List of numeral systems
Numeral systems |
---|
Hindu–Arabic numeral system |
East Asian |
Alphabetic |
Former |
Positional systems by base |
Non-standard positional numeral systems |
List of numeral systems |
This is a list of numeral systems, that is, writing systems for expressing numbers.
By culture / time period
Name | Base | Sample | Approx. first appearance | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Prehistoric numerals | 35,000 BC | ||||||||||||||||||
Babylonian numerals | 60 | 3100 BC | |||||||||||||||||
Egyptian numerals | 10 |
or
| 3000 BC | ||||||||||||||||
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 | c1500 BC | ||||||||||||||||
Chinese numerals, Japanese numerals, Korean numerals (Sino-Korean) | 10 | 〇/零 一 二 三 四 五 六 七 八 九 十 | |||||||||||||||||
Roman numerals | 10 | N Ⅰ Ⅱ Ⅲ Ⅳ Ⅴ Ⅵ Ⅶ Ⅷ Ⅸ Ⅹ L C D M | 1000 BC | ||||||||||||||||
Hebrew numerals | 10 | א ב ג ד ה ו ז ח ט י כ ל מ נ ס ע פ צ | 800 BC | ||||||||||||||||
Indian Numerals | 10 | Tamil ௦ ௧ ௨ ௩ ௪ ௫ ௬ ௭ ௮ ௯
Devanagari 0 १ २ ३ ४ ५ ६ ७ ८ ९ |
750 BC – 690 BC | ||||||||||||||||
Greek numerals | 10 | ō α β γ δ ε ϝ ζ η θ ι ο Αʹ Βʹ Γʹ Δʹ Εʹ Ϛʹ Ζʹ Ηʹ Θʹ | Before 5th century BC | ||||||||||||||||
Chinese rod numerals | 10 | 𝍠 𝍡 𝍢 𝍣 𝍤 𝍥 𝍦 𝍧 𝍨 𝍩 | 1st century | ||||||||||||||||
Phoenician numerals | 10 | 𐤙 𐤘 𐤗 𐤛𐤛𐤛 𐤛𐤛𐤚 𐤛𐤛𐤖 𐤛𐤛 𐤛𐤚 𐤛𐤖 𐤛 𐤚 𐤖 [1] | Before 250 AD[2] | ||||||||||||||||
Thai numerals | 10 | ๐ ๑ ๒ ๓ ๔ ๕ ๖ ๗ ๘ ๙ | 7th century[3] | ||||||||||||||||
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 | before 8th century | ||||||||||||||||
Eastern Arabic numerals | 10 | ٩ ٨ ٧ ٦ ٥ ٤ ٣ ٢ ١ ٠ | 8th century | ||||||||||||||||
Western Arabic numerals | 10 | 0 1 2 3 4 5 6 7 8 9 | 9th century | ||||||||||||||||
Burmese numerals | 10 | ၀ ၁ ၂ ၃ ၄ ၅ ၆ ၇ ၈ ၉ | 11th century[4] | ||||||||||||||||
Maya numerals | 20 | <15th century | |||||||||||||||||
Muisca numerals | 20 | <15th century | |||||||||||||||||
Aztec numerals | 20 | 16th century | |||||||||||||||||
John Napier's Location arithmetic | 2 | a b ab c ac bc abc d ad bd abd cd acd bcd abcd | 1617 in Rabdology, a non-positional binary system |
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
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.[5]
Base | Name | Usage |
---|---|---|
2 | Binary | Digital computing |
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 and Hilbert curves; Chumashan languages, and Kharosthi numerals |
5 | Quinary | Gumatj, 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 |
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 |
10 | Decimal | Most widely used by modern civilizations[6][7][8] |
11 | Undecimal | Jokingly proposed during the French Revolution to settle a dispute between those proposing a shift to duodecimal and those who were content with decimal |
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; Telephone number |
13 | Tridecimal | Conway base 13 function |
14 | Tetradecimal | Programming for the HP 9100A/B calculator[9] and image processing applications[10] |
15 | Pentadecimal | Telephony routing over IP, and the Huli language |
16 | Hexadecimal | Base16 encoding; compact notation for binary data; tonal system; ounce and pound |
18 | Octodecimal | Base18 encoding |
20 | Vigesimal | Basque, Celtic, Maya, Muisca, Inuit, Yoruba, Tlingit, and Dzongkha numerals; Santali, and Ainu languages |
23 | Trivigesimal | Kalam language, Kobon language |
24 | Tetravigesimal | 24-hour clock timekeeping; Kaugel language |
26 | Hexavigesimal | Base 26 encoding; sometimes used for encryption or ciphering.[11] |
27 | Heptavigesimal | Telefol and Oksapmin 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,[12] to provide a concise encoding of alphabetic strings,[13] or as the basis for a form of gematria.[14] |
30 | Trigesimal | The Natural Area Code |
32 | Duotrigesimal | Base32 encoding and the Ngiti language |
33 | Tritrigesimal | Use of letters (except I, O, Q) with digits in vehicle registration plates of Hong Kong |
36 | Hexatrigesimal | Base36 encoding; use of letters with digits |
52 | Duoquinquagesimal | Base52 encoding, a variant of Base62 without vowels[15] |
56 | Hexaquinquagesimal | Base56 encoding, a variant of Base58[16] |
57 | Heptaquinquagesimal | Base57 encoding, a variant of Base62 excluding I, O, l, U, and u[17] |
58 | Octoquinquagesimal | Base58 encoding |
60 | Sexagesimal | Babylonian numerals; NewBase60 encoding, similar to Base62, excluding I, O, and l, but including _(underscore);[18]degrees-minutes-seconds and hours-minutes-seconds measurement systems; Ekari and Sumerian languages |
61 | Unsexagesimal | NewBase61 encoding, variant of NewBase60 with a space[19] |
62 | Duosexagesimal | Base62 encoding, using 0-9, A-Z, and a-z[20] |
64 | Tetrasexagesimal | Base64 encoding; I Ching in China |
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. |
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.[21] |
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.[22] |
94 | Tetranonagesimal | Base94 encoding, using all of ASCII printable characters.[23] |
95 | Pentanonagesimal | Base95 encoding, a variant of Base94 with the addition of the Space character.[24] |
256 | Ducentahexaquinquagesimal | Base256 encoding |
Non-standard positional numeral systems
Bijective numeration
Base | Name | Usage |
---|---|---|
1 | Unary (Bijective base-1) | Tally marks |
10 | Bijective base-10 | |
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.[25] |
Signed-digit representation
Base | Name | Usage |
---|---|---|
2 | Balanced binary (Non-adjacent form) | |
3 | Balanced ternary | Ternary computers |
5 | Balanced quinary | |
9 | Balanced nonary | |
10 | Balanced decimal | John Colson Augustin Cauchy |
Negative bases
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 | |
−10 | Negadecimal |
Complex bases
Base | Name | Usage |
---|---|---|
2i | Quater-imaginary base | |
−1 ± i | Twindragon base | Twindragon fractal shape |
Non-integer bases
Base | Name | Usage |
---|---|---|
φ | Golden ratio base | Early Beta encoder[26] |
e | Base | Lowest radix economy |
Other
Non-positional notation
All known numeral systems developed before the Babylonian numerals are non-positional.[27]
Trivia
- In this Youtube video, Matt Parker jokingly invented a base-1082 system (From a Fermi estimate of the number of atoms in the universe). He uses this numeral system to describe how many atom carry-overs (i.e. number of parallel universes, or digits in this radix) it takes to have a hypothetical supercomputer generate all possible 256×256 grayscale images, when the root universe lasts 1017 seconds (another Fermi estimate). This turns out to be 1925.
See also
- Radix
- Radix economy
- Table of bases
- List of numeral system topics
- List of numbers in various languages (cardinal number names)
- Numeral prefix
References
- ↑ 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.
- ↑ Cajori, Florian (Sep 1928). A History Of Mathematical Notations Vol I. The Open Court Company. p. 18. Retrieved 5 June 2017.
- ↑ Chrisomalis, Stephen (2010). Numerical Notation: A Comparative History. Cambridge University Press. p. 200. ISBN 9780521878180.
- ↑ "Burmese/Myanmar script and pronunciation". Omniglot. Retrieved 5 June 2017.
- ↑ For the mixed roots of the word "hexadecimal", see Epp, Susanna (2010), Discrete Mathematics with Applications (4th ed.), Cengage Learning, p. 91, ISBN 9781133168669.
- ↑ The History of Arithmetic, Louis Charles Karpinski, 200pp, Rand McNally & Company, 1925.
- ↑ Histoire universelle des chiffres, Georges Ifrah, Robert Laffont, 1994.
- ↑ 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
- ↑ HP Museum
- ↑ Free Patents Online
- ↑ http://www.dcode.fr/base-26-cipher
- ↑ Grannis, Shaun J.; Overhage, J. Marc; McDonald, Clement J. (2002), "Analysis of identifier performance using a deterministic linkage algorithm", Proc AMIA Symp., pp. 305–309, PMC 2244404 , PMID 12463836.
- ↑ Stephens, Kenneth Rod (1996), Visual Basic Algorithms: A Developer's Sourcebook of Ready-to-run Code, Wiley, p. 215, ISBN 9780471134183.
- ↑ Sallows, Lee (1993), "Base 27: the key to a new gematria", Word Ways, 26 (2): 67–77.
- ↑ "Base52". Retrieved 2016-01-03.
- ↑ "Base56". Retrieved 2016-01-03.
- ↑ "Base57". Retrieved 2016-01-03.
- ↑ "NewBase60". Retrieved 2016-01-03.
- ↑ "NewBase61". Retrieved 2016-01-03.
- ↑ https://www.kerstner.at/2012/07/shortening-strings-using-base-62-encoding/. Missing or empty
|title=
(help) - ↑ "Base92". Retrieved 2016-01-03.
- ↑ "Base93". Retrieved 2017-02-13.
- ↑ "Base94". Retrieved 2016-01-03.
- ↑ "base95 Numeric System". Retrieved 2016-01-03.
- ↑ Nasar, Sylvia (2001). A Beautiful Mind. Simon and Schuster. pp. 333–6. ISBN 0-7432-2457-4.
- ↑ 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
- ↑ 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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