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." 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. 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 Base Sample Approx. First AppearanceProto-cuneiform numerals 1060 Indus numerals unknown Proto-Elamite numerals 1060 Sumerian numerals 1060 Egyptian numerals 10 Z1 V20 V1 M12 D50 I8 I7 C11 Babylonian numerals 1060 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  ( ) Chinese numeralsJapanese numeralsKorean 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) Roman numerals 510 I V X L C D M Hebrew numerals 10 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

Persian ۰ ۱ ۲ ۳ ۴ ۵ ۶ ۷ ۸ ۹ Attic numerals 10 Ι 𐅃 Δ 𐅄 Η 𐅅 Χ 𐅆 Μ 𐅇 Greek numerals 10 ō α β γ δ ε ϝ ζ η θ ι ο Αʹ Βʹ Γʹ Δʹ Εʹ Ϛʹ Ζʹ Ηʹ Θʹ Kharosthi numerals410𐩇 𐩆 𐩅 𐩄 𐩃 𐩂 𐩁 𐩀 Phoenician numerals 10 𐤙 𐤘 𐤗 𐤛𐤛𐤛 𐤛𐤛𐤚 𐤛𐤛𐤖 𐤛𐤛 𐤛𐤚 𐤛𐤖 𐤛 𐤚 𐤖 Chinese rod numerals 10 𝍠 𝍡 𝍢 𝍣 𝍤 𝍥 𝍦 𝍧 𝍨 𝍩 Coptic numerals 10 Ⲁ Ⲃ Ⲅ Ⲇ Ⲉ Ⲋ Ⲍ Ⲏ Ⲑ Ge'ez numerals 10 ፩ ፪ ፫ ፬ ፭ ፮ ፯ ፰ ፱ ፲ ፳ ፴ ፵ ፶ ፷ ፸ ፹ ፺ ፻ ፼ 15th century (Modern Style) Maya numerals 520 𝋠 𝋡 𝋢 𝋣 𝋤 𝋥 𝋦 𝋧 𝋨 𝋩 𝋪 𝋫 𝋬 𝋭 𝋮 𝋯 𝋰 𝋱 𝋲 𝋳 Armenian numerals 10 Ա Բ Գ Դ Ե Զ Է Ը Թ Ժ Khmer numerals 10 ០ ១ ២ ៣ ៤ ៥ ៦ ៧ ៨ ៩ Thai numerals 10 ๐ ๑ ๒ ๓ ๔ ๕ ๖ ๗ ๘ ๙ 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 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) Eastern Arabic numerals 10 ٩ ٨ ٧ ٦ ٥ ٤ ٣ ٢ ١ ٠ Vietnamese numerals (Chữ Nôm) 10 𠬠 𠄩 𠀧 𦊚 𠄼 𦒹 𦉱 𠔭 𠃩 Western Arabic numerals 10 0 1 2 3 4 5 6 7 8 9 Glagolitic numerals 10 Ⰰ Ⰱ Ⰲ Ⰳ Ⰴ Ⰵ Ⰶ Ⰷ Ⰸ ... Cyrillic numerals 10 а в г д е ѕ з и ѳ і ... Rumi numerals 10 Burmese numerals 10 ၀ ၁ ၂ ၃ ၄ ၅ ၆ ၇ ၈ ၉ Tangut numerals 10 Cistercian numerals 10 Muisca numerals 20 Korean numerals (Hangul) 10 영 일 이 삼 사 오 육 칠 팔 구 Aztec numerals 20 (1, 5, 20, 100, 400, 800, 8000) Sinhala numerals 10 ෦ ෧ ෨ ෩ ෪ ෫ ෬ ෭ ෮ ෯ 𑇡 𑇢 𑇣𑇤 𑇥 𑇦 𑇧 𑇨 𑇩 𑇪 𑇫 𑇬 𑇭 𑇮 𑇯 𑇰 𑇱 𑇲 𑇳 𑇴 Pentadic runes 10 Cherokee numerals 10 Vai numerals 10 ꘠ ꘡ ꘢ ꘣ ꘤ ꘥ ꘦ ꘧ ꘨ ꘩ Bamum numerals 10 ꛯ ꛦ ꛧ ꛨ ꛩ ꛪ ꛫ ꛬ ꛭ ꛮ Mende Kikakui numerals 10 𞣏 𞣎 𞣍 𞣌 𞣋 𞣊 𞣉 𞣈 𞣇 Osmanya numerals 10 𐒠 𐒡 𐒢 𐒣 𐒤 𐒥 𐒦 𐒧 𐒨 𐒩 Medefaidrin numerals 20 𖺀 𖺁/𖺔 𖺂/𖺕 𖺃/𖺖 𖺄 𖺅 𖺆 𖺇 𖺈 𖺉 𖺊 𖺋 𖺌 𖺍 𖺎 𖺏 𖺐 𖺑 𖺒 𖺓 N'Ko numerals 10 ߉ ߈ ߇ ߆ ߅ ߄ ߃ ߂ ߁ ߀ Hmong numerals 10 Garay numerals 10 Adlam numerals 10 𞥙 𞥘 𞥗 𞥖 𞥕 𞥔 𞥓 𞥒 𞥑 𞥐 Kaktovik numerals 520 𝋀 𝋁 𝋂 𝋃 𝋄 𝋅 𝋆 𝋇 𝋈 𝋉 𝋊 𝋋 𝋌 𝋍 𝋎 𝋏 𝋐 𝋑 𝋒 𝋓 Sundanese numerals10᮰ ᮱ ᮲ ᮳ ᮴ ᮵ ᮶ ᮷ ᮸ ᮹20th century (1996)

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.For the mixed roots of the word "hexadecimal", see . There have been some proposals for standardisation.Multiplication Tables of Various Bases, p. 45, Michael Thomas de Vlieger, Dozenal Society of America

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 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), 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. 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.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, , John Wiley and Sons Inc., New York, 2000. Translated from the French by David Bellos, E.F. Harding, Sophie Wood and Ian Monk 11 Undecimal, unodecimal, undenary A base-11 number system was mistakenly attributed to the Māori (New Zealand) in the 19th century and one was reported to be used by the Pangwa (Tanzania) in the 20th century, but was not confirmed by later research and is believed to also be an error. 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 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, tridecimal Conway's base 13 function. 14 Quattuordecimal, quadrodecimal Programming for the HP 9100A/B calculatorHP 9100A/B programming, HP Museum and image processing applications. 15 Quindecimal, pentadecimal Telephony routing over IP, and the Huli language. 16 Hexadecimal, sexadecimal, sedecimal Compact notation for binary data; tonal system of Nystrom. 17 Heptadecimal, septendecimal 18 Octodecimal 19 Undevicesimal, nonadecimal 20 Vigesimal Basque, Celtic, Muisca, Inuit, Yoruba, Tlingit, and Dzongkha numerals; Santali, and Ainu languages. 5&20 Quinary-vigesimal 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, 24 Quadravigesimal 24-hour clock timekeeping; Greek alphabet; Kaugel language. 25 Compact notation for quinary. 26 Hexavigesimal Sometimes used for encryption or ciphering, 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) and SID (sequence identification, an indexing system of PCR amplicons in forensics). 27 Telefol, Oksapmin,. Wambon, and HewaNauka i Zhizn, 1992, issue 3, p. 48. 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,. to provide a concise encoding of alphabetic strings,. or as the basis for a form of gematria.. Compact notation for ternary. 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. 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 , with each character standing for 5 bits. 34 The smallest base where all unitfractions from to have periods of 4 or shorter. 36 Hexatrigesimal 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). 64 Used to encode computer (binary) data into a relatively compact string, with each character standing for 6 bits (). 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 to terminate.

Non-standard positional numeral systems

Bijective numeration Unary, or bijective base1, 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.

Signed-digit representation Base Name Usage 2 Balanced binary (Non-adjacent form) 3 Balanced ternary Ternary computers 10 Balanced decimal John ColsonAugustin Cauchy

Complex bases 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

Non-integer bases Base Name Usage φ Golden ratio base early Beta encoder e Base best radix economy

Mixed radix 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, as are many developed later, such as the Roman numerals. The French Cistercian monks created their own numeral system.

See also

References