Part of a series on |
Numeral systems |
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List of numeral systems |
There are many different numeral systems, that is, writing systems for expressing numbers.
Name | Base | Sample | Approx. First Appearance | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Proto-cuneiform numerals | 10+60 | c. 3500–2000 BCE | ||||||||||
Indus numerals | c. 3500–1900 BCE | |||||||||||
Proto-Elamite numerals | 10+60 | 3,100 BCE | ||||||||||
Sumerian numerals | 10+60 | 3,100 BCE | ||||||||||
Egyptian numerals | 10 |
|
3,000 BCE | |||||||||
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) |
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 | |||||||||
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 ௦ ௧ ௨ ௩ ௪ ௫ ௬ ௭ ௮ ௯ Malayalam ൦ ൧ ൨ ൩ ൪ ൫ ൬ ൭ ൮ ൯ Kannada ೦ ೧ ೨ ೩ ೪ ೫ ೬ ೭ ೮ ೯ Telugu ౦ ౧ ౨ ౩ ౪ ౫ ౬ ౭ ౮ ౯ Odia ୦ ୧ ୨ ୩ ୪ ୫ ୬ ୭ ୮ ୯ Bengali ০ ১ ২ ৩ ৪ ৫ ৬ ৭ ৮ ৯ Devanagari ० १ २ ३ ४ ५ ६ ७ ८ ९ Punjabi ੦ ੧ ੨ ੩ ੪ ੫ ੬ ੭ ੮ ੯ Gujarati ૦ ૧ ૨ ૩ ૪ ૫ ૬ ૭ ૮ ૯ Hindustani ۰ ۱ ۲ ۳ ۴ ۵ ۶ ۷ ۸ ۹ |
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 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 | |||||||||
Coptic numerals | 10 | Ⲁ Ⲃ Ⲅ Ⲇ Ⲉ Ⲋ Ⲍ Ⲏ Ⲑ | 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 |
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 | 영 일 이 삼 사 오 육 칠 팔 구 | 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 | |||||||||
Pentadic runes | 10 | ![]() |
19th Century | |||||||||
Cherokee numerals | 10 | 19th Century (1820s) | ||||||||||
Osmanya numerals | 10 | 𐒠 𐒡 𐒢 𐒣 𐒤 𐒥 𐒦 𐒧 𐒨 𐒩 | 20th Century (1920s) | |||||||||
Kaktovik numerals | 5+20 | 𝋀 𝋁 𝋂 𝋃 𝋄 𝋅 𝋆 𝋇 𝋈 𝋉 𝋊 𝋋 𝋌 𝋍 𝋎 𝋏 𝋐 𝋑 𝋒 𝋓 | 20th Century (1994) |
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.
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 | 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[citation needed] | 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 | Base 9 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 | Base 13 encoding; Conway base 13 function. |
14 | Tetradecimal[citation needed] | Programming for the HP 9100A/B calculator[13] and image processing applications;[14] pound and stone. |
15 | Pentadecimal[citation needed] | Telephony routing over IP, and the Huli language. |
16 | Hexadecimal
(also known as sexadecimal) |
Base 16 encoding; compact notation for binary data; tonal system; ounce and pound. |
17 | Heptadecimal[citation needed] | Base 17 encoding. |
18 | Octodecimal[citation needed] | Base 18 encoding; a base such that 7n is palindromic for n = 3, 4, 6, 9. |
19 | Enneadecimal[citation needed] | Base 19 encoding. |
20 | Vigesimal | Basque, Celtic, Maya, Muisca, Inuit, Yoruba, Tlingit, and Dzongkha numerals; Santali, and Ainu languages; shilling and pound |
21 | Unvigesimal[citation needed] | Base 21 encoding; also the smallest base where all of 1/2 to 1/18 have periods of 4 or shorter. |
22 | Duovigesimal[citation needed] | Base 22 encoding. |
23 | Trivigesimal[citation needed] | Kalam language,[15] Kobon language[citation needed] |
24 | Tetravigesimal[citation needed] | 24-hour clock timekeeping; Kaugel language. |
25 | Pentavigesimal[citation needed] | Base 25 encoding; sometimes used as compact notation for quinary. |
26 | Hexavigesimal[citation needed] | Base 26 encoding; sometimes used for encryption or ciphering,[16] using all letters in the English alphabet |
27 | Heptavigesimal,[citation needed] 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[citation needed] | Base 28 encoding; months timekeeping. |
29 | Enneavigesimal[citation needed] | Base 29 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[citation needed] | Base 31 encoding. |
32 | Duotrigesimal | Base 32 encoding; the Ngiti language. |
33 | Tritrigesimal[citation needed] | Use of letters (except I, O, Q) with digits in vehicle registration plates of Hong Kong. |
34 | Tetratrigesimal[citation needed] | 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[citation needed] | Using all numbers and all letters except O. |
36 | Hexatrigesimal[citation needed] | Base 36 encoding; use of letters with digits. |
37 | Heptatrigesimal[citation needed] | Base 37 encoding; using all numbers and all letters of the Spanish alphabet. |
38 | Octotrigesimal[citation needed] | Base 38 encoding; use all duodecimal digits and all letters. |
39 | Enneatrigesimal[citation needed] | Base 39 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[citation needed] | Base 42 encoding; largest base for which all minimal primes are known. |
45 | Pentaquadragesimal[citation needed] | Base 45 encoding. |
47 | Septaquadragesimal[citation needed] | Smallest base for which no generalized Wieferich primes are known. |
48 | Octoquadragesimal[citation needed] | Base 48 encoding. |
49 | Enneaquadragesimal[citation needed] | Compact notation for septenary. |
50 | Quinquagesimal | Base 50 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[citation needed] | Base 52 encoding, a variant of base 62 without vowels except Y and y[22] or a variant of base 26 using all lower and upper case letters. |
54 | Tetraquinquagesimal[citation needed] | Base 54 encoding. |
56 | Hexaquinquagesimal[citation needed] | Base 56 encoding, a variant of base 58.[23] |
57 | Heptaquinquagesimal[citation needed] | Base 57 encoding, a variant of base 62 excluding I, O, l, U, and u[24] or I, 1, l, 0, and O.[25] |
58 | Octoquinquagesimal[citation needed] | Base 58 encoding, a variant of base 62 excluding 0 (zero), I (capital i), O (capital o) and l (lower case L).[26] |
60 | Sexagesimal | Babylonian numerals; New base 60 encoding, similar to base 62, excluding I, O, and l, but including _(underscore);[27] degrees-minutes-seconds and hours-minutes-seconds measurement systems; Ekari and Sumerian |
62 | Duosexagesimal[citation needed] | Base 62 encoding, using 0–9, A–Z, and a–z. |
64 | Tetrasexagesimal | Base 64 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[citation needed] | Base72 encoding; the smallest base >2 such that no three-digit narcissistic number exists. |
80 | Octogesimal | Base80 encoding; Supyire as a sub-base. |
81 | Unoctogesimal[citation needed] | Base 81 encoding, using as 81=34 is related to ternary. |
85 | Pentoctogesimal[citation needed] | 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[citation needed] | 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[citation needed] | Base 91 encoding, using all ASCII except "-" (0x2D), "\" (0x5C), and "'" (0x27); one variant uses "\" (0x5C) in place of """ (0x22). |
92 | Duononagesimal[citation needed] | Base 92 encoding, using all of ASCII except for "`" (0x60) and """ (0x22) due to confusability.[28] |
93 | Trinonagesimal[citation needed] | Base 93 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[citation needed] | Base 94 encoding, using all of ASCII printable characters.[30] |
95 | Pentanonagesimal[citation needed] | Base 95 encoding, a variant of base 94 with the addition of the Space character.[31] |
96 | Hexanonagesimal[citation needed] | Base 96 encoding, using all of ASCII printable characters as well as the two extra duodecimal digits. |
97 | Septanonagesimal[citation needed] | 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[citation needed] | Base 120 encoding. |
121 | Centeunvigesimal[citation needed] | Related to base 11. |
125 | Centepentavigesimal[citation needed] | Related to base 5. |
128 | Centeoctovigesimal[citation needed] | Using as 128=27. |
144 | Centetetraquadragesimal[citation needed] | Two duodecimal digits. |
169 | Centenovemsexagesimal[citation needed] | Two Tridecimal digits. |
185 | Centepentoctogesimal[citation needed] | Smallest base which is not perfect power (where generalized repunits can be factored algebraically) for which no generalized repunit primes are known. |
196 | Centehexanonagesimal[citation needed] | Two tetradecimal digits. |
200 | Duocentesimal[citation needed] | Base 200 encoding. |
210 | Duocentedecimal[citation needed] | Smallest base such that all of 1/2 to 1/10 terminate. |
216 | Duocentehexidecimal[citation needed] | related to base 6. |
225 | Duocentepentavigesimal[citation needed] | Two pentadecimal digits. |
256 | Duocentehexaquinquagesimal[citation needed] | Base 256 encoding, as 256=28. |
300 | Trecentesimal[citation needed] | Base 300 encoding. |
360 | Trecentosexagesimal[citation needed] | Degrees for angle. |
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] |
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 |
The common names of the negative base numeral systems are formed using the prefix nega-, giving names such as:[citation needed]
Base | Name | Usage |
---|---|---|
−2 | Negabinary | |
−3 | Negaternary | |
−4 | Negaquaternary | |
−5 | Negaquinary | |
−6 | Negasenary | |
−8 | Negaoctal | |
−10 | Negadecimal | |
−12 | Negaduodecimal | |
−16 | Negahexadecimal |
Base | Name | Usage |
---|---|---|
2i | Quater-imaginary base | related to base −4 and base 16 |
Base | related to base −2 and base 4 | |
Base | related to base 2 | |
Base | related to base 8 | |
Base | 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 |
Base | Name | Usage |
---|---|---|
Base | a rational non-integer base | |
Base | related to duodecimal | |
Base | related to decimal | |
Base | related to base 2 | |
Base | related to base 3 | |
Base | ||
Base | ||
Base | usage in 12-tone equal temperament musical system | |
Base | ||
Base | a negative rational non-integer base | |
Base | a negative non-integer base, related to base 2 | |
Base | related to decimal | |
Base | related to duodecimal | |
φ | Golden ratio base | Early Beta encoder[33] |
ρ | Plastic number base | |
ψ | Supergolden ratio base | |
Silver ratio base | ||
e | Base | Lowest radix economy |
π | Base | |
eπ | Base | |
Base |
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 |
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.