Editing Numerical digit

{{Short description|Symbols used to write numbers}}
{{Use dmy dates|date=September 2023}}
[[File:Hindu–Arabic numerals.svg|upright=1.5|thumb|alt=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 [[number]]s in [[positional notation]], such as the common [[base&nbsp;10]]. The name "digit" originates from the [[Latin]] ''digiti'' meaning fingers.<ref>{{cite web |url=http://dictionary.reference.com/browse/digit?s=t |title="Digit" Origin |publisher=[[dictionary.com]] |access-date=23 May 2015}}</ref>

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

==Overview==
In a basic digital system, a [[numeral system|numeral]] is a sequence of digits, which may be of arbitrary length. Each position in the sequence has a [[positional notation|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===
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 [[10 (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===
The [[decimal]] numeral system uses a [[decimal separator]], commonly a [[period (punctuation)|period]] in English, or a [[comma]] in other [[Europe]]an languages,<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Decimal Point|url=https://mathworld.wolfram.com/DecimalPoint.html|access-date=2020-07-22|website=mathworld.wolfram.com|language=en}}</ref> to denote the "ones place" or "units place",<ref>{{Cite book|author=Snyder, Barbara Bode|title=Practical math for the technician : the basics|date=1991|publisher=Prentice Hall|isbn=0-13-251513-X|location=Englewood Cliffs, N.J.|pages=225|oclc=22345295|quote=units or ones place}}</ref><ref name="Rickoff1888">{{cite book|author=Andrew Jackson Rickoff|title=Numbers Applied|url=https://books.google.com/books?id=IYvSWIw3oxUC&pg=PA5|year=1888|publisher=D. Appleton & Company|pages=5–|quote=units' or ones' place}}</ref><ref name="McClymondsJones1905">{{cite book|author1=John William McClymonds|author2=D. R. Jones|title=Elementary Arithmetic|url=https://books.google.com/books?id=xwYAAAAAYAAJ&pg=PA17|year=1905|publisher=R.L. Telfer|pages=17–18|quote=units' or ones' place}}</ref> 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 [[radix|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&nbsp;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'';<ref name="JohnsonLendsey1967">{{cite book|author1=Richard E. Johnson|url=https://books.google.com/books?id=W4AXAQAAMAAJ|title=Introductory Algebra for College Students|author2=Lona Lee Lendsey|author3=William E. Slesnick|publisher=Addison-Wesley Publishing Company|year=1967|page=30|quote=units' or ones', digit}}</ref><ref name="PierceTebeaux1983">{{cite book|author1=R. C. Pierce|author2=W. J. Tebeaux|title=Operational Mathematics for Business|url=https://books.google.com/books?id=ng11FOHjNmcC|year=1983|publisher=Wadsworth Publishing Company|isbn=978-0-534-01235-9|page=29|quote=ones or units digit}}</ref><ref name="Sobel1985a">{{cite book|author=Max A. Sobel|title=Harper & Row algebra one|url=https://books.google.com/books?id=f3Y51BtCOKMC|year=1985|publisher=Harper & Row|isbn=978-0-06-544000-3|page=282|quote=ones, or units, digit}}</ref>
:the '''1''' to the left of the ones place is in the tens place, and is called the ''tens digit'';<ref name="Sobel1985b">{{cite book|author=Max A. Sobel|title=Harper & Row algebra one|url=https://books.google.com/books?id=f3Y51BtCOKMC|year=1985|publisher=Harper & Row|isbn=978-0-06-544000-3|page=277|quote=every two-digit number can be expressed as 10t+u when t is the tens digit}}</ref>
:the '''3''' is to the right of the ones place, so it is in the tenths place, and is called the ''tenths digit'';<ref name=":0">{{Cite book|author=Taggart, Robert|title=Mathematics. Decimals and percents|date=2000|publisher=J. Weston Walch|isbn=0-8251-4178-8|location=Portland, Me.|pages=51–54|oclc=47352965}}</ref>
:the '''4''' to the right of the tenths place is in the hundredths place, and is called the ''hundredths digit''.<ref name=":0" />

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 {{nowrap|''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&nbsp;10),
:the '''1''' is second to the left of the separator, so based on calculation, its value is,

:<math>n - 1 = 2 - 1 = 1</math>

:<math>1 \times 10^1 = 10</math>

:the '''4''' is second to the right of the separator, so based on calculation its value is,

:<math>n = -2</math>

:<math>4 \times 10^{-2} = \frac{4}{100}</math>

==History==
{{main|History of the Hindu–Arabic numeral system}}
<div style="float:right;">
{| class="wikitable zebra"
|-
!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
|}
</div>
The first true written [[positional numeral system]] is considered to be the [[Hindu–Arabic numeral system]]. This system was established by the 7th&nbsp;century in India,<ref name="O'Connor and Robertson">O'Connor, J. J. and Robertson, E. F. [http://www-history.mcs.st-andrews.ac.uk/HistTopics/Arabic_numerals.html Arabic Numerals]. January 2001. Retrieved on 2007-02-20.</ref> 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.<ref>{{cite web|url=https://www.ams.org/featurecolumn/archive/india-zero.html |title=All for Nought |work=Feature Column |author=Bill Casselman |author-link=Bill Casselman (mathematician) |publisher=AMS |date=February 2007}}</ref> The original numerals were very similar to the modern ones, even down to the [[glyph]]s used to represent digits.<ref name="O'Connor and Robertson"/>

[[Image:Maya.svg|thumb|left|150px|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 {{Lang|la|[[Liber Abaci]]}}). They began to enter common use in the 15th&nbsp;century.<ref>{{Cite web|last=Bradley|first=Jeremy|title=How Arabic Numbers Were Invented|url=https://www.theclassroom.com/how-to-identify-numbers-on-brass-from-india-12082499.html|access-date=2020-07-22|website=www.theclassroom.com}}</ref> By the end of the 20th&nbsp;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===
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&nbsp;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 numerals|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 [[China|Chinese]] and [[Japan]]ese 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 [[Chinese numerals#Suzhou numerals|Suzhou numerals]] are variants of rod numerals.

{| class="wikitable" style="text-align:center"
|+ Rod numerals (vertical)
|-
! style="width:50px" | 0
! style="width:50px" | 1
! style="width:50px" | 2
! style="width:50px" | 3
! style="width:50px" | 4
! style="width:50px" | 5
! style="width:50px" | 6
! style="width:50px" | 7
! style="width:50px" | 8
! style="width:50px" | 9
|-
| [[Image:Counting rod 0.png]]
| [[Image:Counting rod v1.png]]
| [[Image:Counting rod v2.png]]
| [[Image:Counting rod v3.png]]
| [[Image:Counting rod v4.png]]
| [[Image:Counting rod v5.png]]
| [[Image:Counting rod v6.png]]
| [[Image:Counting rod v7.png]]
| [[Image:Counting rod v8.png]]
| [[Image:Counting rod v9.png]]
|-
! style="width:50px" | −0
! style="width:50px" | −1
! style="width:50px" | −2
! style="width:50px" | −3
! style="width:50px" | −4
! style="width:50px" | −5
! style="width:50px" | −6
! style="width:50px" | −7
! style="width:50px" | −8
! style="width:50px" | −9
|-
| [[Image:Counting rod -0.png]]
| [[Image:Counting rod v-1.png]]
| [[Image:Counting rod v-2.png]]
| [[Image:Counting rod v-3.png]]
| [[Image:Counting rod v-4.png]]
| [[Image:Counting rod v-5.png]]
| [[Image:Counting rod v-6.png]]
| [[Image:Counting rod v-7.png]]
| [[Image:Counting rod v-8.png]]
| [[Image:Counting rod v-9.png]]
|}

==Modern digital systems==
===In computer science===
The [[Binary numeral system|binary]] (base&nbsp;2), [[octal]] (base&nbsp;8), and [[hexadecimal]] (base&nbsp;16) systems, extensively used in [[computer science]], all follow the conventions of the [[Hindu–Arabic numeral system]].<ref>{{Cite book|last=Ravichandran|first=D.|url=https://books.google.com/books?id=EHNOHAjXdQcC&q=octal|title=Introduction To Computers And Communication|date=2001-07-01|publisher=Tata McGraw-Hill Education|isbn=978-0-07-043565-0|language=en|pages=24–47}}</ref> 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.<ref>{{Cite web|title=Hexadecimals|url=https://www.mathsisfun.com/hexadecimals.html|access-date=2020-07-22|website=www.mathsisfun.com}}</ref> 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===
The [[Ternary numeral system|ternary]] and [[balanced ternary]] systems have sometimes been used. They are both base&nbsp;3 systems.<ref>{{Cite web|date=2019-10-30|url=http://bit-player.org/wp-content/extras/bph-publications/AmSci-2001-11-Hayes-ternary.pdf|archive-url=https://web.archive.org/web/20191030114823/http://bit-player.org/wp-content/extras/bph-publications/AmSci-2001-11-Hayes-ternary.pdf|title=Third Base|archive-date=2019-10-30|access-date=2020-07-22}}</ref>

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.<ref>{{Cite web|title=Development of ternary computers at Moscow State University. Russian Virtual Computer Museum|url=https://www.computer-museum.ru/english/setun.htm|access-date=2020-07-22|website=www.computer-museum.ru}}</ref>

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 [[signed-digit representation#Negative numerals|negative numerals]]. The concept of signed-digit representation has also been taken up in [[computer design]].

==Digits in mathematics==
Despite the essential role of digits in describing numbers, they are relatively unimportant to modern [[mathematics]].<ref>{{Cite web|last=Kirillov|first=A.A.|title=What are numbers?|url=https://www.math.upenn.edu/~kirillov/MATH480-S08/WN1.pdf|website=math.upenn.|page=2|quote=True, if you open a modern mathematical journal and try to read any article, it is very probable that you will see no numbers at all.}}</ref> Nevertheless, there are a few important mathematical concepts that make use of the representation of a number as a sequence of digits.

===Digital roots===
{{main|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.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Digital Root|url=https://mathworld.wolfram.com/DigitalRoot.html|access-date=2020-07-22|website=mathworld.wolfram.com|language=en}}</ref>

===Casting out nines===
{{main|Casting out nines}}
[[Casting out nines]] is a procedure for checking arithmetic done by hand. To describe it, let <math>f(x)</math> represent the [[digital root]] of <math>x</math>, as described above. Casting out nines makes use of the fact that if <math>A + B = C</math>, then <math>f(f(A) + f(B)) = f(C)</math>. 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.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Casting Out Nines|url=https://mathworld.wolfram.com/CastingOutNines.html|access-date=2020-07-22|website=mathworld.wolfram.com|language=en}}</ref>

===Repunits and repdigits===
{{main|Repunit}}
Repunits are integers that are represented with only the digit 1. For example, 1111 (one thousand, one hundred and eleven) is a repunit. [[Repdigit]]s are a generalization of repunits; they are integers represented by repeated instances of the same digit. For example, 333 is a repdigit. The [[prime number|primality]] of repunits is of interest to mathematicians.<ref>{{MathWorld|urlname=Repunit|title=Repunit}}</ref>

===Palindromic numbers and Lychrel numbers===
{{main|Palindromic number}}
Palindromic numbers are numbers that read the same when their digits are reversed.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Palindromic Number|url=https://mathworld.wolfram.com/PalindromicNumber.html|access-date=2020-07-22|website=mathworld.wolfram.com|language=en}}</ref> 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.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Lychrel Number|url=https://mathworld.wolfram.com/LychrelNumber.html|access-date=2020-07-22|website=mathworld.wolfram.com|language=en}}</ref> The question of whether there are any Lychrel numbers in base&nbsp;10 is an open problem in [[recreational mathematics]]; the smallest candidate is [[196 (number)|196]].<ref>{{Cite book|last1=Garcia|first1=Stephan Ramon|url=https://books.google.com/books?id=7qCdDwAAQBAJ&q=Lychrel+196&pg=PA104|title=100 Years of Math Milestones: The Pi Mu Epsilon Centennial Collection|last2=Miller|first2=Steven J.|date=2019-06-13|publisher=American Mathematical Soc.|isbn=978-1-4704-3652-0|pages=104–105|language=en}}</ref>

==History of ancient numbers==
{{Main|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.<ref>{{Cite book|author=Saxe, Geoffrey B.|title=Cultural development of mathematical ideas : Papua New Guinea studies|date=2012|publisher=Cambridge University Press|others=Esmonde, Indigo.|isbn=978-1-139-55157-1|location=Cambridge|pages=44–45|quote=The Okspamin body system includes 27 body parts...|oclc=811060760}}</ref>

To preserve numerical information, [[Tally marks|tallies]] carved in wood, bone, and stone have been used since prehistoric times.<ref>{{Cite book|author=Tuniz, C. (Claudio)|title=Humans : an unauthorized biography|others=Tiberi Vipraio, Patrizia, Haydock, Juliet|date=24 May 2016|isbn=978-3-319-31021-3|location=Switzerland|pages=101|oclc=951076018|quote=...even notches cut into sticks made out of wood, bone or other materials dating back 30,000 years (often referred to as "notched tallies").}}</ref> Stone age cultures, including ancient [[Indigenous peoples of the Americas|indigenous American]] groups, used tallies for gambling, personal services, and trade-goods.

A method of preserving numeric information in clay was invented by the [[Sumer]]ians between 8000 and 3500&nbsp;BC.<ref>{{Cite book|author=Ifrah, Georges|title=From one to zero : a universal history of numbers|date=1985|publisher=Viking|isbn=0-670-37395-8|location=New York|pages=154|oclc=11237558|quote=And so, by the beginning of the third millennium B.C., the Sumerians and Elamites had adopted the practice of recording numerical information on small, usually rectangular clay tablets }}</ref> This was done with small clay tokens of various shapes that were strung like beads on a string. Beginning about 3500&nbsp;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&nbsp;&nbsp;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.<ref>{{Cite book|url=https://books.google.com/books?id=qxP0yJa2G6oC&q=he+vertical+wedge+and+the+chevron&pg=PA226|title=London Encyclopædia, Or, Universal Dictionary of Science, Art, Literature, and Practical Mechanics: Comprising a Popular View of the Present State of Knowledge; Illustrated by Numerous Engravings and Appropriate Diagrams|date=1845|publisher=T. Tegg|pages=226|language=en}}</ref> This sexagesimal number system was fully developed at the beginning of the Old Babylonia period (about 1950&nbsp;BC) and became standard in Babylonia.<ref>{{Cite book|last=Neugebauer|first=O.|url=https://books.google.com/books?id=v1bmBwAAQBAJ&q=sexagesimal+number+system+was+fully+developed+at+the+beginning+of+the+Old+Babylonia+period|title=Astronomy and History Selected Essays|date=2013-11-11|publisher=Springer Science & Business Media|isbn=978-1-4612-5559-8|language=en}}</ref>

[[Sexagesimal]] numerals were a [[mixed radix]] system that retained the alternating base&nbsp;10 and base&nbsp;6 in a sequence of cuneiform vertical wedges and chevrons. By 1950&nbsp;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 [[angle]]s (degrees).<ref>{{Cite book|chapter=Sexagesimal System|place=Berlin/Heidelberg|publisher=Springer-Verlag|doi=10.1007/978-1-4020-4425-0_9055 |title=Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures |date=2008 |last1=Powell |first1=Marvin A. |pages=1998–1999 |isbn=978-1-4020-4559-2 }}</ref>

==History of modern numbers==
In [[China]], armies and provisions were counted using modular tallies of [[prime number]]s. 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.<ref>{{Cite book|author=Knuth, Donald Ervin|title=The art of computer programming|year=1998 |publisher=Addison-Wesley Pub. Co|isbn=0-201-03809-9|location=Reading, Mass.|oclc=823849|quote=The advantages of a modular representation are that addition, subtraction, and multiplication are very simple}}</ref> 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]].<ref>{{Cite book|last1=Echtle|first1=Klaus|url=https://books.google.com/books?id=bzw15Ew_iOoC&q=modern+times+modular+arithmetic++digital+signal+processing.&pg=PA439|title=Dependable Computing - EDCC-1: First European Dependable Computing Conference, Berlin, Germany, October 4-6, 1994. Proceedings|last2=Hammer|first2=Dieter|last3=Powell|first3=David|date=1994-09-21|publisher=Springer Science & Business Media|isbn=978-3-540-58426-1|pages=439|language=en}}</ref>

The oldest Greek system was that of the [[Attic numerals]],<ref>{{Cite book|author=Woodhead, A. G. (Arthur Geoffrey)|title=The study of Greek inscriptions|date=1981|publisher=Cambridge University Press|isbn=0-521-23188-4|edition=2nd|location=Cambridge|pages=109–110|oclc=7736343}}</ref> but in the 4th century BC they began to use a quasidecimal alphabetic system (see [[Greek numerals]]).<ref>{{Cite book|last=Ushakov|first=Igor|url=https://books.google.com/books?id=4cXOAwAAQBAJ&q=quasidecimal+alphabetic+system+greek&pg=PA17|title=In the Beginning Was the Number (2)|date=22 June 2012 |publisher=Lulu.com|isbn=978-1-105-88317-0|language=en}}</ref> Jews began using a similar system ([[Hebrew numerals]]), with the oldest examples known being coins from around 100&nbsp;BC.<ref>{{Cite book|author=Chrisomalis, Stephen|title=Numerical notation : a comparative history|date=2010|publisher=Cambridge University Press|isbn=978-0-511-67683-3|location=Cambridge|pages=157|oclc=630115876|quote=The first safely dated instance in which the use of Hebrew alphabetic numerals is certain is on coins from the reign of Hasmonean king Alexander Janneus(103 to 76 BC)...}}</ref>

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|Roman numerals system]] remained in common use in Europe until [[positional notation]] came into common use in the 16th&nbsp;century.<ref>{{Cite book|last=Silvercloud|first=Terry David|url=https://books.google.com/books?id=Zy-ODwAAQBAJ&q=Roman+numerals+system+remained+in+common+use&pg=PA152|title=The Shape of God: Secrets, Tales, and Legends of the Dawn Warriors|date=2007|publisher=Terry David Silvercloud|isbn=978-1-4251-0836-6|pages=152|language=en}}</ref>

The [[Maya numerals|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]].<ref>{{citation |title=Modern Mathematics |first1=Ruric E. |last1=Wheeler |first2=Ed R. |last2=Wheeler |publisher=Kendall Hunt |year=2001 |isbn=9780787290627 |page=130 |url=https://books.google.com/books?id=azSPh9SBwwEC&pg=PA130}}.</ref> 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]].<ref>{{Cite book|last=Swami|first=Devamrita|url=https://books.google.com/books?id=5JRdIkxETUsC&q=Maya+length+of+the+solar+year+and+the+orbit+of+Venus&pg=PT304|title=Searching for Vedic India|date=2002|publisher=The Bhaktivedanta Book Trust|isbn=978-0-89213-350-5|language=en|quote=Maya astronomy finely calculated both the duration of the solar year and the synodical revolution of Venus}}</ref>

The Incan Empire ran a large command economy using [[quipu]], tallies made by knotting colored fibers.<ref>{{Cite web|title=Quipu {{!}} Incan counting tool|url=https://www.britannica.com/technology/quipu|access-date=2020-07-23|website=Encyclopedia Britannica|language=en}}</ref> Knowledge of the encodings of the knots and colors was suppressed by the [[Spain|Spanish]] [[conquistador]]s in the 16th&nbsp;century, and has not survived although simple quipu-like recording devices are still used in the [[Andes|Andean]] region.

Some authorities believe that positional arithmetic began with the wide use of [[counting rods]] in China.<ref>{{Cite book|last=Chen|first=Sheng-Hong|url=https://books.google.com/books?id=K3lhDwAAQBAJ&q=positional+arithmetic+began+with+the+wide+use+of+counting+rods+in+China&pg=PA8|title=Computational Geomechanics and Hydraulic Structures|date=2018-06-21|publisher=Springer|isbn=978-981-10-8135-4|pages=8|language=en|quote=… definitely before 400 BC they possessed a similar positional notation based on the ancient counting rods.}}</ref> 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]].<ref>{{Cite web|title=Foundations of mathematics – The reexamination of infinity|url=https://www.britannica.com/science/foundations-of-mathematics|access-date=2020-07-23|website=Encyclopædia Britannica|language=en}}</ref>

The modern positional Arabic numeral system was developed by [[Indian mathematics|mathematicians in India]], and passed on to [[Islamic mathematics|Muslim mathematicians]], along with astronomical tables brought to [[Baghdad]] by an Indian ambassador around 773.<ref>{{Cite book|url=https://books.google.com/books?id=uM0sRPoABq8C&q=astronomical+tables+brought+to+Baghdad+by+an+Indian+ambassador+around+773&pg=PA626|title=The Encyclopedia Britannica|date=1899|pages=626|language=en}}</ref>

From [[India subcontinent|India]], the thriving trade between Islamic sultans and Africa carried the concept to [[Cairo]]. Arabic mathematicians extended the system to include [[Decimal|decimal fractions]], and [[Muḥammad ibn Mūsā al-Ḵwārizmī]] wrote an important work about it in the 9th&nbsp;&nbsp;century.<ref>{{Cite book|author=Struik, Dirk J. (Dirk Jan)|title=A concise history of mathematics|date=1967|publisher=Dover Publications|isbn=0-486-60255-9|edition=3d rev.|location=New York|oclc=635553}}</ref> The modern [[Arabic numerals]] were introduced to Europe with the translation of this work in the 12th&nbsp;century in Spain and [[Leonardo of Pisa]]'s ''Liber Abaci'' of 1201.<ref>{{Cite book|last=Sigler|first=Laurence|url=https://books.google.com/books?id=PilhoGJeKBUC&q=Leonardo+of+Pisa's+Liber+Abaci+of+1201|title=Fibonacci's Liber Abaci: A Translation into Modern English of Leonardo Pisano's Book of Calculation|date=2003-11-11|publisher=Springer Science & Business Media|isbn=978-0-387-40737-1|language=en}}</ref> In Europe, the complete Indian system with the zero was derived from the Arabs in the 12th&nbsp;century.<ref>{{Cite book|author=Deming, David|title=Science and technology in world history. Volume 1, The ancient world and classical civilization|date=2010|publisher=McFarland & Co|isbn=978-0-7864-5657-4|location=Jefferson, N.C.|pages=86|oclc=650873991}}</ref>

The [[binary numeral system|binary system]] (base 2) was propagated in the 17th&nbsp;century by [[Gottfried Leibniz]].<ref name=":1">{{Cite book|last=Yanushkevich|first= Svetlana N.|author-link=Svetlana Yanushkevich|title=Introduction to logic design|date=2008|publisher=CRC Press|others=Shmerko, Vlad P.|isbn=978-1-4200-6094-2|location=Boca Raton|pages=56|oclc=144226528}}</ref> Leibniz had developed the concept early in his career, and had revisited it when he reviewed a copy of the ''[[I Ching]]'' from China.<ref>{{Cite book|author=Sloane, Sarah|title=The I Ching for writers : finding the page inside you|date=2005|publisher=New World Library|isbn=1-57731-496-4|location=Novato, Calif.|pages=9|oclc=56672043}}</ref> Binary numbers came into common use in the 20th&nbsp;century because of computer applications.<ref name=":1" />

===<span id="popular"></span>Numerals in most popular systems===
{| class="wikitable" summary="Numerals in many different writing systems"
!West Arabic
! 0
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! 8
! 9
|-
!Asomiya (Assamese); [[Bengali language|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 language|Persian]]
| ٠
| ١
| ٢
| ٣
| ۴
| ۵
| ۶
| ٧
| ٨
| ٩
|-
! [[Gurmukhi]]
| ੦
| ੧
| ੨
| ੩
| ੪
| ੫
| ੬
| ੭
| ੮
| ੯
|-
! [[Urdu]]
| {{Urdu numeral||15}}
| {{Urdu numeral|1|15}}
| {{Urdu numeral|2|15}}
| {{Urdu numeral|3|15}}
| {{Urdu numeral|4|15}}
| {{Urdu numeral|5|15}}
| {{Urdu numeral|6|15}}
| {{Urdu numeral|7|15}}
| {{Urdu numeral|8|15}}
| {{Urdu numeral|9|15}}
|-
! [[Chinese language|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 language|Gujarati]]
| ૦
| ૧
| ૨
| ૩
| ૪
| ૫
| ૬
| ૭
| ૮
| ૯
|-
! Hieroglyphic Egyptian
| 
| 𓏺
| 𓏻
| 𓏼
| 𓏽
| 𓏾
| 𓏿
| 𓐀
| 𓐁
| 𓐂
|-
! [[Japanese numerals|Japanese]] (everyday)
| {{lang|ja|〇}}
| {{lang|ja|一}}
| {{lang|ja|二}}
| {{lang|ja|三}}
| {{lang|ja|四}}
| {{lang|ja|五}}
| {{lang|ja|六}}
| {{lang|ja|七}}
| {{lang|ja|八}}
| {{lang|ja|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 language|Khmer]] (Cambodia)
| ០
| ១
| ២
| ៣
| ៤
| ៥
| ៦
| ៧
| ៨
| ៩

|-
! [[Lao language|Lao]]
| ໐
| ໑
| ໒
| ໓
| ໔
| ໕
| ໖
| ໗
| ໘
| ໙
|-
! [[Limbu language|Limbu]]
| {{lang|lif|᥆}}
| {{lang|lif|᥇}}
| {{lang|lif|᥈}}
| {{lang|lif|᥉}}
| {{lang|lif|᥊}}
| {{lang|lif|᥋}}
| {{lang|lif|᥌}}
| {{lang|lif|᥍}}
| {{lang|lif|᥎}}
| {{lang|lif|᥏}}
|-
! [[Malayalam]]
| ൦
| ൧
| ൨
| ൩
| ൪
| ൫
| ൬
| ൭
| ൮
| ൯
|-
! [[Mongolian alphabet|Mongolian]]
| ᠐
| ᠑
| ᠒
| ᠓
| ᠔
| ᠕
| ᠖
| ᠗
| ᠘
| ᠙
|-
! [[Burmese script|Burmese]]
| ၀
| ၁
| ၂
| ၃
| ၄
| ၅
| ၆
| ၇
| ၈
| ၉
|-
! [[Oriya alphabet|Oriya]]
| ୦
| ୧
| ୨
| ୩
| ୪
| ୫
| ୬
| ୭
| ୮
| ୯
|-
! [[Roman numerals|Roman]]
|
| I
| II
| III
| IV
| V
| VI
| VII
| VIII
| IX
|-
|-
! [[Shan language|Shan]]
| ႐
| ႑
| ႒
| ႓
| ႔
| ႕
| ႖
| ႗
| ႘
| ႙
|-
! [[Sinhala numerals|Sinhala]]
|
| 𑇡
| 𑇢
| 𑇣
| 𑇤
| 𑇥
| 𑇦
| 𑇧
| 𑇨
| 𑇩
|-
! [[Tamil language|Tamil]]
| ௦
| ௧
| ௨
| ௩
| ௪
| ௫
| ௬
| ௭
| ௮
| ௯
|-
! [[Telugu language|Telugu]]
| ౦
| ౧
| ౨
| ౩
| ౪
| ౫
| ౬
| ౭
| ౮
| ౯
|-
! [[Thai numerals|Thai]]
| ๐
| ๑
| ๒
| ๓
| ๔
| ๕
| ๖
| ๗
| ๘
| ๙
|-
! [[Tibetan alphabet|Tibetan]]
| ༠
| ༡
| ༢
| ༣
| ༤
| ༥
| ༦
| ༧
| ༨
| ༩
|-
! [[New Tai Lue alphabet|New Tai Lue]]
| ᧐	
| ᧑
| 	᧒
| 	᧓
| 	᧔
| 	᧕
| 	᧖
| 	᧗
| 	᧘
| 	᧙
|-
! [[Javanese script|Javanese]]
| ꧐
| ꧑
| ꧒
| ꧓
| ꧔
| ꧕
| ꧖
| ꧗
| ꧘
| ꧙
|-
|}

===Additional numerals===
{| class="wikitable" summary="Additional numerals used in Chinese"
!
! 1
! 5
! 10
! 20
! 30
! 40
! 50
! 60
! 70
! 80
! 90
! 100
! 500
! 1000
! 10000
! 10<sup>8</sup>
|-
! [[Chinese numerals|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 numerals|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 script|Geʽez]]
| ፩ 
| ፭ 
| ፲
| ፳
| ፴
| ፵
| ፶
| ፷
| ፸
| ፹
| ፺
| ፻
| ፭፻ 
| ፲፻
| ፼
| ፼፼
|-
! [[Roman numerals|Roman]]
| I
| V
| X
| XX
| XXX
| XL
| L
| LX
| LXX
| LXXX
| XC
| C
| D
| M
| <span style="text-decoration:overline;">X</span>
|
|}

==See also==
* [[List of numeral systems]]
* [[List of numeral system topics]]
* [[Binary digit]]
* [[Hexadecimal digit]]
* [[Natural unit of information]]
* [[Abacus]]
* [[Significant figures]]
* [[Text figures]]
* [[Alphabetic numeral system]]

==References==
{{Reflist}}

{{Authority control}}

{{DEFAULTSORT:Numerical Digit}}
[[Category:Numeral systems]]