non-integer bases[edit]

Base | Name | Usage
--- | --- | ---
3 2 {\displaystyle {\frac {3}{2}}} | Base 3 2 {\displaystyle {\frac {3}{2}}} | a rational non-integer base
4 3 {\displaystyle {\frac {4}{3}}} | Base 4 3 {\displaystyle {\frac {4}{3}}} | related to duodecimal
5 2 {\displaystyle {\frac {5}{2}}} | Base 5 2 {\displaystyle {\frac {5}{2}}} | related to decimal
2 {\displaystyle {\sqrt {2}}} | Base 2 {\displaystyle {\sqrt {2}}} | related to base 2
3 {\displaystyle {\sqrt {3}}} | Base 3 {\displaystyle {\sqrt {3}}} | related to base 3
2 3 {\displaystyle {\sqrt[{3}]{2}}} | Base 2 3 {\displaystyle {\sqrt[{3}]{2}}} |
2 4 {\displaystyle {\sqrt[{4}]{2}}} | Base 2 4 {\displaystyle {\sqrt[{4}]{2}}} |
2 12 {\displaystyle {\sqrt[{12}]{2}}} | Base 2 12 {\displaystyle {\sqrt[{12}]{2}}} | usage in 12-tone equal temperament musical system
2 2 {\displaystyle 2{\sqrt {2}}} | Base 2 2 {\displaystyle 2{\sqrt {2}}} |
− 3 2 {\displaystyle -{\frac {3}{2}}} | Base − 3 2 {\displaystyle -{\frac {3}{2}}} | a negative rational non-integer base
− 2 {\displaystyle -{\sqrt {2}}} | Base − 2 {\displaystyle -{\sqrt {2}}} | a negative non-integer base, related to base 2
10 {\displaystyle {\sqrt {10}}} | Base 10 {\displaystyle {\sqrt {10}}} | related to decimal
2 3 {\displaystyle 2{\sqrt {3}}} | Base 2 3 {\displaystyle 2{\sqrt {3}}} | related to duodecimal
φ | Golden ratio base | Early Beta encoder[33]
ρ | Plastic number base |
ψ | Supergolden ratio base |
1 + 2 {\displaystyle 1+{\sqrt {2}}} | Silver ratio base |
e | Base e {\displaystyle e} | Lowest radix economy
π | Base π {\displaystyle \pi } |
e π | Base e π {\displaystyle e\pi } |
e π {\displaystyle e^{\pi }} | Base e π {\displaystyle e^{\pi }} |

#### n-adic 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 base-n notation
- Asymmetric numeral systems optimized for non-uniform probability distribution of symbols
- Combinatorial number system

### Non-positional notation[edit]

All known numeral systems developed before the Babylonian numerals are non-positional,[34] as are many developed later, such as the Roman numerals. The French Cistercian monks created their own numeral system.

## See also[edit]

- List of numbers in various languages (cardinal number names)
- List of numeral system topics
- Numeral prefix
- Radix
- Radix economy
- Table of bases

## References[edit]

1. ^
   Everson, Michael (2007-07-25). "Proposal to add two numbers for the Phoenician script" (PDF). UTC Document Register. L2/07-206 (WG2 N3284): Unicode Consortium.`{{cite web}}`: CS1 maint: location (link)
2. ^ Cajori, Florian (Sep 1928). A History Of Mathematical Notations Vol I. The Open Court Company. p. 18. Retrieved 5 June 2017.
3. ^ Chrisomalis, Stephen (2010-01-18). Numerical Notation: A Comparative History. ISBN 9781139485333.
4. ^ Chrisomalis, Stephen (2010). Numerical Notation: A Comparative History. Cambridge University Press. p. 200. ISBN 9780521878180.
5. ^ "Burmese/Myanmar script and pronunciation". Omniglot. Retrieved 5 June 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. ^ http://www.numberbases.com/terms/BaseNames.pdf[bare URL PDF]
8. ^ The History of Arithmetic, Louis Charles Karpinski, 200pp, Rand McNally & Company, 1925.
9. ^ Histoire universelle des chiffres, Georges Ifrah, Robert Laffont, 1994.
10. ^ 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
11. ^ 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 24 July 2020.
12. ^ Thomas, N.W (1920). "Duodecimal base of numeration". Man. 20 (1): 56–60. doi:10.2307/2840036. JSTOR 2840036. Retrieved 25 July 2020.
13. ^ HP 9100A/B programming, HP Museum
14. ^ Free Patents Online
15. ^ 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.
16. ^ "Base 26 Cipher (Number ⬌ Words) - Online Decoder, Encoder".
17. ^ 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.
18. ^ 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..
19. ^ 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.
20. ^ Stephens, Kenneth Rod (1996), Visual Basic Algorithms: A Developer's Sourcebook of Ready-to-run Code, Wiley, p. 215, ISBN 9780471134183.
21. ^ Sallows, Lee (1993), "Base 27: the key to a new gematria", Word Ways, 26 (2): 67–77.
22. ^ "Base52". Retrieved 2016-01-03.
23. ^ "Base56". Retrieved 2016-01-03.
24. ^ "Base57". Retrieved 2016-01-03.
25. ^ "Base57". Retrieved 2019-01-22.
26. ^ "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
27. ^ "NewBase60". Retrieved 2016-01-03.
28. ^ "Base92". Retrieved 2016-01-03.
29. ^ "Base93". 26 September 2013. Retrieved 2017-02-13.
30. ^ "Base94". Retrieved 2016-01-03.
31. ^ "base95 Numeric System". Retrieved 2016-01-03.
32. ^ Nasar, Sylvia (2001). A Beautiful Mind. Simon and Schuster. pp. 333–6. ISBN 0-7432-2457-4.
33. ^ 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
34. ^ Chrisomalis calls the Babylonian system "the first positional system ever" in Chrisomalis, Stephen (2010), Numerical Notation: A Comparative History, Cambridge University Press, p. 254, ISBN 9781139485333.

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