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

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
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
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) 零壹貳參肆伍陸柒捌玖拾佰仟萬億 (Financial, T. Chinese) 零壹贰叁肆伍陆柒捌玖拾佰仟萬億 (Financial, S. Chinese)	1,300 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 Tibetan ༠ ༡ ༢ ༣ ༤ ༥ ༦ ༧ ༨ ༩ 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	<250 BCE
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)
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
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
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		20th Century (1994)

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. There have been some proposals for standardisation.

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	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
11	Undecimal, unodecimal	A base-11 number system was attributed to the Māori (New Zealand) in the 19th century and the Pangwa (Tanzania) in the 20th century. 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	Tredecimal, tridecimal	Base 13 encoding; Conway base 13 function.
14	Quattuordecimal, quadrodecimal	Programming for the HP 9100A/B calculator and image processing applications; pound and stone.
15	Quindecimal, pentadecimal	Telephony routing over IP, and the Huli language.
16	Hexadecimal (also known as sexadecimal and sedecimal)	Base 16 encoding; compact notation for binary data; tonal system; ounce and pound.
17	Septendecimal, heptadecimal	Base 17 encoding.
18	Octodecimal	Base 18 encoding; a base such that 7n is palindromic for n = 3, 4, 6, 9.
19	Undevicesimal, nonadecimal	Base 19 encoding.
20	Vigesimal	Basque, Celtic, Muisca, Inuit, Yoruba, Tlingit, and Dzongkha numerals; Santali, and Ainu languages; shilling and pound
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		Base 21 encoding; also the smallest base where all of ⁠1/2⁠ to ⁠1/18⁠ have periods of 4 or shorter.
22		Base 22 encoding.
23		Kalam language, Kobon language
24		24-hour clock timekeeping; Kaugel language.
25		Base 25 encoding; sometimes used as compact notation for quinary.
26	Hexavigesimal	Base 26 encoding; sometimes used for encryption or ciphering, using all letters in the English alphabet
27	Septemvigesimal	Telefol and Oksapmin 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.
28		Base 28 encoding; months timekeeping.
29		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		Base 31 encoding.
32	Duotrigesimal	Base 32 encoding; 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		Using all numbers and all letters except O.
36		Base 36 encoding; use of letters with digits.
37		Base 37 encoding; using all numbers and all letters of the Spanish alphabet.
38		Base 38 encoding; use all duodecimal digits and all letters.
39		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		Base 42 encoding; largest base for which all minimal primes are known.
45		Base 45 encoding.
47		Smallest base for which no generalized Wieferich primes are known.
48		Base 48 encoding.
49		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		Base 52 encoding, a variant of base 62 without vowels except Y and y or a variant of base 26 using all lower and upper case letters.
54		Base 54 encoding.
56		Base 56 encoding, a variant of base 58.
57		Base 57 encoding, a variant of base 62 excluding I, O, l, U, and u or I, 1, l, 0, and O.
58		Base 58 encoding, a variant of base 62 excluding 0 (zero), I (capital i), O (capital o) and l (lower case L).
60	Sexagesimal	Babylonian numerals; New base 60 encoding, similar to base 62, excluding I, O, and l, but including _(underscore); degrees-minutes-seconds and hours-minutes-seconds measurement systems; Ekari and Sumerian
62		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		Base72 encoding; the smallest base >2 such that no three-digit narcissistic number exists.
80	Octogesimal	Base80 encoding; Supyire as a sub-base.
81		Base 81 encoding, using as 81=34 is related to ternary.
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).
91		Base 91 encoding, using all ASCII except "-" (0x2D), "\" (0x5C), and "'" (0x27); one variant uses "\" (0x5C) in place of """ (0x22).
92		Base 92 encoding, using all of ASCII except for "`" (0x60) and """ (0x22) due to confusability.
93		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.
94		Base 94 encoding, using all of ASCII printable characters.
95		Base 95 encoding, a variant of base 94 with the addition of the Space character.
96		Base 96 encoding, using all of ASCII printable characters as well as the two extra duodecimal digits.
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.
120		Base 120 encoding.
121		Related to base 11.
125		Related to base 5.
128		Using as 128=27.
144		Two duodecimal digits.
169		Two Tridecimal digits.
185		Smallest base which is not perfect power (where generalized repunits can be factored algebraically) for which no generalized repunit primes are known.
196		Two tetradecimal digits.
200		Base 200 encoding.
210		Smallest base such that all of ⁠1/2⁠ to ⁠1/10⁠ terminate.
216		related to base 6.
225		Two pentadecimal digits.
256		Base 256 encoding, as 256=28.
300		Base 300 encoding.
360		Degrees for angle.

Non-standard positional numeral systems

Bijective numeration

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.

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

Negative bases

The common names of the negative base numeral systems are formed using the prefix nega-, giving names such as:

Base	Name	Usage
−2	Negabinary
−3	Negaternary
−4	Negaquaternary
−5	Negaquinary
−6	Negasenary
−8	Negaoctal
−10	Negadecimal
−12	Negaduodecimal
−16	Negahexadecimal

Complex bases

Base	Name	Usage
2i	Quater-imaginary base	related to base −4 and base 16
${\sqrt {2}}i$	Base ${\sqrt {2}}i$	related to base −2 and base 4
${\sqrt[{4}]{2}}i$	Base ${\sqrt[{4}]{2}}i$	related to base 2
$2\omega$	Base $2\omega$	related to base 8
${\sqrt[{3}]{2}}\omega$	Base ${\sqrt[{3}]{2}}\omega$	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
${\frac {3}{2}}$	Base ${\frac {3}{2}}$	a rational non-integer base
${\frac {4}{3}}$	Base ${\frac {4}{3}}$	related to duodecimal
${\frac {5}{2}}$	Base ${\frac {5}{2}}$	related to decimal
${\sqrt {2}}$	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 ${\sqrt[{4}]{2}}$
${\sqrt[{12}]{2}}$	Base ${\sqrt[{12}]{2}}$	usage in 12-tone equal temperament musical system
$2{\sqrt {2}}$	Base $2{\sqrt {2}}$
$-{\frac {3}{2}}$	Base $-{\frac {3}{2}}$	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
φ	Golden ratio base	Early Beta encoder
ρ	Plastic number base
ψ	Supergolden ratio base
$1+{\sqrt {2}}$	Silver ratio base
e	Base $e$	Lowest radix economy
π	Base $\pi$
eπ	Base $e\pi$
$e^{\pi }$	Base $e^{\pi }$

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,<ref>Chrisomalis calls the Babylonian system "the first positional system ever"}} 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
• Table of bases – 0 to 74 in base 2 to 36
• Timeline of numerals and arithmetic