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Numbers written from 0 to 9
The ten digits of the Arabic numerals, in order of value

A numerical digit (often shortened to just digit) or numeral is a single symbol used alone (such as "1"), or in combinations (such as "15"), to represent numbers in positional notation, such as the common base 10. The name "digit" originates from the Latin digiti meaning fingers.[1]

For any numeral system with an integer base, the number of different digits required is the absolute value of the base. For example, decimal (base 10) requires ten digits (0 to 9), and binary (base 2) requires only two digits (0 and 1). Bases greater than 10 require more than 10 digits, for instance hexadecimal (base 16) requires 16 digits (usually 0 to 9 and A to F).

Overview

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In a basic digital system, a numeral is a sequence of digits, which may be of arbitrary length. Each position in the sequence has a place value, and each digit has a value. The value of the numeral is computed by multiplying each digit in the sequence by its place value, and summing the results.

Digital values

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Each digit in a number system represents an integer. For example, in decimal the digit "1" represents the integer one, and in the hexadecimal system, the letter "A" represents the number ten. A positional number system has one unique digit for each integer from zero up to, but not including, the radix of the number system.

Thus in the positional decimal system, the numbers 0 to 9 can be expressed using their respective numerals "0" to "9" in the rightmost "units" position. The number 12 is expressed with the numeral "2" in the units position, and with the numeral "1" in the "tens" position, to the left of the "2" while the number 312 is expressed with three numerals: "3" in the "hundreds" position, "1" in the "tens" position, and "2" in the "units" position.

Computation of place values

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The decimal numeral system uses a decimal separator, commonly a period in English, or a comma in other European languages,[2] to denote the "ones place" or "units place",[3][4][5] which has a place value one. Each successive place to the left of this has a place value equal to the place value of the previous digit times the base. Similarly, each successive place to the right of the separator has a place value equal to the place value of the previous digit divided by the base. For example, in the numeral 10.34 (written in base 10),

the 0 is immediately to the left of the separator, so it is in the ones or units place, and is called the units digit or ones digit;[6][7][8]
the 1 to the left of the ones place is in the tens place, and is called the tens digit;[9]
the 3 is to the right of the ones place, so it is in the tenths place, and is called the tenths digit;[10]
the 4 to the right of the tenths place is in the hundredths place, and is called the hundredths digit.[10]

The total value of the number is 1 ten, 0 ones, 3 tenths, and 4 hundredths. The zero, which contributes no value to the number, indicates that the 1 is in the tens place rather than the ones place.

The place value of any given digit in a numeral can be given by a simple calculation, which in itself is a complement to the logic behind numeral systems. The calculation involves the multiplication of the given digit by the base raised by the exponent n − 1, where n represents the position of the digit from the separator; the value of n is positive (+), but this is only if the digit is to the left of the separator. And to the right, the digit is multiplied by the base raised by a negative (−) n. For example, in the number 10.34 (written in base 10),

the 1 is second to the left of the separator, so based on calculation, its value is,
the 4 is second to the right of the separator, so based on calculation its value is,

History

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Main article: History of the Hindu–Arabic numeral system
Western Arabic 0 1 2 3 4 5 6 7 8 9
Eastern Arabic ٠ ١ ٢ ٣ ٤ ٥ ٦ ٧ ٨ ٩
Persian ۰ ۱ ۲ ۳ ۴ ۵ ۶ ۷ ۸ ۹
Devanagari ० १ २ ३ ४ ५ ६ ७ ८ ९
Kadamba ೦ ೧ ೨ ೩ ೪ ೫ ೬ ೭ ೮ ೯

The first true written positional numeral system is considered to be the Hindu–Arabic numeral system. This system was established by the 7th century in India,[11] but was not yet in its modern form because the use of the digit zero had not yet been widely accepted. Instead of a zero sometimes the digits were marked with dots to indicate their significance, or a space was used as a placeholder. The first widely acknowledged use of zero was in 876.[12] The original numerals were very similar to the modern ones, even down to the glyphs used to represent digits.[11]

The digits of the Maya numeral system

By the 13th century, Western Arabic numerals were accepted in European mathematical circles (Fibonacci used them in his Liber Abaci). They began to enter common use in the 15th century.[13] By the end of the 20th century virtually all non-computerized calculations in the world were done with Arabic numerals, which have replaced native numeral systems in most cultures.

Other historical numeral systems using digits

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The exact age of the Maya numerals is unclear, but it is possible that it is older than the Hindu–Arabic system. The system was vigesimal (base 20), so it has twenty digits. The Mayas used a shell symbol to represent zero. Numerals were written vertically, with the ones place at the bottom. The Mayas had no equivalent of the modern decimal separator, so their system could not represent fractions.

The Thai numeral system is identical to the Hindu–Arabic numeral system except for the symbols used to represent digits. The use of these digits is less common in Thailand than it once was, but they are still used alongside Arabic numerals.

The rod numerals, the written forms of counting rods once used by Chinese and Japanese mathematicians, are a decimal positional system able to represent not only zero but also negative numbers. Counting rods themselves predate the Hindu–Arabic numeral system. The Suzhou numerals are variants of rod numerals.

Rod numerals (vertical)
0 1 2 3 4 5 6 7 8 9
−0 −1 −2 −3 −4 −5 −6 −7 −8 −9

Modern digital systems

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In computer science

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The binary (base 2), octal (base 8), and hexadecimal (base 16) systems, extensively used in computer science, all follow the conventions of the Hindu–Arabic numeral system.[14] The binary system uses only the digits "0" and "1", while the octal system uses the digits from "0" through "7". The hexadecimal system uses all the digits from the decimal system, plus the letters "A" through "F", which represent the numbers 10 to 15 respectively.[15] When the binary system is used, the term "bit(s)" is typically used as an alternative for "digit(s)", being a portmanteau of the term "binary digit".

Unusual systems

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The ternary and balanced ternary systems have sometimes been used. They are both base 3 systems.[16]

Balanced ternary is unusual in having the digit values 1, 0 and −1. Balanced ternary turns out to have some useful properties and the system has been used in the experimental Russian Setun computers.[17]

Several authors in the last 300 years have noted a facility of positional notation that amounts to a modified decimal representation. Some advantages are cited for use of numerical digits that represent negative values. In 1840 Augustin-Louis Cauchy advocated use of signed-digit representation of numbers, and in 1928 Florian Cajori presented his collection of references for negative numerals. The concept of signed-digit representation has also been taken up in computer design.

Digits in mathematics

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Despite the essential role of digits in describing numbers, they are relatively unimportant to modern mathematics.[18] Nevertheless, there are a few important mathematical concepts that make use of the representation of a number as a sequence of digits.

Digital roots

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Main article: Digital root

The digital root is the single-digit number obtained by summing the digits of a given number, then summing the digits of the result, and so on until a single-digit number is obtained.[19]

Casting out nines

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Main article: Casting out nines

Casting out nines is a procedure for checking arithmetic done by hand. To describe it, let represent the digital root of , as described above. Casting out nines makes use of the fact that if , then . In the process of casting out nines, both sides of the latter equation are computed, and if they are not equal, the original addition must have been faulty.[20]

Repunits and repdigits

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Main article: Repunit

Repunits are integers that are represented with only the digit 1. For example, 1111 (one thousand, one hundred and eleven) is a repunit. Repdigits are a generalization of repunits; they are integers represented by repeated instances of the same digit. For example, 333 is a repdigit. The primality of repunits is of interest to mathematicians.[21]

Palindromic numbers and Lychrel numbers

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Main article: Palindromic number

Palindromic numbers are numbers that read the same when their digits are reversed.[22] A Lychrel number is a positive integer that never yields a palindromic number when subjected to the iterative process of being added to itself with digits reversed.[23] The question of whether there are any Lychrel numbers in base 10 is an open problem in recreational mathematics; the smallest candidate is 196.[24]

History of ancient numbers

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Main article: History of writing ancient numbers

Counting aids, especially the use of body parts (counting on fingers), were certainly used in prehistoric times as today. There are many variations. Besides counting ten fingers, some cultures have counted knuckles, the space between fingers, and toes as well as fingers. The Oksapmin culture of New Guinea uses a system of 27 upper body locations to represent numbers.[25]

To preserve numerical information, tallies carved in wood, bone, and stone have been used since prehistoric times.[26] Stone age cultures, including ancient indigenous American groups, used tallies for gambling, personal services, and trade-goods.

A method of preserving numeric information in clay was invented by the Sumerians between 8000 and 3500 BC.[27] This was done with small clay tokens of various shapes that were strung like beads on a string. Beginning about 3500 BC, clay tokens were gradually replaced by number signs impressed with a round stylus at different angles in clay tablets (originally containers for tokens) which were then baked. About 3100  BC, written numbers were dissociated from the things being counted and became abstract numerals.

Between 2700 and 2000 BC, in Sumer, the round stylus was gradually replaced by a reed stylus that was used to press wedge-shaped cuneiform signs in clay. These cuneiform number signs resembled the round number signs they replaced and retained the additive sign-value notation of the round number signs. These systems gradually converged on a common sexagesimal number system; this was a place-value system consisting of only two impressed marks, the vertical wedge and the chevron, which could also represent fractions.[28] This sexagesimal number system was fully developed at the beginning of the Old Babylonia period (about 1950 BC) and became standard in Babylonia.[29]

Sexagesimal numerals were a mixed radix system that retained the alternating base 10 and base 6 in a sequence of cuneiform vertical wedges and chevrons. By 1950 BC, this was a positional notation system. Sexagesimal numerals came to be widely used in commerce, but were also used in astronomical and other calculations. This system was exported from Babylonia and used throughout Mesopotamia, and by every Mediterranean nation that used standard Babylonian units of measure and counting, including the Greeks, Romans and Egyptians. Babylonian-style sexagesimal numeration is still used in modern societies to measure time (minutes per hour) and angles (degrees).[30]

History of modern numbers

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In China, armies and provisions were counted using modular tallies of prime numbers. Unique numbers of troops and measures of rice appear as unique combinations of these tallies. A great convenience of modular arithmetic is that it is easy to multiply.[31] This makes use of modular arithmetic for provisions especially attractive. Conventional tallies are quite difficult to multiply and divide. In modern times modular arithmetic is sometimes used in digital signal processing.[32]

The oldest Greek system was that of the Attic numerals,[33] but in the 4th century BC they began to use a quasidecimal alphabetic system (see Greek numerals).[34] Jews began using a similar system (Hebrew numerals), with the oldest examples known being coins from around 100 BC.[35]

The Roman empire used tallies written on wax, papyrus and stone, and roughly followed the Greek custom of assigning letters to various numbers. The Roman numerals system remained in common use in Europe until positional notation came into common use in the 16th century.[36]

The Maya of Central America used a mixed base 18 and base 20 system, possibly inherited from the Olmec, including advanced features such as positional notation and a zero.[37] They used this system to make advanced astronomical calculations, including highly accurate calculations of the length of the solar year and the orbit of Venus.[38]

The Incan Empire ran a large command economy using quipu, tallies made by knotting colored fibers.[39] Knowledge of the encodings of the knots and colors was suppressed by the Spanish conquistadors in the 16th century, and has not survived although simple quipu-like recording devices are still used in the Andean region.

Some authorities believe that positional arithmetic began with the wide use of counting rods in China.[40] The earliest written positional records seem to be rod calculus results in China around 400. Zero was first used in India in the 7th century CE by Brahmagupta.[41]

The modern positional Arabic numeral system was developed by mathematicians in India, and passed on to Muslim mathematicians, along with astronomical tables brought to Baghdad by an Indian ambassador around 773.[42]

From India, the thriving trade between Islamic sultans and Africa carried the concept to Cairo. Arabic mathematicians extended the system to include decimal fractions, and Muḥammad ibn Mūsā al-Ḵwārizmī wrote an important work about it in the 9th  century.[43] The modern Arabic numerals were introduced to Europe with the translation of this work in the 12th century in Spain and Leonardo of Pisa's Liber Abaci of 1201.[44] In Europe, the complete Indian system with the zero was derived from the Arabs in the 12th century.[45]

The binary system (base 2) was propagated in the 17th century by Gottfried Leibniz.[46] Leibniz had developed the concept early in his career, and had revisited it when he reviewed a copy of the I Ching from China.[47] Binary numbers came into common use in the 20th century because of computer applications.[46]

[edit]
West Arabic 0 1 2 3 4 5 6 7 8 9
Asomiya (Assamese); Bengali ০ ১ ২ ৩ ৪ ৫ ৬ ৭ ৮ ৯
Devanagari ० १ २ ३ ४ ५ ६ ७ ८ ९
East Arabic ٠ ١ ٢ ٣ ٤ ٥ ٦ ٧ ٨ ٩
Persian ٠ ١ ٢ ٣ ۴ ۵ ۶ ٧ ٨ ٩
Gurmukhi ੦ ੧ ੨ ੩ ੪ ੫ ੬ ੭ ੮ ੯
Urdu ۰ ۱ ۲ ۳ ۴ ۵ ۶ ۷ ۸ ۹
Chinese (everyday) 〇 一 二 三 四 五 六 七 八 九
Chinese (Traditional) 零 壹 貳 叄 肆 伍 陸 柒 捌 玖
Chinese (Simplified) 零 壹 贰 叁 肆 伍 陆 柒 捌 玖
Chinese (Suzhou) 〇 〡 〢 〣 〤 〥 〦 〧 〨 〩
Ge'ez (Ethiopic) ፩ ፪ ፫ ፬ ፭ ፮ ፯ ፰ ፱
Gujarati ૦ ૧ ૨ ૩ ૪ ૫ ૬ ૭ ૮ ૯
Hieroglyphic Egyptian 𓏺 𓏻 𓏼 𓏽 𓏾 𓏿 𓐀 𓐁 𓐂
Japanese (everyday) 〇 一 二 三 四 五 六 七 八 九
Japanese (formal) 零 壱 弐 参 四 五 六 七 八 九
Kannada ೦ ೧ ೨ ೩ ೪ ೫ ೬ ೭ ೮ ೯
Khmer (Cambodia) ០ ១ ២ ៣ ៤ ៥ ៦ ៧ ៨ ៩
Lao ໐ ໑ ໒ ໓ ໔ ໕ ໖ ໗ ໘ ໙
Limbu ᥆ ᥇ ᥈ ᥉ ᥊ ᥋ ᥌ ᥍ ᥎ ᥏
Malayalam ൦ ൧ ൨ ൩ ൪ ൫ ൬ ൭ ൮ ൯
Mongolian ᠐ ᠑ ᠒ ᠓ ᠔ ᠕ ᠖ ᠗ ᠘ ᠙
Burmese ၀ ၁ ၂ ၃ ၄ ၅ ၆ ၇ ၈ ၉
Oriya ୦ ୧ ୨ ୩ ୪ ୫ ୬ ୭ ୮ ୯
Roman I II III IV V VI VII VIII IX
Shan ႐ ႑ ႒ ႓ ႔ ႕ ႖ ႗ ႘ ႙
Sinhala 𑇡 𑇢 𑇣 𑇤 𑇥 𑇦 𑇧 𑇨 𑇩
Tamil ௦ ௧ ௨ ௩ ௪ ௫ ௬ ௭ ௮ ௯
Telugu ౦ ౧ ౨ ౩ ౪ ౫ ౬ ౭ ౮ ౯
Thai ๐ ๑ ๒ ๓ ๔ ๕ ๖ ๗ ๘ ๙
Tibetan ༠ ༡ ༢ ༣ ༤ ༥ ༦ ༧ ༨ ༩
New Tai Lue ᧐ ᧑ ᧒ ᧓ ᧔ ᧕ ᧖ ᧗ ᧘ ᧙
Javanese ꧐ ꧑ ꧒ ꧓ ꧔ ꧕ ꧖ ꧗ ꧘ ꧙

Additional numerals

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1 5 10 20 30 40 50 60 70 80 90 100 500 1000 10000 108
Chinese (ordinary) 一 五 十 二十 三十 四十 五十 六十 七十 八十 九十 百 五百 千 万 亿
Chinese (financial) 壹 伍 拾 贰拾 叁拾 肆拾 伍拾 陆拾 柒拾 捌拾 玖拾 佰 伍佰 仟 萬 億
Geʽez ፩ ፭ ፲ ፳ ፴ ፵ ፶ ፷ ፸ ፹ ፺ ፻ ፭፻ ፲፻ ፼ ፼፼
Roman I V X XX XXX XL L LX LXX LXXX XC C D M X

See also

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References

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Numerical digit

View on Grokipedia
from Grokipedia
A numerical digit is a single symbol used to represent an integer value from zero up to one less than the base of a positional numeral system, serving as the basic building block for constructing numerals that denote larger numbers. In the widely used decimal system (base-10), the ten digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, which combine through positional notation to express any non-negative integer. The concept of digits has ancient origins, with early numeral systems emerging in civilizations such as around 3500–3000 BCE, where wedge-shaped symbols on clay tablets represented quantities in a base-60 system. The modern digits 0 through 9 evolved from the of ancient , which originated in the 3rd century BCE. Indian mathematicians later developed the place-value system incorporating zero as a placeholder by the 5th to 7th centuries CE. These Indian digits spread to the by the 9th century through scholars like , who refined and documented the system, and were later introduced to via trade and academic exchanges in the 10th to 12th centuries, gradually replacing for their efficiency in arithmetic. The English term "digit" derives from the Latin digitus, meaning "finger," alluding to the historical use of fingers for up to ten. Beyond the system, digits vary by base: binary (base-2) employs only 0 and 1 for digital computing and logic; ternary (base-3) uses 0, 1, and 2; and (base-16) extends to 0-9 plus A-F (representing 10-15) for efficient representation in programming. Digits underpin essential mathematical operations, including , , and , and play a critical role in fields like , where binary digits (bits) form the foundation of all digital information processing.

Fundamentals

Definition and Properties

A numerical digit is a single symbol that represents one of the integer values from to b−1b-1 in a given base-bb , where b>1b > 1 is the base or . These symbols form the basic units for expressing numbers within that . In the decimal (base 10), the digits are the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, each corresponding to values zero through nine. In the (base 2), the digits are simply 0 and 1, representing zero and one, respectively. For the (base 16), the digits include 0 through 9 for values zero to nine, and additional symbols such as A, B, C, D, E, and F for values ten through fifteen. Key properties of numerical digits include their finiteness: each base-bb system employs exactly bb distinct symbols, limiting the set to a fixed, non-infinite collection. Additionally, digit symbols maintain independence from the writing direction of the ambient text; they are consistently rendered from left to right, regardless of whether the surrounding script flows right-to-left, as in or Hebrew. Digits function as the essential building blocks of larger numbers, combining in sequences to denote quantities via . A crucial distinction exists between a digit's glyph—the specific visual or graphical form of the symbol—and the abstract numerical value it denotes. For instance, the glyph 3 (in ) is the written mark, while its value is the integer three, which remains constant across contexts where the symbol is used appropriately.

Positional Notation and Place Value

, also known as place-value notation, is a in which the value of a digit depends on its position relative to other digits in the number, with each position corresponding to a power of the system's base. In contrast, non-positional systems, such as additive numeral systems like , assign fixed values to symbols regardless of their order, where the total value is the sum of individual symbol values, often with limited subtractive rules for efficiency. This positional approach allows for compact representation of using a of symbols, known as digits, each representing values from to one less than the base. In positional systems, place values are computed as successive powers of the base b. For example, in the decimal system (base 10), the rightmost position represents the units place with value 100=110^0 = 1, the next to the left is the tens place with 101=1010^1 = 10, followed by the hundreds place with 102=10010^2 = 100, and so on. Each subsequent position to the left multiplies the previous place value by the base, enabling the scaling of digit contributions based on their location. The total value VV of a multi-digit number in with digits dn,dn−1,…,d0d_n, d_{n-1}, \dots, d_0 and base bb is given by the formula: V=dn⋅bn+dn−1⋅bn−1+⋯+d1⋅b1+d0⋅b0V = d_n \cdot b^n + d_{n-1} \cdot b^{n-1} + \dots + d_1 \cdot b^1 + d_0 \cdot b^0 where each did_i is a digit satisfying 0≤di<b0 \leq d_i < b. This summation reflects how positions amplify the intrinsic value of each digit through exponentiation. For instance, the decimal number 123 expands to 1⋅102+2⋅101+3⋅100=100+20+3=1231 \cdot 10^2 + 2 \cdot 10^1 + 3 \cdot 10^0 = 100 + 20 + 3 = 123, demonstrating how the position of each digit determines its contribution to the overall value. Similarly, in base 5, the number 342_5 equals 3⋅52+4⋅51+2⋅50=75+20+2=97103 \cdot 5^2 + 4 \cdot 5^1 + 2 \cdot 5^0 = 75 + 20 + 2 = 97_{10}.

Historical Development

Ancient Numeral Systems

One of the earliest known numeral systems emerged in ancient Mesopotamia around 2000 BCE, utilizing cuneiform script to represent numbers in a base-60 (sexagesimal) framework. This system employed two primary symbols—a vertical wedge for units of 1 and a chevron-like symbol for 10—with quantities formed additively by repetition, such as five wedges for 5 or combinations for higher values up to 59. While primarily additive and non-positional, it incorporated rudimentary positional elements for larger numbers, where context or spacing indicated multiples of 60, though the absence of a zero symbol led to ambiguities in interpretation. This sexagesimal approach facilitated advanced calculations in astronomy, timekeeping, and commerce, influencing later divisions like 60 seconds in a minute. In ancient Egypt, hieroglyphic numerals developed concurrently around 3000 BCE as a base-10 system that was strictly non-positional and additive, relying on distinct symbols for each power of 10 rather than abstract digits. The basic unit was a single vertical stroke for 1, repeated up to nine times; a heel bone represented 10, a coiled rope 100, a lotus flower 1,000, a pointing finger 10,000, a seated frog or tadpole 100,000, and a god with raised arms 1,000,000, with numbers formed by juxtaposing these symbols without regard to order. This system supported practical applications in architecture, taxation, and pyramid construction but required extensive repetition for large values, limiting efficiency for complex arithmetic. A cursive variant, hieratic numerals, later simplified these for everyday scribal use. Roman numerals, originating by the 1st century BCE in the Italic region under Etruscan influence, formed a base-10 system that was largely non-positional and additive, though it included quasi-positional subtractive principles for efficiency. Core symbols included I for 1, V for 5, X for 10, L for 50, C for 100, D for 500, and M for 1,000, with values like 4 written as IV (subtracting 1 from 5) and 9 as IX, while larger numbers added symbols sequentially, such as VIII for 8. This notation evolved from tally marks and finger-counting methods used in trade and military contexts, prioritizing simplicity over algebraic manipulation, and persisted in Europe for centuries due to its utility in inscriptions and record-keeping despite limitations in multiplication and division. In the Indian subcontinent, early numerals in the Brahmi script appeared around the 3rd century BCE, marking a pivotal shift toward a base-10 system with distinct symbols for digits 1 through 9 that served as precursors to modern Hindu-Arabic numerals. These angular glyphs, inscribed on ashoka pillars and coins, were initially used additively like their predecessors but gradually incorporated positional notation, where the placement of digits indicated powers of 10, enabling compact representation of large numbers. This innovation arose amid the Mauryan Empire's administrative needs, evolving from earlier Indus Valley symbols and facilitating advancements in trade and astronomy. The concept of zero as a placeholder emerged gradually, with early uses in the 3rd–4th centuries CE and formal rules by Brahmagupta in the 7th century CE; a key development occurred in the 5th century CE, when the mathematician Aryabhata employed zero in his positional system, as detailed in his Aryabhatiya, allowing for unambiguous representation of absent values and transforming arithmetic into a fully operational framework.

Transition to Modern Digits

The positional decimal numeral system, originating in ancient India, spread to the Islamic world during the 9th century CE through the works of Persian mathematician Muhammad ibn Musa al-Khwarizmi, whose treatise On the Calculation with Hindu Numerals introduced the digits 0 through 9 and their arithmetic operations to Arabic scholars. This adoption facilitated advancements in algebra and commerce across the , with al-Khwarizmi's text serving as a key conduit for the system's transmission from Indian . In Europe, the Hindu-Arabic numerals gained prominence in the early 13th century through Italian mathematician Leonardo of Pisa, known as , who detailed their use in his 1202 book Liber Abaci, promoting them over for their efficiency in multiplication and division among merchants. 's work, drawing from encountered during his travels, included practical examples like accounting problems, accelerating the numerals' integration into European trade and scholarship by the late Middle Ages. The shapes of these digits evolved gradually from the angular forms of the in ancient India (circa 3rd century BCE) through intermediate Arabic variants, such as the Eastern and Western Arabic forms, to the more rounded Western European glyphs by the Renaissance. For instance, the digit 4 transitioned from an open triangular shape in early Brahmi and Arabic "ghubar" numerals to both open and closed variants in medieval manuscripts, with the closed form (resembling a triangle with a crossbar) becoming dominant in printed texts. Similarly, the digit 7 simplified from a Brahmi double stroke to a single angled line with a crossbar in modern Western usage, reflecting adaptations for cursive writing and legibility across cultural transmissions. The invention of the movable-type printing press by Johannes Gutenberg around 1450 played a pivotal role in standardizing these digit glyphs across Europe, as consistent metal typefaces in incunabula (early printed books) reduced regional variations and promoted uniform adoption of the Hindu-Arabic forms. By the end of the 15th century, printed mathematical texts and commercial documents increasingly featured these standardized symbols, solidifying their place in Western notation up to the 16th century. Parallel to this Indo-Arabic lineage, other historical systems demonstrated positional principles, such as the Mayan base-20 (vigesimal) numerals, which used dots for units, bars for fives, and a shell for zero, emerging during the Classic Period (c. 250–900 CE) in Mesoamerica for calendrical and astronomical calculations. In East Asia, Chinese rod numerals formed a positional decimal system using horizontal and vertical rods on counting boards, dating from the 4th to 2nd centuries BCE, which supported advanced computations in texts like The Nine Chapters on the Mathematical Art.

Common Numeral Systems

Decimal System

The decimal system, also known as the base-10 numeral system, is the predominant positional notation for representing numbers worldwide, employing ten distinct digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. In this system, each digit's value depends on its position relative to the others, with powers of 10 determining place values, and the digit 0 functioning primarily as a placeholder to maintain positional integrity without adding numerical value. This structure enables efficient representation of both integers and fractions, separated by a decimal point for the latter. Numbers in the decimal system are conventionally written and read from left to right, beginning with the most significant digit (the highest place value) on the left and proceeding to the least significant digit on the right. For instance, the number 123 is interpreted as 1 × 10² + 2 × 10¹ + 3 × 10⁰. Leading zeros are generally omitted in integer notation to avoid redundancy, as they do not alter the value (e.g., 0123 is written as 123), but a leading zero is included before the decimal point for numbers less than 1 to clarify the position (e.g., 0.123). Trailing zeros after the decimal point, however, are retained to denote precision, such as in measurements where 1.230 indicates three decimal places of accuracy. Arithmetic operations in the decimal system follow aligned place-value addition or subtraction, as exemplified by the addition 123 + 456 = 579, where digits are summed column by column from right to left, carrying over as needed. Verbalizing numbers also adheres to structured rules; in English, for example, 2025 is expressed as "two thousand twenty-five," grouping digits in threes from the right and naming powers of 1,000 accordingly. Cultural variations appear in the typographic forms of these digits, distinguishing Western Arabic numerals (the familiar 0-9 shapes used globally in the West) from Eastern Arabic numerals (٠-٩), which retain more angular, distinct forms and are prevalent in typography across the Middle East and parts of Asia. The Western forms evolved in North Africa and Spain, facilitating their adoption in Europe, while Eastern variants persisted in regions like Persia, reflecting regional scribal practices and historical divergences in numeral evolution.

Non-Decimal Bases

Non-decimal bases employ positional notation with a radix other than 10, requiring a distinct set of digits to represent values from 0 up to one less than the base. The operates in base-2, utilizing only the digits 0 and 1 to encode all numbers. Octal functions in base-8, employing digits 0 through 7. uses base-16, incorporating digits 0-9 followed by A for 10, B for 11, C for 12, D for 13, E for 14, and F for 15. Conversion from decimal to a non-decimal base follows an iterative process of repeated division by the target base, where the remainders form the digits from least to most significant. For example, converting 10 in decimal to yields 1010: divide 10 by 2 to get quotient 5 and remainder 0; divide 5 by 2 to get 2 and 1; divide 2 by 2 to get 1 and 0; divide 1 by 2 to get 0 and 1, reading remainders upward as 1010. Mathematically, for a decimal number nn and base bb, the digits dkd_k are obtained via dk=nmod  bd_k = n \mod b and n=⌊n/b⌋n = \lfloor n / b \rfloor, repeating until n=0n = 0, with the sequence of dkd_k reversed for the final representation. Other non-decimal bases appear in practical measurements, such as base-12 (dozenal), which requires digits 0-9 along with X for 10 and E for 11. Base-12 facilitates divisions by 2, 3, 4, and 6, offering advantages in fractional arithmetic over base-10 due to its greater number of divisors. Base-60, or sexagesimal, structures time with 60 minutes per hour and 60 seconds per minute, and angles with 60 arcminutes per degree and 60 arcseconds per arcminute. Higher bases like hexadecimal provide compactness, representing the same value with fewer symbols than lower bases such as binary, which results in longer strings for equivalent numbers. Conversely, lower bases like binary emphasize simplicity in fundamental operations, though at the cost of increased length for larger values.

Applications in Computing

Digits in Binary and Hexadecimal

In digital electronics, binary digits, commonly known as bits, form the fundamental unit of information, limited to two possible states: 0 and 1, which represent off and on electrical conditions in circuits such as switches or transistors. The term "bit" originated from "binary digit," coined by John W. Tukey and first used by Claude Shannon in his 1948 paper on information theory. This binary foundation enables all digital processing, as computers manipulate vast sequences of these bits to encode and compute data. The reliance on binary digits traces back to early electronic computers in the 1940s, where vacuum tubes served as the primary switching elements to realize these on/off states; for instance, the Atanasoff-Berry Computer (1942) used vacuum tubes to store and process binary digits electronically. By the mid-1950s, the shift to transistors revolutionized hardware while preserving binary as the core representation: the (Transistor DIgital Computer), completed in 1955 by Bell Laboratories, was the first fully transistorized general-purpose computer and operated entirely on binary digits for its logic and storage. This evolution from fragile vacuum tubes—prone to frequent failures in machines like —to more reliable, compact transistors enabled scalable binary-based computing that dominates modern systems. Hexadecimal notation offers a human-readable shorthand for binary data, grouping four bits into a single hex digit ranging from 0-9 or A-F; for example, the binary pattern 1010 (ten in decimal) equates to A in hexadecimal. This 4-bit-to-1-hex mapping, often called a nibble, reduces the verbosity of binary strings, making it ideal for representing machine code, registers, and debugging in software development. In memory storage, computers organize data into bytes—units of eight binary digits—allowing 256 possible values per byte (from 00000000 to 11111111 in binary). Memory addresses, which identify byte locations, are conventionally denoted in hexadecimal for brevity; for instance, the address 0xFF points to the 256th byte (255 in zero-based decimal indexing). This hex addressing simplifies navigation in low-level programming and hardware documentation. Bitwise operations on binary digits enable precise low-level manipulations essential for computing tasks like graphics rendering, cryptography, and optimization. Shifts relocate bits within a word—for example, a left shift by n positions multiplies the value by 2^n, as seen in binary patterns like 00000001 shifted left once becoming 00000010. Masks, typically using the bitwise AND operator, isolate or clear specific bits; applying a mask of 00001111 to a byte preserves only the lower four bits while setting the upper four to zero. These operations, performed directly on hardware registers, underpin efficient data processing in binary systems.

Encoding and Representation

In digital systems, numerical digits are encoded as characters to facilitate text processing and display. The American Standard Code for Information Interchange (ASCII), a 7-bit encoding standard, maps the decimal digits 0 through 9 to the integer values 48 through 57, respectively (hexadecimal 0x30 to 0x39). This assignment allows digits to be stored and transmitted as binary data while preserving their human-readable form in text streams. Unicode, a universal character encoding standard, incorporates the ASCII digits unchanged in its Basic Latin block at code points U+0030 through U+0039, ensuring backward compatibility with ASCII systems. Beyond these, Unicode defines multiple sets of decimal digits tailored to various scripts, such as the Arabic-Indic digits (U+0660–U+0669) used in Arabic and Persian texts, and the Devanagari digits (U+0966–U+096F) for Hindi and related languages. These encodings support internationalization by allowing digits to visually match surrounding script styles without altering their numerical value. Binary-coded decimal (BCD) provides an alternative representation where each decimal digit is encoded independently using a fixed-width binary field, typically four bits, directly mirroring the digit's value (e.g., decimal 5 becomes 0101 in binary). This method, often implemented as 8421-weighted code, avoids the need for binary-to-decimal conversion during arithmetic operations on decimal numbers, making it useful in financial and legacy computing applications. For error detection in decimal-based data transmission or storage, checksum methods frequently rely on summing the digits modulo 10 to generate a check digit, ensuring the total sum is divisible by 10 for validation. The Luhn algorithm, a widely adopted variant, weights alternate digits by doubling them before summing, then applies modulo 10 to detect single-digit errors and some transposition mistakes in identifiers like credit card numbers. Seven-segment displays represent decimal digits through the selective illumination of seven linear segments arranged in a figure-eight pattern, plus an optional decimal point, enabling compact visual output in devices such as calculators and digital meters. Each digit corresponds to a unique combination of lit segments; for instance, digit 0 activates six segments (all except the middle), while digit 7 activates the top, upper-right, and lower-right segments. These displays are driven by BCD inputs via decoder circuits to map binary codes to segment patterns efficiently. Modern extensions in Unicode include full-width digits (U+FF10–U+FF19), which are proportionally wider to align with East Asian ideographic characters in typesetting, contrasting the half-width variants. Additionally, Unicode supports emoji representations of digits through keycap sequences, such as the keycap digit one (U+0031 U+FE0F U+20E3, displayed as 1️⃣), which enclose the digit in a stylized button for use in digital messaging and interfaces.

Mathematical Properties

Digital Root and Divisibility Rules

The digital root of a positive integer nn is obtained by repeatedly summing its digits until a single digit is reached. For example, the digital root of 123 is calculated as 1+2+3=61 + 2 + 3 = 6. This process is mathematically equivalent to nmod  9n \mod 9, with the exception that if n≡0(mod9)n \equiv 0 \pmod{9} and n≠0n \neq 0, the digital root is 9 rather than 0. The equivalence arises because in base 10, 10≡1(mod9)10 \equiv 1 \pmod{9}, so 10k≡1(mod9)10^k \equiv 1 \pmod{9} for any non-negative integer kk. Thus, any positive integer n=dm10m+⋯+d110+d0n = d_m 10^m + \cdots + d_1 10 + d_0 satisfies n≡dm+⋯+d1+d0(mod9)n \equiv d_m + \cdots + d_1 + d_0 \pmod{9}, and repeating the sum preserves the congruence until a single digit is obtained. An explicit formula for the digital root of n>0n > 0 is dr(n)=1+(n−1)mod  9dr(n) = 1 + (n - 1) \mod 9. Digital roots facilitate divisibility rules in base 10. A number is divisible by 3 if the sum of its digits is divisible by 3, which is equivalent to its being 0, 3, 6, or 9. Similarly, a number is divisible by 9 if its is 9 (or 0 for n=0n=0). The "" method, a practical application, involves subtracting multiples of 9 from the number (or ignoring digits that sum to 9) to check these conditions efficiently. For instance, consider 729: 7+2+9=187 + 2 + 9 = 18, then 1+8=91 + 8 = 9, so the is 9. Since 729≡9(mod9)729 \equiv 9 \pmod{9}, it is divisible by 9 (729=9×81729 = 9 \times 81). This follows from the congruence: 729=7⋅102+2⋅10+9≡7⋅1+2⋅1+9⋅1=18≡0(mod9)729 = 7 \cdot 10^2 + 2 \cdot 10 + 9 \equiv 7 \cdot 1 + 2 \cdot 1 + 9 \cdot 1 = 18 \equiv 0 \pmod{9}, confirming divisibility.

Special Digit Sequences

Special digit sequences refer to numbers characterized by repetitive or symmetric patterns in their decimal representations, exhibiting unique mathematical properties such as primality or resistance to palindromic transformation. These sequences include , repdigits, palindromic numbers, and Lychrel numbers, each defined by specific digit arrangements that influence their and behavior under iterative operations. Repunits are numbers consisting entirely of the digit 1 repeated n times, formally defined as Rn=10n−19R_n = \frac{10^n - 1}{9}. For example, R2=11R_2 = 11, R3=111R_3 = 111, and R6=111111R_6 = 111111. Their primality varies: R2=11R_2 = 11 is prime, while R3=3×37R_3 = 3 \times 37 and R6=3×7×11×13×37R_6 = 3 \times 7 \times 11 \times 13 \times 37 are composite. Repdigits generalize repunits to any repeated digit d from 1 to 9, forming numbers like 777 for n=3. A repdigit of length n with digit d is given by d×Rnd \times R_n, so its prime factorization incorporates the factors of R_n scaled by d, leading to patterns where composite n often yields composite repdigits unless d and R_n align to produce a prime. For instance, 777 = 3 \times 7 \times 37, sharing factors with R_3. Palindromic numbers are those that read the same forwards and backwards, such as 121. In base b, they exhibit reflectional in their digit expansion. In base 10, the count of n-digit palindromic numbers is 9×10⌊(n−1)/2⌋9 \times 10^{\lfloor (n-1)/2 \rfloor}; for n=3, there are 90 such numbers (e.g., 101 to 999 with mirrored digits). Generation in other bases follows similar mirroring, with every being palindromic in bases like 1 or N+1 for number N. Lychrel numbers are candidates that fail to yield a through the iterative reverse-and-add process, where a number is added to its digit reversal repeatedly. For example, starting with 196: 196 + 691 = 887, 887 + 788 = 1675, and continuing yields no known after millions of iterations, making 196 a prominent Lychrel candidate in base 10.

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