A numerical digit (often shortened to just digit) is a single symbol used alone (such as "2") or in combinations (such as "25"), to represent
numbers in a
positional
Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or decimal system). More generally, a positional system is a numeral system in which the ...
numeral system. The name "digit" comes from the fact that the ten digits (
Latin ''digiti'' meaning fingers) of the hands correspond to the ten symbols of the common
base 10 numeral system, i.e. the decimal (ancient Latin adjective ''decem'' meaning ten) digits.
For a given numeral system with an integer
base, the number of different digits required is given by the
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
of the base. For example, the decimal system (base 10) requires ten digits (0 through to 9), whereas the
binary system
A binary system is a system of two astronomical bodies which are close enough that their gravitational attraction causes them to orbit each other around a barycenter ''(also see animated examples)''. More restrictive definitions require that thi ...
(base 2) requires two digits (0 and 1).
Overview
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
Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or decimal system). More generally, a positional system is a numeral system in which the ...
, 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
1 (one, unit, unity) is a number representing a single or the only entity. 1 is also a numerical digit and represents a single unit of counting or measurement. For example, a line segment of ''unit length'' is a line segment of length 1. I ...
, and in the
hexadecimal system, the letter "A" represents the number
ten
Ten, TEN or 10 may refer to:
* 10, an even natural number following 9 and preceding 11
* one of the years 10 BC, AD 10, 1910 and 2010
* October, the tenth month of the year
Places
* Mount Ten, in Vietnam
* Tongren Fenghuang Airport (IATA cod ...
. A
positional number system has one unique digit for each integer from
zero
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usuall ...
up to, but not including, the
radix
In a positional numeral system, the radix or base is the number of unique digits, including the digit zero, used to represent numbers. For example, for the decimal/denary system (the most common system in use today) the radix (base number) is t ...
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 can be 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 can be expressed by 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
A decimal separator is a symbol used to separate the integer part from the fractional part of a number written in decimal form (e.g., "." in 12.45). Different countries officially designate different symbols for use as the separator. The cho ...
, commonly a
period
Period may refer to:
Common uses
* Era, a length or span of time
* Full stop (or period), a punctuation mark
Arts, entertainment, and media
* Period (music), a concept in musical composition
* Periodic sentence (or rhetorical period), a concept ...
in English, or a
comma
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
in other
European languages, to denote the "ones place" or "units place",
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'';
:the 1 to the left of the ones place is in the tens place, and is called the ''tens digit'';
:the 3 is to the right of the ones place, so it is in the tenths place, and is called the ''tenths digit'';
:the 4 to the right of the tenths place is in the hundredths place, and is called the ''hundredths digit''.
The total value of the number is 1 ten, 0 ones, 3 tenths, and 4 hundredths. Note that 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 , 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
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,
[O'Connor, J. J. and Robertson, E. F]
Arabic Numerals
January 2001. Retrieved on 2007-02-20. but was not yet in its modern form because the use of the digit
zero
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usuall ...
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. The original numerals were very similar to the modern ones, even down to the
glyph
A glyph () is any kind of purposeful mark. In typography, a glyph is "the specific shape, design, or representation of a character". It is a particular graphical representation, in a particular typeface, of an element of written language. A g ...
s used to represent digits.
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. 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
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
A decimal separator is a symbol used to separate the integer part from the fractional part of a number written in decimal form (e.g., "." in 12.45). Different countries officially designate different symbols for use as the separator. The cho ...
, 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
Chinese can refer to:
* Something related to China
* Chinese people, people of Chinese nationality, citizenship, and/or ethnicity
**''Zhonghua minzu'', the supra-ethnic concept of the Chinese nation
** List of ethnic groups in China, people of v ...
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
The Suzhou numerals, also known as ' (), is a numeral system used in China before the introduction of Arabic numerals. The Suzhou numerals are also known as ' (), ' (), ' (), ' () and ' ().
History
The Suzhou numeral system is the only survivin ...
are variants of rod numerals.
Modern digital systems
In computer science
The
binary
Binary may refer to:
Science and technology Mathematics
* Binary number, a representation of numbers using only two digits (0 and 1)
* Binary function, a function that takes two arguments
* Binary operation, a mathematical operation that ta ...
(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. 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. When the binary system is used, the term "bit(s)" is typically used as a alternative for "digit(s)", being a portmanteau of the term "binary digit". Similar terms exist for other number systems, such as "trit(s)" for a ternary system and "dit(s) for the decimal system, although less frequently used.
Unusual systems
The
ternary and
balanced ternary systems have sometimes been used. They are both base 3 systems.
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
Setun (russian: Сетунь) was a computer developed in 1958 at Moscow State University. It was built under the leadership of Sergei Sobolev and Nikolay Brusentsov. It was the most modern ternary computer, using the balanced ternary numeral sys ...
computers.
Several authors in the last 300 years have noted a facility of
positional notation
Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or decimal system). More generally, a positional system is a numeral system in which the ...
that amounts to a ''modified''
decimal representation
A decimal representation of a non-negative real number is its expression as a sequence of symbols consisting of decimal digits traditionally written with a single separator:
r = b_k b_\ldots b_0.a_1a_2\ldots
Here is the decimal separator, i ...
. Some advantages are cited for use of numerical digits that represent negative values. In 1840
Augustin-Louis Cauchy
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...
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
In computer engineering, computer architecture is a description of the structure of a computer system made from component parts. It can sometimes be a high-level description that ignores details of the implementation. At a more detailed level, the ...
.
Digits in mathematics
Despite the essential role of digits in describing numbers, they are relatively unimportant to modern
mathematics. Nevertheless, there are a few important mathematical concepts that make use of the representation of a number as a sequence of digits.
Digital roots
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.
Casting out nines
Casting out nines
Casting out nines is any of three arithmetical procedures:
*Adding the decimal digits of a positive whole number, while optionally ignoring any 9s or digits which sum to a multiple of 9. The result of this procedure is a number which is smaller th ...
is a procedure for checking arithmetic done by hand. To describe it, let
represent the
digital root
The digital root (also repeated digital sum) of a natural number in a given radix is the (single digit) value obtained by an iterative process of summing digits, on each iteration using the result from the previous iteration to compute a digit s ...
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.
Repunits and repdigits
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
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
of repunits is of interest to mathematicians.
Palindromic numbers and Lychrel numbers
Palindromic numbers are numbers that read the same when their digits are reversed. A
Lychrel number
A Lychrel number is a natural number that cannot form a palindrome through the iterative process of repeatedly reversing its digits and adding the resulting numbers. This process is sometimes called the ''196-algorithm'', after the most famous numb ...
is a positive integer that never yields a palindromic number when subjected to the iterative process of being added to itself with digits reversed. The question of whether there are any Lychrel numbers in base 10 is an open problem in
recreational mathematics; the smallest candidate is
196
Year 196 ( CXCVI) was a leap year starting on Thursday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Dexter and Messalla (or, less frequently, year 949 ''Ab urbe condita ...
.
History of 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.
To preserve numerical information,
tallies carved in wood, bone, and stone have been used since prehistoric times. 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. 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
Sexagesimal, also known as base 60 or sexagenary, is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form� ...
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. This sexagesimal number system was fully developed at the beginning of the Old Babylonia period (about 1950 BC) and became standard in Babylonia.
Sexagesimal
Sexagesimal, also known as base 60 or sexagenary, is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form� ...
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
Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or decimal system). More generally, a positional system is a numeral system in which the ...
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
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles a ...
s (degrees).
History of modern numbers
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
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his bo ...
is that it is easy to multiply. 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.
The oldest Greek system was that of the
Attic numerals
The Attic numerals are a symbolic number notation used by the ancient Greeks. They were also known as Herodianic numerals because they were first described in a 2nd-century manuscript by Herodian; or as acrophonic numerals (from acrophony) ...
, but in the 4th century BC they began to use a quasidecimal alphabetic system (see
Greek numerals). Jews began using a similar system (
Hebrew numerals), with the oldest examples known being coins from around 100 BC.
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
Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or decimal system). More generally, a positional system is a numeral system in which the ...
came into common use in the 16th century.
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
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usuall ...
. 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.
The Incan Empire ran a large command economy using
quipu, tallies made by knotting colored fibers. Knowledge of the encodings of the knots and colors was suppressed by the
Spanish
Spanish might refer to:
* Items from or related to Spain:
**Spaniards are a nation and ethnic group indigenous to Spain
**Spanish language, spoken in Spain and many Latin American countries
**Spanish cuisine
Other places
* Spanish, Ontario, Can ...
conquistador
Conquistadors (, ) or conquistadores (, ; meaning 'conquerors') were the explorer-soldiers of the Spanish and Portuguese Empires of the 15th and 16th centuries. During the Age of Discovery, conquistadors sailed beyond Europe to the Americas, ...
s in the 16th century, and has not survived although simple quipu-like recording devices are still used in the
Andean
The Andes, Andes Mountains or Andean Mountains (; ) are the longest continental mountain range in the world, forming a continuous highland along the western edge of South America. The range is long, wide (widest between 18°S – 20°S ...
region.
Some authorities believe that positional arithmetic began with the wide use of
counting rods in China. 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
Brahmagupta ( – ) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical tr ...
.
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
Baghdad (; ar, بَغْدَاد , ) is the capital of Iraq and the second-largest city in the Arab world after Cairo. It is located on the Tigris near the ruins of the ancient city of Babylon and the Sassanid Persian capital of Ctesiphon ...
by an Indian ambassador around 773.
From
India, the thriving trade between Islamic sultans and Africa carried the concept to
Cairo
Cairo ( ; ar, القاهرة, al-Qāhirah, ) is the capital of Egypt and its largest city, home to 10 million people. It is also part of the largest urban agglomeration in Africa, the Arab world and the Middle East: The Greater Cairo metro ...
. Arabic mathematicians extended the system to include
decimal fractions
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
, and
Muḥammad ibn Mūsā al-Ḵwārizmī
Muḥammad ibn Mūsā al-Khwārizmī ( ar, محمد بن موسى الخوارزمي, Muḥammad ibn Musā al-Khwārazmi; ), or al-Khwarizmi, was a Persian polymath from Khwarazm, who produced vastly influential works in mathematics, astronom ...
wrote an important work about it in the 9th century. The modern
Arabic numerals
Arabic numerals are the ten numerical digits: , , , , , , , , and . They are the most commonly used symbols to write decimal numbers. They are also used for writing numbers in other systems such as octal, and for writing identifiers such as ...
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. In Europe, the complete Indian system with the zero was derived from the Arabs in the 12th century.
The
binary system
A binary system is a system of two astronomical bodies which are close enough that their gravitational attraction causes them to orbit each other around a barycenter ''(also see animated examples)''. More restrictive definitions require that thi ...
(base 2), was propagated in the 17th century by
Gottfried Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mat ...
.
Leibniz had developed the concept early in his career, and had revisited it when he reviewed a copy of the
I Ching from China.
Binary numbers came into common use in the 20th century because of computer applications.
Numerals in most popular systems
Additional numerals
See also
*
Hexadecimal
*
Binary digit
Binary may refer to:
Science and technology Mathematics
* Binary number, a representation of numbers using only two digits (0 and 1)
* Binary function, a function that takes two arguments
* Binary operation, a mathematical operation that t ...
(
bit
The bit is the most basic unit of information in computing and digital communications. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represented ...
),
Quantum binary digit (
qubit)
*
Ternary digit (
trit),
Quantum ternary digit (
qutrit
A qutrit (or quantum trit) is a unit of quantum information that is realized by a 3-level quantum system, that may be in a superposition of three mutually orthogonal quantum states.
The qutrit is analogous to the classical radix-3 trit, just as ...
)
*Decimal digit (dit (unit), dit)
*Hexadecimal digit (Hexit (computing), Hexit)
*Natural digit (nat (unit), nat, nit (unit of information), nit)
*Naperian digit (nepit (unit), nepit)
*Significant digit
*Large numbers
*Text figures
*Abacus
*History of large numbers
*List of numeral system topics
Numeral notation in various scripts
*
Arabic numerals
Arabic numerals are the ten numerical digits: , , , , , , , , and . They are the most commonly used symbols to write decimal numbers. They are also used for writing numbers in other systems such as octal, and for writing identifiers such as ...
*Armenian numerals
*Babylonian numerals
*Balinese numerals
*Bengali numerals
*Burmese numerals
*Chinese numerals
*Cistercian numerals
*Dzongkha numerals
*Eastern Arabic numerals
*Georgian numerals
*
Greek numerals
*Gurmukhi numerals
*
Hebrew numerals
*Hokkien numerals
*Indian numerals
*Japanese numerals
*Javanese numerals
*Khmer numerals
*Korean numerals
*Lao numerals
*Mayan numerals
*Mongolian numerals
*Quipu
*Rod numerals
*Roman numerals
*Sinhala numerals
*Suzhou numerals
*Tamil numerals
*Thai numerals
*Vietnamese numerals
References
{{DEFAULTSORT:Numerical Digit
Numeral systems