List of numeral systems

For different types of numbers, such as rational numbers, real numbers, complex numbers, etc, see List of types of numbers.

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]: 38 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	Base	Sample	Approx. First Appearance
Proto-cuneiform numerals	10&60		c. 3500–2000 BCE
Indus numerals	unknown[2]		c. 3500–1900 BCE[2]
Proto-Elamite numerals	10&60		3100 BCE
Sumerian numerals	10&60		3100 BCE
Egyptian numerals	10		3000 BCE
Babylonian numerals	10&60		2000 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  ( )	1500 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) 〇一二三四五六七八九十百千万亿 (Default, Simplified Chinese)	1300 BCE
Roman numerals	5&10	I V X L C D M	1000 BCE[1]
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	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	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
Kharosthi numerals	4&10	𐩇 𐩆 𐩅 𐩄 𐩃 𐩂 𐩁 𐩀	<400–250 BCE[3]
Phoenician numerals	10	𐤙 𐤘 𐤗 𐤛𐤛𐤛 𐤛𐤛𐤚 𐤛𐤛𐤖 𐤛𐤛 𐤛𐤚 𐤛𐤖 𐤛 𐤚 𐤖 [4]	<250 BCE[5]
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 ፼ [6]	3rd–4th Century 15th Century (Modern Style)[7]: 135–136
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[8]
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
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)	late 7th/early 8th Century[9]
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[10]
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)
Vai numerals	10	꘠ ꘡ ꘢ ꘣ ꘤ ꘥ ꘦ ꘧ ꘨ ꘩ [11]	19th Century (1832)[12]
Bamum numerals	10	ꛯ ꛦ ꛧ ꛨ ꛩ ꛪ ꛫ ꛬ ꛭ ꛮ [13]	19th Century (1896)[12]
Mende Kikakui numerals	10	𞣏 𞣎 𞣍 𞣌 𞣋 𞣊 𞣉 𞣈 𞣇 [14]	20th Century (1917)[15]
Osmanya numerals	10	𐒠 𐒡 𐒢 𐒣 𐒤 𐒥 𐒦 𐒧 𐒨 𐒩	20th Century (1920s)
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 [16]	20th Century (1930s)[17]
N'Ko numerals	10	߉ ߈ ߇ ߆ ߅ ߄ ߃ ߂ ߁ ߀ [18]	20th Century (1949)[19]
Hmong numerals	10	𖭐 𖭑 𖭒 𖭓 𖭔 𖭕 𖭖 𖭗 𖭘 𖭑𖭐	20th Century (1959)
Garay numerals	10	[20]	20th Century (1961)[21]
Adlam numerals	10	𞥙 𞥘 𞥗 𞥖 𞥕 𞥔 𞥓 𞥒 𞥑 𞥐 [22]	20th Century (1989)[23]
Kaktovik numerals	5&20	𝋀 𝋁 𝋂 𝋃 𝋄 𝋅 𝋆 𝋇 𝋈 𝋉 𝋊 𝋋 𝋌 𝋍 𝋎 𝋏 𝋐 𝋑 𝋒 𝋓 [24]	20th Century (1994)[25]
Sundanese numerals	10	᮰ ᮱ ᮲ ᮳ ᮴ ᮵ ᮶ ᮷ ᮸ ᮹	20th Century (1996)[26]

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

A binary clock might use LEDs to express binary values. In this clock, each column of LEDs shows a binary-coded decimal numeral of the traditional sexagesimal time.

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.^[27] There have been some proposals for standardisation.^[28]

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[29]	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, seximal	Diceware, Ndom, Kanum, and Proto-Uralic language (suspected)
7	Septimal, Septenary[30]	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, nonal	Compact notation for ternary
10	Decimal, denary	Most widely used by contemporary societies[31][32][33]
11	Undecimal, unodecimal, undenary	A base-11 number system was attributed to the Māori (New Zealand) in the 19th century[34] and the Pangwa (Tanzania) in the 20th century.[35] 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.[36][37][38] 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[39][40]	Conway base 13 function.
14	Quattuordecimal, quadrodecimal[39][40]	Programming for the HP 9100A/B calculator[41] and image processing applications;[42] pound and stone.
15	Quindecimal, pentadecimal[43][40]	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[43][40]
18	Octodecimal[43][40]	A base in which 7n is palindromic for n = 3, 4, 6, 9.
19	Undevicesimal, nonadecimal[43][40]
20	Vigesimal	Basque, Celtic, Muisca, Inuit, Yoruba, Tlingit, and Dzongkha numerals; Santali, and Ainu languages; shilling and pound
5&20	Quinary-vigesimal[44][45][46]	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"[44]
21		The smallest base in which all fractions ⁠1/2⁠ to ⁠1/18⁠ have periods of 4 or shorter.
23		Kalam language,[47] Kobon language[citation needed]
24	Quadravigesimal[48]	24-hour clock timekeeping; Greek alphabet; Kaugel language.
25		Sometimes used as compact notation for quinary.
26	Hexavigesimal[48][49]	Sometimes used for encryption or ciphering,[50] using all letters in the English alphabet
27	Septemvigesimal	Telefol,[47] Oksapmin,[51] Wambon,[52] and Hewa[53] 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,[54] to provide a concise encoding of alphabetic strings,[55] or as the basis for a form of gematria.[56] Compact notation for ternary.
28		Months timekeeping.
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.
32	Duotrigesimal	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 ⁠1/2⁠ terminates and all of ⁠1/2⁠ to ⁠1/18⁠ 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.
36	Hexatrigesimal[57][58]	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;[59] similar to base 26 but distinguishing upper- and lower-case letters.
56		A variant of base 58.[clarification needed][60]
57		Covers base 62 apart from I, O, l, U, and u,[61] or I, 1, l, 0, and O.[62]
58		Covers base 62 apart from 0 (zero), I (capital i), O (capital o) and l (lower case L).[63]
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]
62		Can be notated with the digits 0–9 and the cased letters A–Z and a–z of the English alphabet.
64	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.[65]
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.[clarification needed]
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 ⁠1/2⁠ to ⁠1/10⁠ terminate.
225		Number expressible with two pentadecimal digits.
256		Number expressible with eight binary digits.
360		Degrees of angle.

Non-standard positional numeral systems

Bijective numeration

Base	Name	Usage
1	Unary (Bijective base‑1)	Tally marks, 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.
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.[66]

Signed-digit representation

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

Complex bases

Base	Name	Usage
2i	Quater-imaginary base	related to base −4 and base 16
${\displaystyle i{\sqrt {2}}}$	Base ${\displaystyle i{\sqrt {2}}}$	related to base −2 and base 4
${\displaystyle i{\sqrt[{4}]{2}}}$	Base ${\displaystyle i{\sqrt[{4}]{2}}}$	related to base 2
${\displaystyle 2\omega }$	Base ${\displaystyle 2\omega }$	related to base 8
${\displaystyle \omega {\sqrt[{3}]{2}}}$	Base ${\displaystyle \omega {\sqrt[{3}]{2}}}$	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

Non-integer bases

Base	Name	Usage
${\displaystyle {\frac {3}{2}}}$	Base ${\displaystyle {\frac {3}{2}}}$	a rational non-integer base
${\displaystyle {\frac {4}{3}}}$	Base ${\displaystyle {\frac {4}{3}}}$	related to duodecimal
${\displaystyle {\frac {5}{2}}}$	Base ${\displaystyle {\frac {5}{2}}}$	related to decimal
${\displaystyle {\sqrt {2}}}$	Base ${\displaystyle {\sqrt {2}}}$	related to base 2
${\displaystyle {\sqrt {3}}}$	Base ${\displaystyle {\sqrt {3}}}$	related to base 3
${\displaystyle {\sqrt[{3}]{2}}}$	Base ${\displaystyle {\sqrt[{3}]{2}}}$
${\displaystyle {\sqrt[{4}]{2}}}$	Base ${\displaystyle {\sqrt[{4}]{2}}}$
${\displaystyle {\sqrt[{12}]{2}}}$	Base ${\displaystyle {\sqrt[{12}]{2}}}$	usage in 12-tone equal temperament musical system
${\displaystyle 2{\sqrt {2}}}$	Base ${\displaystyle 2{\sqrt {2}}}$
${\displaystyle -{\frac {3}{2}}}$	Base ${\displaystyle -{\frac {3}{2}}}$	a negative rational non-integer base
${\displaystyle -{\sqrt {2}}}$	Base ${\displaystyle -{\sqrt {2}}}$	a negative non-integer base, related to base 2
${\displaystyle {\sqrt {10}}}$	Base ${\displaystyle {\sqrt {10}}}$	related to decimal
${\displaystyle 2{\sqrt {3}}}$	Base ${\displaystyle 2{\sqrt {3}}}$	related to duodecimal
φ	Golden ratio base	early Beta encoder[67]
ρ	Plastic number base
ψ	Supergolden ratio base
${\displaystyle 1+{\sqrt {2}}}$	Silver ratio base
e	Base ${\displaystyle e}$	best radix economy [citation needed]
π	Base ${\displaystyle \pi }$
eπ	Base ${\displaystyle e\pi }$
${\displaystyle e^{\pi }}$	Base ${\displaystyle e^{\pi }}$

n-adic number

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

Mixed radix

• Factorial number system {1, 2, 3, 4, 5, 6, ...}
• Even double factorial number system {2, 4, 6, 8, 10, 12, ...}
• Odd double factorial number system {1, 3, 5, 7, 9, 11, ...}
• Primorial number system {2, 3, 5, 7, 11, 13, ...}
• Fibonorial number system {1, 2, 3, 5, 8, 13, ...}
• {60, 60, 24, 7} in timekeeping
• {60, 60, 24, 30 (or 31 or 28 or 29), 12, 10, 10, 10} in timekeeping
• (12, 20) traditional English monetary system (£sd)
• (20, 18, 13) Maya timekeeping

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,^[68] as are many developed later, such as the Roman numerals. The French Cistercian monks created their own numeral system.

See also

• History of ancient numeral systems – Symbols representing numbers
• History of the Hindu–Arabic numeral system
• List of numeral system topics
• Numeral prefix – Prefix derived from numerals or other numbers
• Radix – Number of digits of a numeral system
• Radix economy – Number of digits needed to express a number in a particular basePages displaying short descriptions of redirect targets
• Timeline of numerals and arithmetic

References

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2. Chrisomalis 2010, pp. 330-333.
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5. Cajori, Florian (September 1928). A History Of Mathematical Notations Vol I. The Open Court Company. p. 18. Retrieved June 5, 2017.
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8. Chrisomalis 2010, p. 200.
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