List of numeral systems

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Numeral systems
Numeral Systems of the World.svg
Hindu–Arabic numeral system
East Asian
Alphabetic
Former
Positional systems by base
Non-standard positional numeral systems
List of numeral systems

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 numerals 10&60 -3500 c. 3500–2000 BCE


Indus numerals -3501 c. 3500–1900 BCE


Proto-Elamite numerals 10&60 -3101 3,100 BCE


Sumerian numerals 10&60 -3100 3,100 BCE


Egyptian numerals 10 <hiero size=10>Z1 V20 V1 M12 D50 I8 I7 C11</hiero> -3000 3,000 BCE


Babylonian numerals 10&60 Babylonian 1.svg 15px 15px 15px 15px 15px 15px 15px 15px Babylonian 10.svg -2000 2,000 BCE


Aegean numerals 10 𐄇 𐄈 𐄉 𐄊 𐄋 𐄌 𐄍 𐄎 𐄏  ( 1 10px|frameless|2 10px|frameless|3 10px|frameless|4 10px|frameless|5 10px|frameless|6 10px|frameless|7 10px|frameless|8 10px|frameless|9 )
𐄐 𐄑 𐄒 𐄓 𐄔 𐄕 𐄖 𐄗 𐄘  ( 10px|frameless|10 10px|frameless|20 10px|frameless|30 10px|frameless|40 10px|frameless|50 10px|frameless|60 10px|frameless|70 10px|frameless|80 10px|frameless|90 )
𐄙 𐄚 𐄛 𐄜 𐄝 𐄞 𐄟 𐄠 𐄡  ( 10px|frameless|100 10px|frameless|200 10px|frameless|300 10px|frameless|400 10px|frameless|500 10px|frameless|600 10px|frameless|700 10px|frameless|800 10px|frameless|900 )
𐄢 𐄣 𐄤 𐄥 𐄦 𐄧 𐄨 𐄩 𐄪  ( 10px|frameless|1000 10px|frameless|2000 10px|frameless|3000 10px|frameless|4000 10px|frameless|5000 10px|frameless|6000 10px|frameless|7000 10px|frameless|8000 10px|frameless|9000 )
𐄫 𐄬 𐄭 𐄮 𐄯 𐄰 𐄱 𐄲 𐄳  ( 10px|frameless|10000 10px|frameless|20000 10px|frameless|30000 10px|frameless|40000 10px|frameless|50000 10px|frameless|60000 10px|frameless|70000 10px|frameless|80000 90000 )		 -1500 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)

 -1300 1,300 BCE


Roman numerals I V X L C D M -1000 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		 -800 800 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


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 ፻		 200 3rd–4th Century


15th Century (Modern Style)[1]

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

[2]

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 699 <8th Century


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

[3]

Tangut numerals 10 Template:Tangut 1036 11th Century (1036)


Cistercian numerals 10 Cistercian numerals.svg 1200 13th Century


Maya numerals 5&20 0 maia.svg 15px 15px 15px 15px 15px 15px 15px 15px 15px 15px 15px 15px 15px 15px 15px 15px 15px 15px 19 maia.svg 1400 <15th Century


Muisca numerals 20 Muisca cyphers acc acosta humboldt zerda.svg 1399 <15th Century


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


Aztec numerals 20 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 Pentimal Runes 1 through 10.svg 1800 19th Century


Cherokee numerals 10 Cherokee Numbers – cropped (1-20).png 1820 19th Century (1820s)


Osmanya numerals 10 𐒠 𐒡 𐒢 𐒣 𐒤 𐒥 𐒦 𐒧 𐒨 𐒩 1921 20th Century (1920s)


Hmong numerals 10 𖭐 𖭑 𖭒 𖭓 𖭔 𖭕 𖭖 𖭗 𖭘 𖭙 1959 20th Century (1959)


Kaktovik numerals 5&20 𝋀 x20px|𝋁 x20px|𝋂 x20px|𝋃 x20px|𝋄 x20px|𝋅 x20px|𝋆 x20px|𝋇 x20px|𝋈 x20px|𝋉 x20px|𝋊 x20px|𝋋 x20px|𝋌 x20px|𝋍 x20px|𝋎 x20px|𝋏 x20px|𝋐 x20px|𝋑 x20px|𝋒 𝋓 1994 20th 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.[4] There have been some proposals for standardisation.[5]

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 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[6] 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[7][8][9]
11 Undecimal, unodecimal, undenary A base-11 number system was attributed to the Māori (New Zealand) in the 19th century[10] and the Pangwa (Tanzania) in the 20th century.[11] 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.[12][13][14] 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[15][16] Conway base 13 function.
14 Quattuordecimal, quadrodecimal[15][16] Programming for the HP 9100A/B calculator[17] and image processing applications;[18] pound and stone.
15 Quindecimal, pentadecimal[19][16] 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[19][16]
18 Octodecimal[19][16] A base in which 7n is palindromic for n = 3, 4, 6, 9.
19 Undevicesimal, nonadecimal[19][16]
20 Vigesimal Basque, Celtic, Muisca, Inuit, Yoruba, Tlingit, and Dzongkha numerals; Santali, and Ainu languages; shilling and pound
5&20 Quinary-vigesimal[20][21][22] 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"[20]
21 The smallest base in which all fractions 1/2 to 1/18 have periods of 4 or shorter.
24 Quadravigesimal[23] 24-hour clock timekeeping; Greek alphabet; Kaugel language.
25 Sometimes used as compact notation for quinary.
26 Hexavigesimal[23][24] Sometimes used for encryption or ciphering,[25] using all letters in the English alphabet
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[26][27] 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;[28] similar to base 26 but distinguishing upper- and lower-case letters.
56 A variant of base 58.[clarification needed][29]
57 Covers base 62 apart from I, O, l, U, and u,[30] or I, 1, l, 0, and O.[31]
58 Covers base 62 apart from 0 (zero), I (capital i), O (capital o) and l (lower case L).[32]
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).[33]
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.[34]
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

Bijective numeration

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.[35]

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
[math]\displaystyle{ i\sqrt{2} }[/math] Base [math]\displaystyle{ i\sqrt{2} }[/math] related to base −2 and base 4
[math]\displaystyle{ i \sqrt[4]{2} }[/math] Base [math]\displaystyle{ i \sqrt[4]{2} }[/math] related to base 2
[math]\displaystyle{ 2 \omega }[/math] Base [math]\displaystyle{ 2 \omega }[/math] related to base 8
[math]\displaystyle{ \omega \sqrt[3]{2} }[/math] Base [math]\displaystyle{ \omega \sqrt[3]{2} }[/math] 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
[math]\displaystyle{ \frac{3}{2} }[/math] Base [math]\displaystyle{ \frac{3}{2} }[/math] a rational non-integer base
[math]\displaystyle{ \frac{4}{3} }[/math] Base [math]\displaystyle{ \frac{4}{3} }[/math] related to duodecimal
[math]\displaystyle{ \frac{5}{2} }[/math] Base [math]\displaystyle{ \frac{5}{2} }[/math] related to decimal
[math]\displaystyle{ \sqrt{2} }[/math] Base [math]\displaystyle{ \sqrt{2} }[/math] related to base 2
[math]\displaystyle{ \sqrt{3} }[/math] Base [math]\displaystyle{ \sqrt{3} }[/math] related to base 3
[math]\displaystyle{ \sqrt[3]{2} }[/math] Base [math]\displaystyle{ \sqrt[3]{2} }[/math]
[math]\displaystyle{ \sqrt[4]{2} }[/math] Base [math]\displaystyle{ \sqrt[4]{2} }[/math]
[math]\displaystyle{ \sqrt[12]{2} }[/math] Base [math]\displaystyle{ \sqrt[12]{2} }[/math] usage in 12-tone equal temperament musical system
[math]\displaystyle{ 2\sqrt{2} }[/math] Base [math]\displaystyle{ 2\sqrt{2} }[/math]
[math]\displaystyle{ -\frac{3}{2} }[/math] Base [math]\displaystyle{ -\frac{3}{2} }[/math] a negative rational non-integer base
[math]\displaystyle{ -\sqrt{2} }[/math] Base [math]\displaystyle{ -\sqrt{2} }[/math] a negative non-integer base, related to base 2
[math]\displaystyle{ \sqrt{10} }[/math] Base [math]\displaystyle{ \sqrt{10} }[/math] related to decimal
[math]\displaystyle{ 2\sqrt{3} }[/math] Base [math]\displaystyle{ 2\sqrt{3} }[/math] related to duodecimal
φ Golden ratio base Early Beta encoder[36]
ρ Plastic number base
ψ Supergolden ratio base
[math]\displaystyle{ 1+\sqrt{2} }[/math] Silver ratio base
e Base [math]\displaystyle{ e }[/math] Lowest radix economy
π Base [math]\displaystyle{ \pi }[/math]
eπ Base [math]\displaystyle{ e\pi }[/math]
[math]\displaystyle{ e^\pi }[/math] Base [math]\displaystyle{ e^\pi }[/math]

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

Non-positional notation

All known numeral systems developed before the Babylonian numerals are non-positional,[37] as are many developed later, such as the Roman numerals. The French Cistercian monks created their own numeral system.

See also


References

  1. ↑ Chrisomalis, Stephen (2010-01-18) (in en). Numerical Notation: A Comparative History. Cambridge University Press. pp. 135–136. ISBN 978-0-521-87818-0. https://books.google.com/books?id=ux--OWgWvBQC&pg=PA135. 
  2. ↑ Chrisomalis 2010, p. 200.
  3. ↑ "Burmese/Myanmar script and pronunciation". http://www.omniglot.com/writing/burmese.htm. 
  4. ↑ 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 .
  5. ↑ Multiplication Tables of Various Bases, p. 45, Michael Thomas de Vlieger, Dozenal Society of America
  6. ↑ "Definition of SEPTENARY" (in en). https://www.merriam-webster.com/dictionary/septenary. 
  7. ↑ The History of Arithmetic, Louis Charles Karpinski, 200pp, Rand McNally & Company, 1925.
  8. ↑ Histoire universelle des chiffres, Georges Ifrah, Robert Laffont, 1994.
  9. ↑ 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
  10. ↑ 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. 
  11. ↑ 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. 
  12. ↑ 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. 
  13. ↑ 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. 
  14. ↑ 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...". 
  15. ↑ 15.0 15.1 Das & Lanjewar 2012, p. 13.
  16. ↑ 16.0 16.1 16.2 16.3 16.4 16.5 Rawat & Sah 2013.
  17. ↑ HP 9100A/B programming, HP Museum
  18. ↑ Free Patents Online
  19. ↑ 19.0 19.1 19.2 19.3 Das & Lanjewar 2012, p. 14.
  20. ↑ 20.0 20.1 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.". 
  21. ↑ 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.' ..." 
  22. ↑ 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.".
  23. ↑ 23.0 23.1 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." 
  24. ↑ Young, Brian; Faris, Tom; Armogida, Luigi (2019). "A nomenclature for sequence-based forensic DNA analysis" (in en). Genetics (Forensic Science International) 42: 14–20. 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 […]". 
  25. ↑ "Base 26 Cipher (Number ⬌ Words) - Online Decoder, Encoder". http://www.dcode.fr/base-26-cipher. 
  26. ↑ 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. https://ieeexplore.ieee.org/document/4082444. "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..." 
  27. ↑ Balagadde1, Robert Ssali; Premchand, Parvataneni (2016). "The Structured Compact Tag-Set for Luganda" (in en). International Journal on Natural Language Computing (IJNLC) 5 (4). 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.". 
  28. ↑ "Base52". https://github.com/atomaka/base52. 
  29. ↑ "Base56". https://gist.github.com/csnover/1110664. 
  30. ↑ "Base57". https://github.com/wyattisimo/base57-ruby. 
  31. ↑ "Base57". https://github.com/skorokithakis/shortuuid. 
  32. ↑ "The Base58 Encoding Scheme". November 27, 2019. https://tools.ietf.org/id/draft-msporny-base58. ""Thanks to Satoshi Nakamoto for inventing the Base58 encoding format"" 
  33. ↑ "NewBase60". http://tantek.pbworks.com/w/page/19402946/NewBase60. 
  34. ↑ "base95 Numeric System". http://www.icerealm.org/FTR/?s=docs&p=base95. 
  35. ↑ Nasar, Sylvia (2001). A Beautiful Mind. Simon and Schuster. pp. 333–6. ISBN 0-7432-2457-4. https://archive.org/details/beautifulmindli00nasa. 
  36. ↑ 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 
  37. ↑ Chrisomalis 2010, p. 254: Chrisomalis calls the Babylonian system "the first positional system ever".



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