# List of numeral systems

(Redirected from Septenary)

This is a list of numeral systems, that is, writing systems for expressing numbers.

## By culture / time period

Name Base Sample Approx. first appearance
Prehistoric numerals 35,000 BC
Babylonian numerals 60 3100 BC
Egyptian numerals 10

or
3000 BC
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
c1500 BC
Chinese numerals, Japanese numerals, Korean numerals (Sino-Korean) 10

〇一二三四五六七八九十百千万亿 (Default, Simplified Chinese)

Roman numerals 10 I V X L C D M 1000 BC
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 ף ץ
800 BC
Indian numerals 10 Tamil ௦ ௧ ௨ ௩ ௪ ௫ ௬ ௭ ௮ ௯

Devanagari ० १ २ ३ ४ ५ ६ ७ ८ ९

750 BC – 690 BC
Greek numerals 10 ō α β γ δ ε ϝ ζ η θ ι
ο Αʹ Βʹ Γʹ Δʹ Εʹ Ϛʹ Ζʹ Ηʹ Θʹ
Before 5th century BC
Cyrillic numerals 10 А҃ В҃ Г҃ Д҃ Е҃ Ѕ҃ З҃ И҃ Ѳ҃ І҃ ... 10th century
Ge'ez numerals - ፩, ፪, ፫, ፬, ፭, ፮, ፯, ፰, ፱
፲, ፳, ፴, ፵, ፶, ፷, ፸, ፹, ፺, ፻
3rd-4th century CE, modern style from 15th century CE[1]
Chinese rod numerals 10 𝍠 𝍡 𝍢 𝍣 𝍤 𝍥 𝍦 𝍧 𝍨 𝍩 1st century
Phoenician numerals 10 𐤙 𐤘 𐤗 𐤛𐤛𐤛 𐤛𐤛𐤚 𐤛𐤛𐤖 𐤛𐤛 𐤛𐤚 𐤛𐤖 𐤛 𐤚 𐤖 [2] Before 250 AD[3]
Thai numerals 10 ๐ ๑ ๒ ๓ ๔ ๕ ๖ ๗ ๘ ๙ 7th century[4]
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 before 8th century
Eastern Arabic numerals 10 ٩ ٨ ٧ ٦ ٥ ٤ ٣ ٢ ١ ٠ 8th century
Western Arabic numerals 10 0 1 2 3 4 5 6 7 8 9 9th century
Burmese numerals 10 ၀ ၁ ၂ ၃ ၄ ၅ ၆ ၇ ၈ ၉ 11th century[5]
Maya numerals 20 <15th century
Muisca numerals 20 <15th century
Aztec numerals 20 16th century

## 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.[6] There have been some proposals for standardisation.[7]

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 Data transmission, DNA bases and Hilbert curves; 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
8 Octal Charles XII of Sweden, Unix-like permissions, Squawk codes, DEC PDP-11, compact notation for binary numbers, Xiantian (I Ching, China)
9 Nonary Base9 encoding
10 Decimal Most widely used by modern civilizations[8][9][10]
11 Undecimal Jokingly proposed during the French Revolution to settle a dispute between those proposing a shift to duodecimal and those who were content with decimal; check digits in ISBN
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
13 Tridecimal Base13 encoding; Conway base 13 function
14 Tetradecimal Programming for the HP 9100A/B calculator[11] and image processing applications[12]; pound and stone
15 Pentadecimal Telephony routing over IP, and the Huli language
16 Hexadecimal Base16 encoding; compact notation for binary data; tonal system; ounce and pound
18 Octodecimal Base18 encoding
20 Vigesimal Basque, Celtic, Maya, Muisca, Inuit, Yoruba, Tlingit, and Dzongkha numerals; Santali, and Ainu languages
21 Unvigesimal Base21 encoding
22 Duovigesimal Base22 encoding
23 Trivigesimal Kalam language, Kobon language
24 Tetravigesimal 24-hour clock timekeeping; Kaugel language
25 Pentavigesimal Base25 encoding
26 Hexavigesimal Base26 encoding; sometimes used for encryption or ciphering.[13], using all letters
27 Heptavigesimal 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,[14] to provide a concise encoding of alphabetic strings,[15] or as the basis for a form of gematria.[16]
28 Octovigesimal Base28 encoding; months timekeeping
29 Enneavigesimal Base29
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 Base32 encoding and the Ngiti language
33 Tritrigesimal Use of letters (except I, O, Q) with digits in vehicle registration plates of Hong Kong
34 Tetratrigesimal Using all numbers and all letters except I and O
35 Pentatrigesimal Using all numbers and all letters except O
36 Hexatrigesimal Base36 encoding; use of letters with digits
38 Octotrigesimal Base38 encoding; use all duodecimal digits and all letters
40 Quadragesimal DEC Radix-50₈ 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.
49 Enneaquadragesimal Related to base 7
50 Quinquagesimal Base50 encoding
52 Duoquinquagesimal Base52 encoding, a variant of Base62 without vowels[17]
54 Tetraquinquagesimal Base54 encoding
56 Hexaquinquagesimal Base56 encoding, a variant of Base58[18]
57 Heptaquinquagesimal Base57 encoding, a variant of Base62 excluding I, O, l, U, and u[19] or I, 1, l, 0, and O [20]
58 Octoquinquagesimal Base58 encoding
60 Sexagesimal Babylonian numerals; NewBase60 encoding, similar to Base62, excluding I, O, and l, but including _(underscore);[21] degrees-minutes-seconds and hours-minutes-seconds measurement systems; Ekari and Sumerian languages
62 Duosexagesimal Base62 encoding, using 0–9, A–Z, and a–z
64 Tetrasexagesimal Base64 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 (for example, YouTube video codes use the hyphen and underscore characters, - and _ to total 64).
72 Duoseptagesimal Base72 encoding
80 Octogesimal Base80 encoding
81 Unoctogesimal Base81 encoding, using as 81=34 is related to ternary
85 Pentoctogesimal 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.
90 Nonagesimal Related to Goormaghtigh conjecture for the generalized repunit numbers.
91 Unnonagesimal Base91 encoding, using all ASCII except "-" (0x2D), "\" (0x5C), and "'" (0x27); one variant uses "\" (0x5C) in place of """ (0x22).
92 Duononagesimal Base92 encoding, using all of ASCII except for "`" (0x60) and """ (0x22) due to confusability.[22]
93 Trinonagesimal Base93 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.[23]
94 Tetranonagesimal Base94 encoding, using all of ASCII printable characters.[24]
95 Pentanonagesimal Base95 encoding, a variant of Base94 with the addition of the Space character.[25]
96 Hexanonagesimal Base96 encoding, using all of ASCII printable characters as well as the two extra duodecimal digits
100 Centesimal As 100=102, these are two decimal digits
120 Centevigesimal Base120 encoding
121 Centeunvigesimal Related to base 11
125 Centepentavigesimal Related to base 5
128 Centeoctovigesimal Using as 128=27
256 Duocentehexaquinquagesimal Base256 encoding, as 256=28
360 Trecentosexagesimal Degrees for angle

### Non-standard positional numeral systems

#### Bijective numeration

Base Name Usage
1 Unary (Bijective base-1) Tally marks
2 Bijective base-2
3 Bijective base-3
4 Bijective base-4
5 Bijective base-5
6 Bijective base-6
8 Bijective base-8
10 Bijective base-10
12 Bijective base-12
16 Bijective base-16
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.[26]

#### Signed-digit representation

Base Name Usage
3 Balanced ternary Ternary computers
4 Balanced quaternary
5 Balanced quinary
6 Balanced senary
7 Balanced sepentary
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

#### Complex bases

Base Name Usage
2i Quater-imaginary base related to base −4 and base 16
${\displaystyle {\sqrt {2}}i}$ Base ${\displaystyle {\sqrt {2}}i}$ related to base −2 and base 4
${\displaystyle {\sqrt[{4}]{2}}i}$ Base ${\displaystyle {\sqrt[{4}]{2}}i}$ related to base 2
${\displaystyle 2\omega }$ Base ${\displaystyle 2\omega }$ related to base 8
${\displaystyle {\sqrt[{3}]{2}}\omega }$ Base ${\displaystyle {\sqrt[{3}]{2}}\omega }$ related to base 2
−1 ± i Twindragon base Twindragon fractal shape, related to base −4 and base 16
1 ± i Nega-Twindragon 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 ${\displaystyle {\sqrt[{12}]{2}}}$ using in music scale
${\displaystyle -{\sqrt {2}}}$ Base ${\displaystyle -{\sqrt {2}}}$ a negative non-integer base
${\displaystyle {\sqrt {10}}}$ 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[27]
ρ Plastic number base
ψ Supergolden ratio base
${\displaystyle 1+{\sqrt {2}}}$ Silver ratio base
e Base ${\displaystyle e}$ Lowest radix economy
π Base ${\displaystyle \pi }$
${\displaystyle e^{\pi }}$ Base ${\displaystyle e^{\pi }}$

Base Name Usage
6 Hexadic number not a field
10 Decadic number not a field
12 Dodecadic number not a field

• 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, ...}
• {60, 60, 24, 7} in timekeeping
• {60, 60, 24, 30 (or 31 or 28 or 29), 12} in timekeeping

### Non-positional notation

All known numeral systems developed before the Babylonian numerals are non-positional,[28] as are many developed later, such as the Roman numerals.

## References

2. ^ Everson, Michael (2007-07-25). "Proposal to add two numbers for the Phoenician script" (PDF). UTC Document Register. L2/07-206 (WG2 N3284): Unicode Consortium.
3. ^ Cajori, Florian (Sep 1928). A History Of Mathematical Notations Vol I. The Open Court Company. p. 18. Retrieved 5 June 2017.
4. ^ Chrisomalis, Stephen (2010). Numerical Notation: A Comparative History. Cambridge University Press. p. 200. ISBN 9780521878180.
5. ^ "Burmese/Myanmar script and pronunciation". Omniglot. Retrieved 5 June 2017.
6. ^ For the mixed roots of the word "hexadecimal", see Epp, Susanna (2010), Discrete Mathematics with Applications (4th ed.), Cengage Learning, p. 91, ISBN 9781133168669.
7. ^ http://www.numberbases.com/terms/BaseNames.pdf
8. ^ The History of Arithmetic, Louis Charles Karpinski, 200pp, Rand McNally & Company, 1925.
9. ^ Histoire universelle des chiffres, Georges Ifrah, Robert Laffont, 1994.
10. ^ The Universal History of Numbers: From prehistory to the invention of the computer, Georges Ifrah, ISBN 0-471-39340-1, John Wiley and Sons Inc., New York, 2000. Translated from the French by David Bellos, E.F. Harding, Sophie Wood and Ian Monk
11. ^ HP 9100A/B programming, HP Museum
12. ^ Free Patents Online
13. ^ http://www.dcode.fr/base-26-cipher
14. ^ Grannis, Shaun J.; Overhage, J. Marc; McDonald, Clement J. (2002), "Analysis of identifier performance using a deterministic linkage algorithm", Proceedings. AMIA Symposium: 305–309, PMC 2244404, PMID 12463836.
15. ^ Stephens, Kenneth Rod (1996), Visual Basic Algorithms: A Developer's Sourcebook of Ready-to-run Code, Wiley, p. 215, ISBN 9780471134183.
16. ^ Sallows, Lee (1993), "Base 27: the key to a new gematria", Word Ways, 26 (2): 67–77.
17. ^ "Base52". Retrieved 2016-01-03.
18. ^ "Base56". Retrieved 2016-01-03.
19. ^ "Base57". Retrieved 2016-01-03.
20. ^ "Base57". Retrieved 2019-01-22.
21. ^ "NewBase60". Retrieved 2016-01-03.
22. ^ "Base92". Retrieved 2016-01-03.
23. ^ "Base93". Retrieved 2017-02-13.
24. ^ "Base94". Retrieved 2016-01-03.
25. ^ "base95 Numeric System". Retrieved 2016-01-03.
26. ^ Nasar, Sylvia (2001). A Beautiful Mind. Simon and Schuster. pp. 333–6. ISBN 0-7432-2457-4.
27. ^ Ward, Rachel (2008), "On Robustness Properties of Beta Encoders and Golden Ratio Encoders", IEEE Transactions on Information Theory, 54 (9): 4324–4334, arXiv:0806.1083, doi:10.1109/TIT.2008.928235
28. ^ Chrisomalis calls the Babylonian system "the first positional system ever" in Chrisomalis, Stephen (2010), Numerical Notation: A Comparative History, Cambridge University Press, p. 254, ISBN 9781139485333.