List of numeral systems
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Template:Short description Template:About Template:Use mdy dates Template:Contains special characters Template:Numeral systems
There are many different numeral systems, that is, writing systems for expressing numbers.
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]Template:Rp 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).
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By type of notation
Template:Anchor 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.[18] There have been some proposals for standardisation.[19]
| 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[20] | 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),[21] 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.[22][23][24] |
| 11 | Undecimal, unodecimal, undenary | A base-11 number system was mistakenly attributed to the Māori (New Zealand) in the 19th century[25] and one was reported to be used by the Pangwa (Tanzania) in the 20th century,[26] but was not confirmed by later research and is believed to also be an error.[27] 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.[28][29][30] |
| 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, tridecimalTemplate:SfnTemplate:Sfn | Conway's base 13 function. |
| 14 | Quattuordecimal, quadrodecimalTemplate:SfnTemplate:Sfn | Programming for the HP 9100A/B calculator[31] and image processing applications.[32] |
| 15 | Quindecimal, pentadecimalTemplate:SfnTemplate:Sfn | Telephony routing over IP, and the Huli language.[27] |
| 16 | Hexadecimal, sexadecimal, sedecimal | Compact notation for binary data; tonal system of Nystrom. |
| 17 | Heptadecimal, septendecimalTemplate:SfnTemplate:Sfn | |
| 18 | OctodecimalTemplate:SfnTemplate:Sfn | |
| 19 | Undevicesimal, nonadecimalTemplate:SfnTemplate:Sfn | |
| 20 | Vigesimal | Basque, Celtic, Muisca, Inuit, Yoruba, Tlingit, and Dzongkha numerals; Santali, and Ainu languages. |
| 5&20 | Quinary-vigesimal[33][34]Template:Sfn | 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."[33] |
| 21 | The smallest base in which all fractions Template:Sfrac to Template:Sfrac have periods of 4 or shorter. | |
| 23 | Kalam language,[35] | |
| 24 | Quadravigesimal[36] | 24-hour clock timekeeping; Greek alphabet; Kaugel language. |
| 25 | Compact notation for quinary. | |
| 26 | Hexavigesimal[36][37] | Sometimes used for encryption or ciphering,[38] 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)[39] and SID (sequence identification, an indexing system of PCR amplicons in forensics).[37] |
| 27 | Telefol,[35] Oksapmin,[40] Wambon,[41] and Hewa[42] 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,[43] to provide a concise encoding of alphabetic strings,[44] or as the basis for a form of gematria.[45] Compact notation for ternary. | |
| 30 | Trigesimal[46] | The Natural Area Code, this is the smallest base such that all of Template:Sfrac to Template:Sfrac terminate, a number n is a regular number if and only if Template:Sfrac 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 Template:IETF RFC, with each character standing for 5 bits. |
| 34 | The smallest base where all unitfractions from Template:Sfrac to Template:Sfrac have periods of 4 or shorter. | |
| 36 | Hexatrigesimal[47][48] | Used to encode large numbers into an alphanumeric string (26 letters, 10 numbers). Compact notation for senary. |
| 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).[49] |
| 64 | Used to encode computer (binary) data into a relatively compact string, with each character standing for 6 bits (Template:IETF RFC). | |
| 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 unit fractions Template:Sfrac to Template:Sfrac terminate. |
Unary, or bijective baseTemplate:Non breaking hyphen1, is used in Tally marks, and 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. Template:Cn
| Base | Name | Usage |
|---|---|---|
| 2 | Balanced binary (Non-adjacent form) | |
| 3 | Balanced ternary | Ternary computers |
| 10 | Balanced decimal | John Colson Augustin Cauchy |
| Base | Name | Usage |
|---|---|---|
| 2i | Quater-imaginary base | related to base −4 and base 16 |
| −1 ± i | Twindragon base | Twindragon fractal shape, related to base −4 and base 16 |
| Base | Name | Usage |
|---|---|---|
| φ | Golden ratio base | early Beta encoder[50] |
| e | Base | best radix economy Template:Citation needed |
- Factorial number system {1, 2, 3, 4, 5, 6, ...}
- Primorial number system {2, 3, 5, 7, 11, 13, ...}
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,Template:Sfn as are many developed later, such as the Roman numerals. The French Cistercian monks created their own numeral system.
See also
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References
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- ↑ Template:Cite webTemplate:Page needed
- ↑ For the mixed roots of the word "hexadecimal", see Template:Citation.
- ↑ Multiplication Tables of Various Bases, p. 45, Michael Thomas de Vlieger, Dozenal Society of America
- ↑ Template:Cite journal
- ↑ 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.
- ↑ 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, Template:Isbn, John Wiley and Sons Inc., New York, 2000. Translated from the French by David Bellos, E.F. Harding, Sophie Wood and Ian Monk
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- ↑ 27.0 27.1 Template:Cite book
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- ↑ HP 9100A/B programming, HP Museum
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- ↑ 33.0 33.1 Template:Cite journal
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- ↑ 35.0 35.1 Template:Cite book
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- ↑ Nauka i Zhizn, 1992, issue 3, p. 48.
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