List of numeral systems

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

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Part of a series on

Numeral systems

Place-value notation

Hindu–Arabic numerals

* Western Arabic
* Eastern Arabic

* * *

* Bengali
* Devanagari
* Gujarati
* Gurmukhi
* Odia
* Sinhala
* Tamil
* Malayalam
* Telugu
* Kannada
* Dzongkha

* * *

* Tibetan
* Balinese
* Burmese
* Javanese
* Khmer
* Lao
* Mongolian
* Sundanese
* Thai

East Asian systems

Contemporary

* Chinese
* Hokkien
* Suzhou
* Japanese
* Korean
* Vietnamese

* * *

Historic

* Counting rods
* Tangut

Other systems

* History

* * *

Ancient

* Babylonian

* * *

Post-classical

* Cistercian
* Mayan
* Muisca
* Pentadic
* Quipu
* Rumi

* * *

Contemporary

* Cherokee
* Kaktovik (Iñupiaq)

By radix/base

Common radices/bases

* 2
* 3
* 4
* 5
* 6
* 8
* 10
* 12
* 16
* 20
* 60

* * *

Non-standard radices/bases

* Bijective (1)
* Signed-digit (balanced ternary)
* Mixed (factorial)
* Negative
* Complex (2i)
* Non-integer (φ)
* Asymmetric

Sign-value notation

Non-alphabetic

* Aegean
* Attic
* Aztec
* Brahmi
* Chuvash
* Egyptian
* Etruscan
* Kharosthi
* Prehistoric counting
* Proto-cuneiform
* Roman
* Tally marks

* * *

Alphabetic

* Abjad
* Armenian
* Alphasyllabic
* Akṣarapallī
* Āryabhaṭa
* Kaṭapayādi
* Coptic
* Cyrillic
* Geʽez
* Georgian
* Glagolitic
* Greek
* Hebrew

List of numeral systems

* v
* t
* e

There are many different numeral systems, that is, writing systems for expressing numbers.

## By culture / time period

[edit]

"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

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|
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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

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

[edit]

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

[edit]

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|

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] | 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 mistakenly attributed to the Māori (New Zealand) in the 19th century[34] and one was reported to be used by the Pangwa (Tanzania) in the 20th century,[35] but was not confirmed by later research and is believed to also be an error.[36] 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.[37][38][39] Featured in popular fiction.[citation needed]
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[40][41] | Conway's base 13 function.
14 | Quattuordecimal, quadrodecimal[40][41] | Programming for the HP 9100A/B calculator[42] and image processing applications.[43]
15 | Quindecimal, pentadecimal[44][41] | Telephony routing over IP, and the Huli language.[36]
16 | Hexadecimal, sexadecimal, sedecimal  | Compact notation for binary data; tonal system of Nystrom.
17 | Septendecimal, heptadecimal[44][41] |
19 | Undevicesimal, nonadecimal[44][41] |
20 | Vigesimal | Basque, Celtic, Muisca, Inuit, Yoruba, Tlingit, and Dzongkha numerals; Santali, and Ainu languages.
5&20 | Quinary-vigesimal[45][46][47] | 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"[45]
21 |  | The smallest base in which all fractions ⁠1/2⁠ to ⁠1/18⁠ have periods of 4 or shorter.
23 |  | Kalam language,[48] Kobon language[citation needed]
24 | Quadravigesimal[49] | 24-hour clock timekeeping; Greek alphabet; Kaugel language.
25 |  | Sometimes used as compact notation for quinary.
26 | Hexavigesimal[49][50] | Sometimes used for encryption or ciphering,[51] using all letters in the English alphabet
27 | Septemvigesimal[citation needed] | Telefol,[48] Oksapmin,[52] Wambon,[53] and Hewa[54] 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,[55] to provide a concise encoding of alphabetic strings,[56] or as the basis for a form of gematria.[57] Compact notation for ternary.
28 |  | Months timekeeping.
30 | Trigesimal[citation needed] | 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 |  | The smallest base where ⁠1/2⁠ terminates and all of ⁠1/2⁠ to ⁠1/18⁠ have periods of 4 or shorter.
36 | Hexatrigesimal[58][59] | 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.
40 | Quadragesimal[citation needed] | 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.[citation needed]
50 | Quinquagesimal[citation needed] | 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).[60]
62 |  | Can be notated with the digits 0–9 and the cased letters A–Z and a–z of the English alphabet.
64 | Tetrasexagesimal[citation needed] | 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[citation needed] | 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[citation needed] | Related to Goormaghtigh conjecture for the generalized repunit numbers (111 in base 90 = 1111111111111 in base 2).
95 |  | Number of printable ASCII characters.[61]
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.
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 fractions ⁠1/2⁠ to ⁠1/10⁠ terminate.

### Non-standard positional numeral systems

[edit]

#### Bijective numeration

[edit]

Base | Name | Usage

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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.[62]

#### Signed-digit representation

[edit]

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

[edit]

Base | Name | Usage

|
|

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

[edit]

Base | Name | Usage

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|

3 2 {\displaystyle {\frac {3}{2}}} | Base  3 2 {\displaystyle {\frac {3}{2}}} | a rational non-integer base
4 3 {\displaystyle {\frac {4}{3}}} | Base  4 3 {\displaystyle {\frac {4}{3}}} | related to duodecimal
5 2 {\displaystyle {\frac {5}{2}}} | Base  5 2 {\displaystyle {\frac {5}{2}}} | related to decimal
2 {\displaystyle {\sqrt {2}}} | Base  2 {\displaystyle {\sqrt {2}}} | related to base 2
3 {\displaystyle {\sqrt {3}}} | Base  3 {\displaystyle {\sqrt {3}}} | related to base 3
2 3 {\displaystyle {\sqrt[{3}]{2}}} | Base  2 3 {\displaystyle {\sqrt[{3}]{2}}} |
2 4 {\displaystyle {\sqrt[{4}]{2}}} | Base  2 4 {\displaystyle {\sqrt[{4}]{2}}} |
2 12 {\displaystyle {\sqrt[{12}]{2}}} | Base  2 12 {\displaystyle {\sqrt[{12}]{2}}} | usage in 12-tone equal temperament musical system
2 2 {\displaystyle 2{\sqrt {2}}} | Base  2 2 {\displaystyle 2{\sqrt {2}}} |
− 3 2 {\displaystyle -{\frac {3}{2}}} | Base  − 3 2 {\displaystyle -{\frac {3}{2}}} | a negative rational non-integer base
− 2 {\displaystyle -{\sqrt {2}}} | Base  − 2 {\displaystyle -{\sqrt {2}}} | a negative non-integer base, related to base 2
10 {\displaystyle {\sqrt {10}}} | Base  10 {\displaystyle {\sqrt {10}}} | related to decimal
2 3 {\displaystyle 2{\sqrt {3}}} | Base  2 3 {\displaystyle 2{\sqrt {3}}} | related to duodecimal
φ | Golden ratio base | early Beta encoder[63]
ρ | Plastic number base |
ψ | Supergolden ratio base |
1 + 2 {\displaystyle 1+{\sqrt {2}}} | Silver ratio base |
e | Base  e {\displaystyle e} | best radix economy [citation needed]
π | Base  π {\displaystyle \pi } |
eπ | Base  e π {\displaystyle e\pi } |
e π {\displaystyle e^{\pi }} | Base  e π {\displaystyle e^{\pi }} |

#### n-adic number

[edit]

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

[edit]

* 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

[edit]

* 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

[edit]

All known numeral systems developed before the Babylonian numerals are non-positional,[64] as are many developed later, such as the Roman numerals. The French Cistercian monks created their own numeral system.

## See also

[edit]

* History of ancient numeral systems
* 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
* List of books on history of number systems

## References

[edit]

1. ^ a b c Chrisomalis, Stephen (2004). "A cognitive typology for numerical notation". Cambridge Archaeological Journal. 14 (1): 37–52\. doi:10.1017/S0959774304000034.
2. ^ a b Chrisomalis 2010, pp. 330-333.
3. ^ Glass, Andrew; Baums, Stefan; Salomon, Richard (September 18, 2003). "Proposal to Encode Kharoṣ ṭhī in Plane 1 of ISO/IEC 10646" (PDF). Unicode.org.
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47. ^ Chrisomalis 2010, p. 200: "The early origin of bar-and-dot numeration alongside the Middle Formative Mesoamerican scripts, the quinary-vigesimal structure of the system, and the general increase in the frequency and complexity of numeral expressions over time all point to its indigenous development.".
48. ^ a b Laycock, Donald (1975). "Observations on Number Systems and Semantics". In Wurm, Stephen (ed.). New Guinea Area Languages and Language Study, I: Papuan Languages and the New Guinea Linguistic Scene. Pacific Linguistics C-38. Canberra: Research School of Pacific Studies, Australian National University. pp. 219–233.
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64. ^ Chrisomalis 2010, p. 254: Chrisomalis calls the Babylonian system "the first positional system ever".

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