There are many different numeral systems, that is, writing systems for expressing numbers.
"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] 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).
| Name | data-sort-type=number | Base | Sample | data-sort-type=number | Approx. First Appearance |
|---|---|---|---|---|---|
| 10&60 | |||||
| Indus numerals | |||||
| Proto-Elamite numerals | 10&60 | ||||
| Sumerian numerals | 10&60 | ||||
| 10 | |||||
| Babylonian numerals | 10&60 | ||||
| 10 | | ||||
| 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) | |||
| 5&10 | I V X L C D M | ||||
| 10 | |||||
| 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 | ||||
| 10 | ō α β γ δ ε ϝ ζ η θ ι ο Αʹ Βʹ Γʹ Δʹ Εʹ Ϛʹ Ζʹ Ηʹ Θʹ | ||||
| Kharosthi numerals | 4&10 | [2] | |||
| 10 | [3] | [4] | |||
| 10 | |||||
| 10 | Ⲁ Ⲃ Ⲅ Ⲇ Ⲉ Ⲋ Ⲍ Ⲏ Ⲑ | ||||
| 10 | ፩ ፪ ፫ ፬ ፭ ፮ ፯ ፰ ፱ ፲ ፳ ፴ ፵ ፶ ፷ ፸ ፹ ፺ ፻ ፼ [5] | 15th Century (Modern Style)[6] | |||
| 10 | Ա Բ Գ Դ Ե Զ Է Ը Թ Ժ | ||||
| 10 | ០ ១ ២ ៣ ៤ ៥ ៦ ៧ ៨ ៩ | ||||
| 10 | ๐ ๑ ๒ ๓ ๔ ๕ ๖ ๗ ๘ ๙ | ||||
| 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 | ||||
| Chinese numerals (financial) | 10 | 零壹貳參肆伍陸柒捌玖拾佰仟萬億 (T. Chinese) 零壹贰叁肆伍陆柒捌玖拾佰仟萬億 (S. Chinese) | [7] | ||
| 10 | ٩ ٨ ٧ ٦ ٥ ٤ ٣ ٢ ١ ٠ | ||||
| Vietnamese numerals (Chữ Nôm) | 10 | ||||
| 10 | 0 1 2 3 4 5 6 7 8 9 | ||||
| 10 | Ⰰ Ⰱ Ⰲ Ⰳ Ⰴ Ⰵ Ⰶ Ⰷ Ⰸ ... | ||||
| 10 | а в г д е ѕ з и ѳ і ... | ||||
| 10 | |||||
| 10 | ၀ ၁ ၂ ၃ ၄ ၅ ၆ ၇ ၈ ၉ | [8] | |||
| 10 | |||||
| 10 | |||||
| 5&20 | |||||
| 20 | |||||
| Korean numerals (Hangul) | 10 | 영 일 이 삼 사 오 육 칠 팔 구 | |||
| 20 | |||||
| 10 | ෦ ෧ ෨ ෩ ෪ ෫ ෬ ෭ ෮ ෯ | ||||
| 10 | |||||
| 10 | |||||
| 10 | ꘠ ꘡ ꘢ ꘣ ꘤ ꘥ ꘦ ꘧ ꘨ ꘩ [9] | ||||
| 10 | ꛯ ꛦ ꛧ ꛨ ꛩ ꛪ ꛫ ꛬ ꛭ ꛮ [10] | ||||
| 10 | [11] | ||||
| 10 | |||||
| 20 | / / / [12] | ||||
| 10 | ߉ ߈ ߇ ߆ ߅ ߄ ߃ ߂ ߁ ߀ [13] | ||||
| 10 | |||||
| 10 | [14] | ||||
| Adlam numerals | 10 | [15] | |||
| 5&20 | [16] | ||||
| Sundanese numerals | 10 | ᮰ ᮱ ᮲ ᮳ ᮴ ᮵ ᮶ ᮷ ᮸ ᮹ | 20th Century (1996)[17] |
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.[18] There have been some proposals for standardisation.[19]
| Base | Name | Usage | |
|---|---|---|---|
| 2 | 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 | Chumashan languages and Kharosthi numerals | ||
| 5 | Gumatj, Ateso, Nunggubuyu, Kuurn Kopan Noot, and Saraveca languages; common count grouping e.g. tally marks | ||
| 6 | Diceware, Ndom, Kanum, and Proto-Uralic language (suspected) | ||
| 7 | Septimal, Septenary[20] | Weeks timekeeping, Western music letter notation | |
| 8 | Charles XII of Sweden, Unix-like permissions, Squawk codes, DEC PDP-11, Yuki, Pame, compact notation for binary numbers, Xiantian (I Ching, China) | ||
| 9 | Compact notation for ternary | ||
| 10 | Decimal, denary | Most widely used by contemporary societies[21] [22] [23] | |
| 11 | Undecimal, unodecimal, undenary | A base-11 number system was attributed to the Māori (New Zealand) in the 19th century[24] and the Pangwa (Tanzania) in the 20th century.[25] 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.[26] [27] [28] 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 | Conway base 13 function. | |
| 14 | Quattuordecimal, quadrodecimal | Programming for the HP 9100A/B calculator[29] and image processing applications;[30] pound and stone. | |
| 15 | Quindecimal, pentadecimal | 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 | ||
| 18 | Octodecimal | A base in which 7n is palindromic for n = 3, 4, 6, 9. | |
| 19 | Undevicesimal, nonadecimal | ||
| 20 | Basque, Celtic, Muisca, Inuit, Yoruba, Tlingit, and Dzongkha numerals; Santali, and Ainu languages; shilling and pound | ||
| 5&20 | Quinary-vigesimal[31] [32] | 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" | |
| 21 | The smallest base in which all fractions to have periods of 4 or shorter. | ||
| 23 | Kalam language,[33] Kobon language | ||
| 24 | Quadravigesimal[34] | 24-hour clock timekeeping; Greek alphabet; Kaugel language. | |
| 25 | Sometimes used as compact notation for quinary. | ||
| 26 | Hexavigesimal[35] | Sometimes used for encryption or ciphering,[36] using all letters in the English alphabet | |
| 27 | Septemvigesimal | Telefol, Oksapmin,[37] Wambon,[38] and Hewa[39] 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,[40] to provide a concise encoding of alphabetic strings,[41] or as the basis for a form of gematria.[42] Compact notation for ternary. | |
| 28 | Months timekeeping. | ||
| 30 | Trigesimal | The Natural Area Code, this is the smallest base such that all of to terminate, a number n is a regular number if and only if terminates in base 30. | |
| 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 terminates and all of to 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. | ||
| Hexatrigesimal[43] [44] | 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;[45] similar to base 26 but distinguishing upper- and lower-case letters. | ||
| 56 | A variant of base 58.[46] | ||
| 57 | Covers base 62 apart from I, O, l, U, and u,[47] or I, 1, l, 0, and O.[48] | ||
| 58 | Covers base 62 apart from 0 (zero), I (capital i), O (capital o) and l (lower case L).[49] | ||
| 60 | 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).[50] | ||
| Can be notated with the digits 0–9 and the cased letters A–Z and a–z of the English alphabet. | |||
| 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.[51] | ||
| 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. | ||
| 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 to terminate. | ||
| 225 | Number expressible with two pentadecimal digits. | ||
| 256 | Number expressible with eight binary digits. | ||
| 360 | Degrees of angle. |
| Base | Name | Usage | |
|---|---|---|---|
| 1 | Unary(Bijectivebase1) | Tally marks, Counting | |
| 10 | To avoid zero | ||
| 26 | Spreadsheet column numeration. Also used by John Nash as part of his obsession with numerology and the uncovering of "hidden" messages.[52] |
| Base | Name | Usage | |
|---|---|---|---|
| 2 | Balanced binary (Non-adjacent form) | ||
| 3 | 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 |
| Base | Name | Usage | |
|---|---|---|---|
| 2i | related to base −4 and base 16 | ||
i\sqrt{2} | Base i\sqrt{2} | related to base −2 and base 4 | |
i\sqrt[4]{2} | Base i\sqrt[4]{2} | related to base 2 | |
2\omega | Base 2\omega | related to base 8 | |
\omega\sqrt[3]{2} | Base \omega\sqrt[3]{2} | 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 \sqrt{2} | related to base 2 | |||||||
\sqrt{3} | Base \sqrt{3} | related to base 3 | ||||||
\sqrt[3]{2} | Base \sqrt[3]{2} | |||||||
\sqrt[4]{2} | Base \sqrt[4]{2} | |||||||
| Base \sqrt[12]{2} | usage in 12-tone equal temperament musical system | |||||||
2\sqrt{2} | Base 2\sqrt{2} | |||||||
| Base
| a negative rational non-integer base | ||||||
-\sqrt{2} | Base -\sqrt{2} | a negative non-integer base, related to base 2 | ||||||
\sqrt{10} | Base \sqrt{10} | related to decimal | ||||||
2\sqrt{3} | Base 2\sqrt{3} | related to duodecimal | ||||||
| early Beta encoder | ||||||||
| Plastic number base | ||||||||
| Supergolden ratio base | ||||||||
| Silver ratio base | ||||||||
| Base e | ||||||||
| Base \pi | ||||||||
| Base e\pi | ||||||||
e\pi | Base e\pi |
| 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, as are many developed later, such as the Roman numerals. The French Cistercian monks created their own numeral system.