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Part of a series on
Numeral systems
Place-value notation
Hindu–Arabic numerals
East Asian systems
Contemporary
Historic
Other systems
Ancient
Post-classical
Contemporary
By radix/base
Common radices/bases
Non-standard radices/bases
Sign-value notation
Non-alphabetic
Alphabetic
List of numeral systems

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
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
Z1V20V1M12D50I8I7C11
 3000 BCE
Babylonian numerals 10&60 2000 BCE
Aegean numerals 10 𐄇 𐄈 𐄉 𐄊 𐄋 𐄌 𐄍 𐄎 𐄏  ( 1 2 3 4 5 6 7 8 9 )
𐄐 𐄑 𐄒 𐄓 𐄔 𐄕 𐄖 𐄗 𐄘  ( 10 20 30 40 50 60 70 80 90 )
𐄙 𐄚 𐄛 𐄜 𐄝 𐄞 𐄟 𐄠 𐄡  ( 100 200 300 400 500 600 700 800 900 )
𐄢 𐄣 𐄤 𐄥 𐄦 𐄧 𐄨 𐄩 𐄪  ( 1000 2000 3000 4000 5000 6000 7000 8000 9000 )
𐄫 𐄬 𐄭 𐄮 𐄯 𐄰 𐄱 𐄲 𐄳  ( 10000 20000 30000 40000 50000 60000 70000 80000 90000 )		 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 Garay numbers[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
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]
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]
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
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
Base related to base −2 and base 4
Base related to base 2
Base related to base 8
Base 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
Base a rational non-integer base
Base related to duodecimal
Base related to decimal
Base related to base 2
Base related to base 3
Base
Base
Base usage in 12-tone equal temperament musical system
Base
Base a negative rational non-integer base
Base a negative non-integer base, related to base 2
Base related to decimal
Base related to duodecimal
φ Golden ratio base early Beta encoder[63]
ρ Plastic number base
ψ Supergolden ratio base
Silver ratio base
e Base best radix economy [citation needed]
π Base
eπ Base
Base

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]

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]

References

[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

List of numeral systems

View on Grokipedia
from Grokipedia
A list of numeral systems catalogs the diverse notations developed across cultures and eras for representing numerical quantities using symbols, digits, or other markers in a structured manner. These systems serve as mathematical frameworks to encode numbers from a defined set, enabling consistent interpretation and computation, and range from ancient to modern digital encodings. Numeral systems are broadly classified into positional (or place-value) types, where a symbol's value depends on its location within the representation—such as the decimal system (base-10, using digits 0-9) prevalent in global commerce and science, or the binary system (base-2, using 0 and 1) fundamental to —and non-positional (or additive/multiplicative) types, where symbols carry fixed values that are summed or scaled independently of position, like (using I, V, X, L, C, D, M) or Egyptian hieroglyphics (employing strokes, coils, and lotuses for powers of 10). Other notable examples include the Babylonian system (base-60, wedge-shaped marks for astronomy and timekeeping), the Mayan vigesimal system (base-20, with dots, bars, and shells for calendars), Greek alphabetic numerals (letters assigning values from 1 to 900), and Brahmi script precursors to modern Indic digits. This compilation highlights how numeral systems reflect cultural, practical, and technological needs, evolving from simple grouping methods like Inca quipus (knotted strings) to or for specialized applications.

Non-Positional Numeral Systems

Unary and Tally Systems

Unary numeral systems, also known as tally systems, represent natural numbers through the repetition of a single, identical symbol for each unit counted, without any positional weighting or base structure. For instance, the number 3 is denoted by three vertical strokes, such as |||, where each mark signifies one unit. These systems trace their origins to prehistoric times, with some of the earliest evidence appearing in etched on animal bones. The , discovered in the and dated to approximately 25,000 years ago, features grouped notches that likely served as a primitive counting tool for tracking quantities, such as lunar phases or small inventories. Such marks played a foundational role in early human , enabling the recording of goods, debts, or events through simple incisions on wood, bone, or stone, as seen in tally sticks from various ancient cultures. Examples of unary and tally systems include vertical strokes in ancient scripts, like the single-line notations in early for small counts, and modern hash marks used in scoring sports or , where five units are often grouped as four vertical lines crossed by a diagonal for the fifth. In casual settings, such as matchstick in , players replicate unary representation by lining up objects to tally points or moves. In contemporary contexts, unary systems find application in , particularly within , where numbers are encoded via repeated function applications—known as Church numerals—to model without traditional arithmetic structures. These encodings highlight unary's role as a baseline for studying non-efficient counting models in . A key limitation of unary and tally systems is their inefficiency for representing large numbers, as the required number of symbols grows linearly with the value, leading to cumbersome notation and practical challenges beyond small counts. This contrasts with positional systems, which achieve compactness through place value.

Additive Numeral Systems

Additive numeral systems represent numbers through the summation of individual symbols or groups, each carrying a fixed value independent of position, allowing for the construction of larger quantities by repetition or combination. This approach contrasts with positional systems by deriving the total value solely from the additive contributions of the symbols used, though some variants incorporate limited subtractive principles to optimize notation, such as placing a smaller value before a larger one to indicate subtraction (e.g., IV representing 5 - 1 = 4). These systems emerged in ancient civilizations as efficient alternatives to pure unary tallies, enabling more compact representations while remaining intuitive for basic counting and enumeration. The Roman numeral system exemplifies additive notation, tracing its origins to the Etruscan numerals of the 8th century BCE, which influenced early Latin adaptations through cultural exchange in the Italic peninsula. By the 1st century BCE, the system standardized around seven core symbols: I for 1, V for 5, X for 10, L for 50, C for 100, D for 500, and M for 1000, with values accumulated additively when symbols appear in descending order of magnitude (e.g., VI for 6 as 5 + 1). Subtractive rules apply when a smaller symbol precedes a larger one, limited to specific pairs like I before V or X (IV = 4, IX = 9), X before L or C (XL = 40, XC = 90), and C before D or M (CD = 400, CM = 900), reducing redundancy while adhering to the additive framework; no more than three consecutive identical symbols are permitted in standard form to maintain brevity. Today, Roman numerals persist in non-computational contexts, such as clock faces, book chapter outlines, and event designations like Super Bowl titles, valued for their classical aesthetic and symbolic endurance despite obsolescence in everyday arithmetic. In , the Milesian or alphabetic numeral system, introduced around the 5th century BCE in and later adopted across Hellenistic regions, assigned numerical values to letters of the Greek alphabet for an additive representation of integers. The first nine letters (α to θ) denote 1 to 9, the next nine (ι to ϙ) represent tens from 10 to 90, and the following nine (ρ to ϡ, with archaic forms) signify hundreds from 100 to 900; thousands are marked by a superscript or prime on the letter (e.g., αʹ for 1000), with summation yielding the total (e.g., ͵αϡϙδʹ for 1994 as 1000 + 900 + 90 + 4). This system superseded the earlier, obsolete acrophonic method, which used initial letters of number words but lacked the alphabetic integration and broader applicability of the Milesian variant. Still employed in modern for ordinal numbering in texts, legal documents, and dates, it highlights the enduring utility of additive principles in specialized cultural contexts. Despite their historical prevalence, additive systems like Roman and Milesian numerals proved cumbersome for complex arithmetic operations, as manipulation required physical rearrangement or abacus aids rather than direct algorithmic computation, rendering multiplication and division particularly laborious. Their inefficiency for large numbers—exemplified by the lengthy string needed for one million in Roman numerals (a thousand Ms)—contributed to their decline after the 15th century, when the more versatile positional Hindu-Arabic system gained dominance in Europe through printing and commerce. For instance, the year 1990 is denoted as MCMXC in Roman numerals (1000 + 900 + 90), illustrating both the subtractive efficiency and the growing verbosity for modern scales.

Acrophonic and Ideographic Systems

Acrophonic numeral systems employ symbols derived from the initial letter or sound of the spoken name for a number, serving as mnemonic aids in early writing traditions. This , evident in various ancient civilizations, facilitated the representation of numerical values through abbreviated forms tied to , often appearing in , inscriptions, and administrative records. In contrast, ideographic systems use pictorial symbols that directly evoke the concept of the quantity, typically structured around powers of ten in an additive manner without positional value. These non-positional approaches were prevalent in societies where arithmetic operations were performed mentally or with auxiliary tools rather than through the notation itself. The Greek acrophonic numerals, originating around the 7th century BCE in and other regions, exemplify this linguistic derivation. Symbols were taken from the first letter of the Greek word for the number: for instance, Π (pi) represented 5 from , Δ (delta) stood for 10 from deka, and Η () denoted 100 from hekaton. Higher values followed similarly, with Μ for 1,000 from myria and special forms for multiples like ΠΔ for 50 (five tens). This system was widely used in commerce, public inscriptions, and weights until the 1st century BCE, when it was largely supplanted by the for its greater efficiency in complex calculations. Egyptian hieroglyphic numerals, dating back to approximately 3000 BCE, relied on ideographic symbols for powers of ten, forming an additive system. A single vertical stroke (|) signified 1, a cattle hobble (∩) represented 10, a coiled (ɔ) for 100, a lotus flower (*) for 1,000, a pointing finger (d) for 10,000, a (p) for 100,000, and a god with raised arms (𓆐) for 1,000,000. Numbers were constructed by repeating these ideograms as needed, grouped horizontally or vertically for readability, such as nine strokes for 9 or four coils for 40. This system evolved into the more fluid script by the Middle Kingdom (c. 2050–1710 BCE), a form used on for administrative purposes, and later demotic, maintaining the ideographic base until the adoption of Greek influences in the Ptolemaic period. Chinese numeral characters trace their origins to oracle bone inscriptions from the Shang Dynasty around 1200 BCE, featuring pictographic forms that evolved into the traditional ideograms still recognized today. Basic symbols include 一 (yī) for 1, 二 (èr) for 2, and 三 (sān) for 3, with 十 (shí) representing 10 as a cross-like form. These characters formed an additive system for composing larger numbers, such as 廿 (niàn) for 20. For accounting and legal contexts, specialized suàn (or accounting) numerals developed later, like 壹 (yī) for 1 and 贰 (èr) for 2, designed to prevent alteration or fraud by using more complex strokes distinct from standard forms; this practice persists in formal financial documents across East Asia. Unlike the positional rod numeral system used for calculations on counting boards, these character-based numerals remained non-positional and ideographic. In ancient non-arithmetic-oriented societies, acrophonic and ideographic systems supported practical needs like tallying goods, recording divinations, and marking monuments, emphasizing symbolic representation over computational efficiency. Their legacy endures in modern , where traditional appear on clocks (e.g., vertical arrangement for hours) and calendars to denote dates and festivals, preserving cultural continuity alongside Arabic digits.

Historical Positional Numeral Systems

Mesopotamian Cuneiform Numerals

The Mesopotamian numeral system emerged in ancient during the late , around 3500 BCE, as part of notations impressed on clay tablets using tokens and numerical signs that evolved from concrete counting methods. This early system transitioned into more abstract forms by circa 3200–3000 BCE, with scribes developing standardized signs for numerals amid the growth of urban administration and trade in southern . The numerals were adapted and refined during the (circa 2334–2154 BCE) and flourished in the Old Babylonian period (circa 2000–1600 BCE), when Akkadian-speaking scribes in cities like and produced extensive mathematical and astronomical texts. Usage persisted through the Neo-Babylonian and Achaemenid eras until approximately 500 BCE, after which it waned in favor of emerging scripts. At its core, the system employed a sexagesimal (base-60) structure with positional notation, allowing representation of large numbers through powers of 60: units (60^0), sixties (60^1), sixtieths (60^2), and higher. Numerals from 1 to 59 were formed by combining two primary symbols—a vertical wedge (representing 1) and a chevron or corner wedge (representing 10)—impressed with a reed stylus on wet clay tablets, which were then baked or sun-dried for durability. For instance, the number 23 would appear as two chevrons and three vertical wedges. Place values were read from right to left, but the system initially lacked a dedicated symbol for zero, leading to ambiguity that scribes resolved through contextual clues, spacing between positions, or the overall tablet content. Later developments around the 3rd century BCE introduced a placeholder symbol (two slanted wedges) to indicate empty positions, though it was not treated as a true numerical zero. A representative example of the notation is the expression for 81, written as "1,21" in modern transcription, where the comma separates place values: this denotes 1×60+21=811 \times 60 + 21 = 81 in decimal terms. Such positional sexagesimal entries appear frequently in administrative records for quantities of grain or labor, as well as in scholarly tablets. The system's flexibility supported mixed radix elements, such as subunits in (e.g., base-10 for smaller measures within base-60 frameworks), but remained anchored in pure for higher . In applications, the numerals underpinned , exemplified by the tablet from circa 1800 BCE, a scribal exercise listing 15 rows of Pythagorean triples (e.g., sides of right-angled triangles) computed using sexagesimal reciprocals and square roots to generate ratios like 119:120:169 (short leg : long leg : ). Astronomy relied heavily on this system for tracking celestial cycles; scribes calculated planetary positions, eclipses, and lunar phases in sexagesimal fractions, dividing the circle into 360 degrees (6×60) and influencing modern timekeeping with per hour and 60 seconds per minute as a direct legacy. These computations appear in astronomical diaries from the 7th century BCE onward, demonstrating predictive models based on periodic observations. The system's influence extended to Greek astronomy via cultural exchanges during the after the Great's conquests (circa 331 BCE), where Babylonian tables informed Ptolemy's Almagest and the adoption of 360-degree circles. It declined as alphabetic scripts like proliferated under Persian and later Hellenistic rule, supplanting for everyday and administrative use by the 1st century CE.

Mesoamerican Numeral Systems

Mesoamerican numeral systems, developed independently in the , represent some of the earliest known positional notations in the , with the Maya system being the most sophisticated and well-documented example. The Maya vigesimal (base-20) system emerged during the Late Preclassic period, with the earliest evidence of positional numeration dating to approximately 400 BCE to 150 CE, allowing for efficient representation of large numbers through place value. This system utilized a true zero symbol, a rarity in ancient numeral systems, enabling precise calculations for astronomy, calendars, and rituals. In the Maya notation, numbers from 1 to 19 were formed using dots (each representing 1) and bars (each representing 5), often arranged vertically with the units place at the bottom; for instance, the number 13 was depicted as two horizontal bars (for 10) topped by three dots (for 3). Higher place values followed powers of 20: the first position represented 1 (20020^0), the second 20 (20120^1), the third 400 (20220^2), the fourth 8,000 (20320^3), and so on, though in calendrical applications like the Long Count, the third position was adjusted to 360 (18 × 20) to align with the solar year. Early notations sometimes employed abstract forms or head glyphs (logograms depicting faces) for numerals 1 through 19, particularly in monumental inscriptions, while the zero was symbolized by a shell-like glyph. This positional structure paralleled developments in the but arose independently in . The system found extensive use in and stelae for recording dates and astronomical data, notably in the , a bark-paper manuscript from the 11th–12th century CE that details eclipse tables, planetary positions, and ritual cycles. Integral to the Long Count calendar, it tracked elapsed days from a mythical creation date denoted as 0.0.0.0.0, corresponding to August 11, 3114 BCE in the , facilitating long-term historical and prophetic records. Among other Mesoamerican cultures, precursors appear in Olmec inscriptions from around 1200–400 BCE, featuring elements in early writing systems, while Zapotec numerals at (ca. 500 BCE) show similar bar-and-dot notations but with less emphasis on zero and positional complexity. The , in the Postclassic period (ca. 1300–1521 CE), partially adopted counting for markets and tributes but relied more on pictographic and additive methods without a fully positional zero. The legacy of these systems endures in modern Maya communities, where vigesimal structures influence spoken numerals and traditional counting practices in languages like Yucatec Maya. Rediscovery occurred in the through archaeological excavations and epigraphic studies, with scholars like Ernst Förstemann deciphering the Dresden Codex's mathematical tables in the , revealing the system's advanced computational role.

Early Indian and Arabic Systems

The , originating around the 3rd century BCE, represent the earliest known decimal numeral system in , appearing in the inscriptions of Emperor Ashoka on pillars and rocks across the subcontinent. These numerals, primarily additive in nature but showing early hierarchical structures for larger values, used simple strokes and symbols for units up to nine, with evidence of grouping for tens and hundreds in some edicts. By the 4th to 6th centuries CE, the Brahmi script evolved into the Gupta numerals during the Gupta Empire, introducing more distinct symbols for digits 1 through 9 and representing zero as a dot (bindu). This period marked a transition toward clearer digit forms that foreshadowed the modern Hindu-Arabic glyphs, used in astronomical and administrative texts. Indian mathematical developments culminated in the full adoption of a positional decimal system by the 7th century CE, with the Bakhshali manuscript—dated through radiocarbon analysis to portions from the 3rd or 4th century CE—featuring the dot zero in practical arithmetic problems involving large numbers and square roots. In 628 CE, Brahmagupta's Brahmasphutasiddhanta formalized rules for arithmetic operations with zero, treating it as a distinct number: for addition, zero plus a quantity equals the quantity; for multiplication, anything times zero is zero; and for division, a quantity divided by zero is undefined, though he provided rules for zero in other contexts. The system reached the Islamic world through Arabic adoption in the 9th century CE, where Muhammad ibn Musa al-Khwarizmi standardized the Eastern Arabic digits (٠-٩) in his treatise On the Calculation with Hindu Numerals (circa 825 CE), explaining the positional decimal method for computation without relying on physical aids like the abacus. This distinguished the written place-value system—where the value of a digit depends on its position—from earlier abacus-based counting, which used beads for temporary calculations but lacked permanent symbolic representation. Al-Khwarizmi introduced the place-value formula explicitly, representing a number nn as: n=∑i=0kdi⋅10in = \sum_{i=0}^{k} d_i \cdot 10^i where did_i are the digits from 0 to 9, enabling efficient handling of in and astronomy. The numerals spread culturally through Persian and Islamic scientific texts, facilitating advancements in fields like and inheritance law, before transmission via trade routes to by the 10th century CE through (Islamic ). Their widespread adoption in was propelled by Leonardo of (Fibonacci) in his 1202 work , which demonstrated their superiority for commerce and calculation, laying the foundation for the modern decimal system.

Modern Positional Numeral Systems

Decimal Numeral System

The numeral system, also known as base-10, is a system that uses ten distinct digits: , 1, 2, 3, 4, 5, 6, 7, 8, and 9. In this system, the value of a number is determined by the sum of each digit multiplied by the appropriate , based on its position relative to the decimal point, with the rightmost digit representing 100=110^0 = 1. The digit serves as a crucial placeholder to indicate the absence of value in a position, enabling compact representation of without ambiguity. For example, the number 123 is interpreted as 1×102+2×101+3×100=100+20+3=1231 \times 10^2 + 2 \times 10^1 + 3 \times 10^0 = 100 + 20 + 3 = 123. The system has roots in Indian-Arabic developments but became globally standardized through later European adoption. Historical standardization of the decimal system in Europe accelerated after the 15th century, particularly with the invention of the printing press by Johannes Gutenberg around 1440, which facilitated the widespread dissemination of arithmetic texts using Hindu-Arabic numerals. Prior to this, Roman numerals dominated, but printed books on commercial arithmetic promoted the shift to the more efficient decimal forms by the 1500s. The Western Arabic digits (0-9 in their modern rounded forms) emerged in the Maghreb and Al-Andalus by the 10th century and were refined in Europe, differing from Eastern Arabic variants (٠-٩) still used in parts of the Middle East and North Africa, which retain more angular shapes. Arithmetic operations in the decimal system rely on standardized rules, such as and by aligning digits by place value and carrying over when sums exceed 9, and via tables memorized up to 9×9=819 \times 9 = 81. For fractions and decimals, a point separates the integer and fractional parts, allowing representation like 3.14 for an of π\pi, where the first decimal place denotes tenths (10−110^{-1}) and subsequent places hundredths, thousandths, and so on. These rules enable efficient computation in everyday tasks, from basic counting to complex calculations. The decimal system underpins global standards, including the International System of Units (SI), where decimal prefixes like kilo- (10310^3) and milli- (10−310^{-3}) scale measurements coherently, as defined by the International Bureau of Weights and Measures (BIPM). It is integral to International Organization for Standardization (ISO) norms, financial reporting (e.g., currency denominations in most countries), and universal education curricula, promoting consistency in trade and science. Regional variants persist, such as the Indian grouping system, where numbers are comma-separated every two digits after the first three (e.g., 1,00,000 for one lakh or 100,000, and 1,00,00,000 for one crore or 10,000,000), as used in official Reserve Bank of India documents. A key challenge in the decimal system's dominance is the cognitive bias toward base-10 arising from humans' ten fingers, which facilitated its early adoption for finger-counting and persists in intuitive number sense, potentially hindering adaptation to other bases. This anthropomorphic foundation influenced reforms like the metric system in the late 18th century, which extended decimal principles to measurements for greater precision and universality, as established by the French Academy of Sciences and later endorsed internationally.

Binary, Octal, and Hexadecimal Systems

The binary numeral system, also known as base-2, represents numbers using only the digits 0 and 1, where each digit's position corresponds to a power of 2, and the value of a number is calculated as the sum of its digits multiplied by these powers: n=∑i=0kdi⋅2in = \sum_{i=0}^{k} d_i \cdot 2^i, with each did_i being 0 or 1. This system was formally described by Gottfried Wilhelm Leibniz in his 1703 essay "Explication de l'Arithmétique Binaire," where he presented it as a tool for arithmetic computation and philosophical insight into creation from nothingness. Leibniz's work built on earlier binary concepts but emphasized its practical arithmetic operations. Later, George Boole's 1854 publication "An Investigation of the Laws of Thought" introduced Boolean algebra, which formalized binary operations like AND, OR, and NOT, laying the groundwork for digital logic circuits that manipulate binary values. In digital electronics, binary is fundamental, as seen in representations like 1011 in binary equaling 11 in decimal (1×2³ + 0×2² + 1×2¹ + 1×2⁰ = 8 + 0 + 2 + 1 = 11). The numeral system, or base-8, uses digits from 0 to 7 and groups binary digits into sets of three for compact representation, since 8 = 2³ aligns naturally with binary hardware. For example, the 101101 equals 55 in (grouping as 101 101 → 5 5). found early application in for file permissions in Unix systems, where three digits encode read (4), write (2), and execute (1) bits for owner, group, and others; the mode 755, for instance, grants full access to the owner and read/execute to others, a convention dating to the original Unix implementation in the 1970s. Hexadecimal, or base-16, extends this grouping to four binary digits (since 16 = 2⁴), using digits 0-9 followed by A-F (representing 10-15) for brevity in handling . A classic example is 0xFF in equaling 255 in (15×16¹ + 15×16⁰ = 240 + 15 = 255), or in binary as 11111111. It became standard in programming for addresses and due to its one-to-one mapping with bytes (8 bits), emerging prominently in the with computers like the System/360. In , triplets like #RRGGBB specify RGB colors, where each pair denotes , , and intensities from 00 to FF, as defined in the W3C CSS Color Module for consistent digital rendering. Conversions between these systems rely on repeated division by the target base for decimal-to-base conversions or by the base for the reverse, with remainders yielding digits from least to most significant. For binary-to- or binary-to-hex, direct grouping of bits (three for , four for hex, padding with leading zeros if needed) provides an efficient mapping without full recalculation. These powers-of-2 systems form the backbone of modern , as outlined in John von Neumann's 1945 "First Draft of a Report on the ," which proposed binary arithmetic units for electronic stored-program computers to enable efficient, reliable and storage at the hardware level. Their use optimizes for transistor-based logic gates, reducing complexity in circuits while enabling scalable representation in everything from processors to .

Other Integer Base Systems

In positional numeral systems with an integer base b≥2b \geq 2, each digit represents a value from 0 to b−1b-1, allowing the representation of any non-negative integer through powers of bb. Converting numbers between such bases involves repeated division by bb to find digits from least to most significant, but non-standard bases pose challenges like the need for custom symbols beyond 0-9 and mental arithmetic adjustments not aligned with decimal familiarity. The ternary system, or base-3, employs digits 0, 1, and 2 to encode numbers efficiently in certain computational contexts. A variant uses digits -1, 0, and 1 (often denoted as -, 0, +), enabling compact representation of negative numbers without a separate . This system powered the Soviet computer, developed at in 1958, which processed 18-trit numbers for a range up to approximately 194 million in magnitude, outperforming equivalent binary machines in digit economy. Base-12, known as or dozenal, utilizes digits 0-9 followed by A for 10 and B (or sometimes X and E) for 11, facilitating divisions by 2, 3, 4, and 6 more naturally than . The Dozenal Society of America advocates its adoption for enhanced divisibility in measurements, citing historical units like the (12 items) and gross (12 dozen, or 144). Such applications appear in counting eggs or imperial dozens, where base-12 simplifies grouping. The system, base-60, persists in modern timekeeping and angular measurement despite its ancient origins. It divides hours into and minutes into 60 seconds, with notation such as 1:23:45 representing 1 hour, 23 minutes, and 45 seconds. Similarly, circles are divided into 360 degrees, reflecting Babylonian approximations of a 360-day year, which favored 60's high divisibility by 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30. Other integer bases appear in cultural contexts, such as (base-5) systems linked to in some indigenous traditions, including certain Papua New Guinean languages that group by fives before scaling to 20 or higher. (base-20) echoes survive in French numerals, where 80 is "quatre-vingts" (four twenties), a vestige of Celtic counting that multiplies scores of 20 for higher values.

Non-Standard Positional Numeral Systems

Bijective Numeration

Bijective numeration is a variant of positional numeral systems in base-k (where k ≥ 1) that employs k digits ranging from 1 to k, deliberately excluding the digit zero to ensure a one-to-one correspondence between positive integers and non-empty finite strings of these digits. The numerical value of a representation dndn-1 ... d1 (with the rightmost digit d1 in the units place) is computed as ∑i=1ndi⋅ki−1\sum_{i=1}^{n} d_i \cdot k^{i-1} where each di satisfies 1 ≤ di ≤ k. This formulation guarantees that every positive integer receives a unique representation without the need for leading zeros or any zero digits whatsoever, as the absence of zero eliminates representational ambiguities inherent in standard base-k systems. The origins of bijective numeration trace back to theoretical explorations in modern mathematics, though the underlying concept of zero-less positional notation may have ancient precedents in non-standard numeral systems. Scholars have proposed that pre-Columbian Mesoamerican cultures, particularly the Maya, potentially developed an initial form of positional notation as bijective in their vigesimal (base-20) framework, predating the explicit introduction of zero to address redundancies in calendars like the Long Count. This hypothesis posits that bijective systems allowed for compact, ambiguity-free counting before zero's invention enhanced flexibility, with the Maya exploiting the resulting multiple representations for calendrical purposes. Representative examples illustrate the system's mechanics clearly. In bijective base-10 (using digits 1 through 9 and a X for 10), the representation "9" equals 9, while adding 1 yields "X" (value 10); further incrementing produces "11" (1 + 1×10 = 11). Similarly, in bijective base-2 (digits 1 and 2), the sequence begins 1 (value 1), 2 (value 2), 11 (1 + 1×2 = 3), 21 (2 + 1×2 = 4), 12 (1 + 2×2 = 5), and 22 (2 + 2×2 = 6), covering all positive integers sequentially without gaps or duplicates. A practical extension appears in bijective base-26 (digits A=1 to Z=26), where 26 is Z, 27 is AA (1 + 1×26), and 52 is AZ (1 \cdot 26 + 26 = 52), demonstrating how the system maps compactly to sequences like alphabet-based labels. Applications of arise primarily in computational contexts requiring dense, unique encodings without , such as generating identifiers in programming or algorithmic string mappings. For instance, it facilitates efficient conversion between numeric indices and variable-length strings in data structures, avoiding the complications of zero-padding in standard bases. Key advantages include the strict uniqueness of representations, which simplifies and eliminates issues like leading , making it ideal for theoretical bijections or compact storage in zero-averse environments. However, disadvantages emerge in arithmetic operations, as the lack of complicates carrying over in and , often necessitating specialized algorithms that increase computational overhead compared to zero-inclusive systems.

Signed-Digit and Balanced Systems

Signed-digit numeral systems extend traditional by permitting digits to take both positive and negative values within a specified range, typically from −r-r to ss in base bb, where rr and ss are non-negative integers chosen to ensure or balance. A number is represented as ∑dibi\sum d_i b^i, with each digit did_i drawn from the signed set, allowing for multiple representations of the same value due to this . This contrasts with standard unsigned systems by enabling properties like carry-free arithmetic operations, where and avoid long propagation chains, as each digit operation is independent. Generalized signed-digit systems unify various redundant representations, supporting efficient parallel processing in hardware implementations. Balanced ternary, a prominent example in base b=3b=3, uses digits −1-1, $0, and &#36;1, often denoted as 1ˉ\bar{1}, $0, &#36;1 or T, 0, 1, providing a symmetric range around zero. The value of a representation like 11ˉ131 \bar{1} 1_3 is calculated as 1⋅32+(−1)⋅31+1⋅30=9−3+1=71 \cdot 3^2 + (-1) \cdot 3^1 + 1 \cdot 3^0 = 9 - 3 + 1 = 7, illustrating how negative digits contribute to the total without requiring a separate . Invented in the 1840s by self-taught Thomas Fowler, who built a based on this system, ensures every has a unique representation, eliminating ambiguities common in other redundant systems. This uniqueness stems from the digit set's minimal redundancy, covering the range −(3n−1)/2-(3^n - 1)/2 to (3n−1)/2(3^n - 1)/2 for nn digits without gaps or overlaps. Applications of signed-digit and balanced systems highlight their computational advantages, particularly in arithmetic efficiency. In balanced ternary, addition and subtraction can proceed without carry or borrow propagation in many cases, as the result of digit sums stays within the digit set or requires only local adjustments, enabling constant-time operations per digit. This property was exploited in Soviet computing prototypes, such as the Setun computer developed in 1958 at by Nikolai Brusentsov, which used logic to achieve lower component counts and energy efficiency compared to binary counterparts—requiring about 30% fewer elements for equivalent functionality. Later models like 70 in the 1970s further demonstrated ternary's potential for reliable, cost-effective computing in resource-constrained environments. Other signed-digit variants, such as balanced in base b=4b=4, employ digits like −1,0,1,2-1, 0, 1, 2 to approximate symmetry, though even bases introduce slight imbalances compared to odd-base systems. These allow multiple representations for , which can be resolved through forms that enforce rules like no adjacent non-zero digits of the same , minimizing hardware in multipliers and adders. The in signed-digit systems relates to broader redundant number frameworks, facilitating fault-tolerant designs and high-speed , but requires conversion algorithms to standard forms for interoperability. Every remains representable, though uniqueness is not always guaranteed without additional constraints.

Negative, Complex, and Non-Integer Bases

Negative bases, also known as negabases, employ a negative as the in positional numeral systems, allowing representation of both positive and negative s using non-negative digits without a separate . In such systems, the value of a number with digits dkdk−1…d1d0d_k d_{k-1} \dots d_1 d_0 in base b=−rb = -r (where r>1r > 1 is an ) is given by ∑i=0kdi(−r)i\sum_{i=0}^k d_i (-r)^i, with digits did_i ranging from 0 to r−1r-1. This approach simplifies arithmetic operations, as and can proceed without distinguishing between positive and negative operands, akin to but integrated into the base itself. Negative-base systems were explored in the mid-20th century for computational efficiency, with early applications in experimental computers during the . A prominent example is negabinary, or base −2-2, which uses digits and 1. The value is computed as ∑i=0kdi(−2)i\sum_{i=0}^k d_i (-2)^i. For instance, the representation 11−211_{-2} equals 1⋅(−2)1+1⋅(−2)0=−2+1=−11 \cdot (-2)^1 + 1 \cdot (-2)^0 = -2 + 1 = -1. Negabinary uniquely represents all integers and was utilized in Polish experimental computers like SKRZAT 1 and BINEG for its ability to handle signed values compactly. Other negative bases, such as base −3-3 (using digits 0, 1, 2), offer similar advantages, enabling balanced representations that avoid sign bits and facilitate hardware implementations without borrow propagation in subtraction. Complex bases extend this concept to radices in the , typically using Gaussian integers for the base to ensure representations converge. A notable case is base −1+i-1 + i, with magnitude 2>1\sqrt{2} > 1
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