Numerical digit
Encyclopedia
Number
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
s in positional
Positional notation
Positional notation or place-value notation is a method of representing or encoding numbers. Positional notation is distinguished from other notations for its use of the same symbol for the different orders of magnitude...
numeral system
Numeral system
A numeral system is a writing system for expressing numbers, that is a mathematical notation for representing numbers of a given set, using graphemes or symbols in a consistent manner....
s. The name "digit" comes from the fact that the 10 digits (ancient Latin
Latin
Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...
digita meaning fingers) of the hands correspond to the 10 symbols of the common base 10 number system, i.e. the decimal (ancient Latin adjective dec. meaning ten) digits.
In a given number system, if the base
Radix
In mathematical numeral systems, the base or radix for the simplest case is the number of unique digits, including zero, that a positional numeral system uses to represent numbers. For example, for the decimal system the radix is ten, because it uses the ten digits from 0 through 9.In any numeral...
is an integer, the number of digits required is always equal to the absolute value
Absolute value
In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...
of the base.
Overview
In a basic digital system, a numeralNumeral system
A numeral system is a writing system for expressing numbers, that is a mathematical notation for representing numbers of a given set, using graphemes or symbols in a consistent manner....
is a sequence of digits, which may be of arbitrary length. Each position in the sequence has a place value
Positional notation
Positional notation or place-value notation is a method of representing or encoding numbers. Positional notation is distinguished from other notations for its use of the same symbol for the different orders of magnitude...
, 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 decimalDecimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....
the digit "1" represents the integer one, and in the hexadecimal
Hexadecimal
In mathematics and computer science, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F to represent values ten to fifteen...
system, the letter "A" represents the number ten
10 (number)
10 is an even natural number following 9 and preceding 11.-In mathematics:Ten is a composite number, its proper divisors being , and...
. A positional number system must have a digit representing the integers from zero up to, but not including, the radix
Radix
In mathematical numeral systems, the base or radix for the simplest case is the number of unique digits, including zero, that a positional numeral system uses to represent numbers. For example, for the decimal system the radix is ten, because it uses the ten digits from 0 through 9.In any numeral...
of the number system.
Computation of place values
The Hindu–Arabic numeral systemHindu–Arabic numeral system
The Hindu–Arabic numeral system or Hindu numeral system is a positional decimal numeral system developed between the 1st and 5th centuries by Indian mathematicians, adopted by Persian and Arab mathematicians , and spread to the western world...
(or the Hindu numeral system) uses a decimal separator
Decimal separator
Different symbols have been and are used for the decimal mark. The choice of symbol for the decimal mark affects the choice of symbol for the thousands separator used in digit grouping. Consequently the latter is treated in this article as well....
, commonly a period in the United Kingdom
United Kingdom
The United Kingdom of Great Britain and Northern IrelandIn the United Kingdom and Dependencies, other languages have been officially recognised as legitimate autochthonous languages under the European Charter for Regional or Minority Languages...
and United States
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
or a comma
Comma
A comma is a type of punctuation mark . The word comes from the Greek komma , which means something cut off or a short clause.Comma may also refer to:* Comma , a type of interval in music theory...
in Europe
Europe
Europe is, by convention, one of the world's seven continents. Comprising the westernmost peninsula of Eurasia, Europe is generally 'divided' from Asia to its east by the watershed divides of the Ural and Caucasus Mountains, the Ural River, the Caspian and Black Seas, and the waterways connecting...
, to denote the "ones 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
Radix
In mathematical numeral systems, the base or radix for the simplest case is the number of unique digits, including zero, that a positional numeral system uses to represent numbers. For example, for the decimal system the radix is ten, because it uses the ten digits from 0 through 9.In any numeral...
. 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 ten),
- the 0 is immediately to the left of the separator, so it is in the ones place, and is called the units 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 compliment 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 ten),
- the 1 is second to the left of the separator, so based on calculation, its value is,
- n - 1 = 2 - 1 = 1
- 1 × 101 = 10
- the 4 is second to the right of the separator, so based on calculation its value is,
- n = -2
- 4 × 10-2 =
History
Hindu–Arabic numeral system
The Hindu–Arabic numeral system or Hindu numeral system is a positional decimal numeral system developed between the 1st and 5th centuries by Indian mathematicians, adopted by Persian and Arab mathematicians , and spread to the western world...
. This system was established by the 7th century, 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, a space was left in the numeral as a placeholder. The first widely acknowledged use of zero was in 876. Although the original Hindu-Arabic system was very similar to the modern one, even down to the glyph
Glyph
A glyph is an element of writing: an individual mark on a written medium that contributes to the meaning of what is written. A glyph is made up of one or more graphemes....
s used to represent digits, the direction of places was reversed, so that place values increased to the right rather than to the left.
Fibonacci
Leonardo Pisano Bigollo also known as Leonardo of Pisa, Leonardo Pisano, Leonardo Bonacci, Leonardo Fibonacci, or, most commonly, simply Fibonacci, was an Italian mathematician, considered by some "the most talented western mathematician of the Middle Ages."Fibonacci is best known to the modern...
used them in his Liber Abaci
Liber Abaci
Liber Abaci is a historic book on arithmetic by Leonardo of Pisa, known later by his nickname Fibonacci...
). 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 numeralsMaya numerals
Maya Numerals were a vigesimal numeral system used by the Pre-Columbian Maya civilization.The numerals are made up of three symbols; zero , one and five...
is unclear, but it is possible that it is older than the Hindu-Arabic system. The system was vigesimal
Vigesimal
The vigesimal or base 20 numeral system is based on twenty .- Places :...
(base twenty), 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
Decimal separator
Different symbols have been and are used for the decimal mark. The choice of symbol for the decimal mark affects the choice of symbol for the thousands separator used in digit grouping. Consequently the latter is treated in this article as well....
, so their system could not represent fractions.
The Thai numeral system
Thai numerals
Thai numerals constitute a numeral system of Thai number names for the Khmer numerals traditionally used in Thailand, also used for the more common Arabic numerals, and which follow the Hindu-Arabic numeral system.-Usage:...
is identical to the Hindu–Arabic numeral system
Hindu–Arabic numeral system
The Hindu–Arabic numeral system or Hindu numeral system is a positional decimal numeral system developed between the 1st and 5th centuries by Indian mathematicians, adopted by Persian and Arab mathematicians , and spread to the western world...
except for the symbols used to represent digits. The use of these digits is less common in Thailand
Thailand
Thailand , officially the Kingdom of Thailand , formerly known as Siam , is a country located at the centre of the Indochina peninsula and Southeast Asia. It is bordered to the north by Burma and Laos, to the east by Laos and Cambodia, to the south by the Gulf of Thailand and Malaysia, and to the...
than it once was, but they are still used alongside Hindu-Arabic numerals.
The rod numerals, the written forms of counting rods
Counting rods
Counting rods are small bars, typically 3–14 cm long, used by mathematicians for calculation in China, Japan, Korea, and Vietnam. They are placed either horizontally or vertically to represent any number and any fraction....
once used by Chinese
China
Chinese civilization may refer to:* China for more general discussion of the country.* Chinese culture* Greater China, the transnational community of ethnic Chinese.* History of China* Sinosphere, the area historically affected by Chinese culture...
and Japan
Japan
Japan is an island nation in East Asia. Located in the Pacific Ocean, it lies to the east of the Sea of Japan, China, North Korea, South Korea and Russia, stretching from the Sea of Okhotsk in the north to the East China Sea and Taiwan in the south...
ese mathematicians, are a decimal positional system able to represent not only zero but also negative numbers. Counting rods themselves predate Hindu-Arabic numeral system. The Suzhou numerals are variants of rod numerals.
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
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| -0 | -1 | -2 | -3 | -4 | -5 | -6 | -7 | -8 | -9 |
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In computer science
The binaryBinary numeral system
The binary numeral system, or base-2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2...
(base 2), octal
Octal
The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. Numerals can be made from binary numerals by grouping consecutive binary digits into groups of three...
(base 8), and hexadecimal
Hexadecimal
In mathematics and computer science, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F to represent values ten to fifteen...
(base 16) systems, extensively used in computer science
Computer science
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
, all follow the conventions of the Hindu–Arabic numeral system
Hindu–Arabic numeral system
The Hindu–Arabic numeral system or Hindu numeral system is a positional decimal numeral system developed between the 1st and 5th centuries by Indian mathematicians, adopted by Persian and Arab mathematicians , and spread to the western world...
. 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.
Unusual systems
The ternaryTernary numeral system
Ternary is the base- numeral system. Analogous to a bit, a ternary digit is a trit . One trit contains \log_2 3 bits of information...
and balanced ternary
Balanced ternary
Balanced ternary is a non-standard positional numeral system , useful for comparison logic. It is a ternary system, but unlike the standard ternary system, the digits have the values −1, 0, and 1...
systems have sometimes been used. They are both base-three 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
Setun was a balanced ternary computer developed in 1958 at Moscow State University. The device was built under the lead of Sergei Sobolev and Nikolay Brusentsov. It was the only modern ternary computer, using three-valued ternary logic instead of two-valued binary logic prevalent in computers...
computers.
Digits in mathematics
Despite the essential role of digits in describing numbers, they are relatively unimportant to modern mathematicsMathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
. 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 ninesCasting out nines
Casting out nines is a sanity check to ensure that hand computations of sums, differences, products, and quotients of integers are correct. By looking at the digital roots of the inputs and outputs, the casting-out-nines method can help one check arithmetic calculations...
is a procedure for checking arithmetic done by hand. To describe it, let
represent the digital rootDigital root
The digital root of a number is the value obtained by an iterative process of summing digits, on each iteration using the result from the previous iteration to compute a digit sum...
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 equationEquation
An equation is a mathematical statement that asserts the equality of two expressions. In modern notation, this is written by placing the expressions on either side of an equals sign , for examplex + 3 = 5\,asserts that x+3 is equal to 5...
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 eleven) is a repunit. RepdigitRepdigit
In recreational mathematics, a repdigit is a natural number composed of repeated instances of the same digit, most often in the decimal numeral system....
s are a generalization of repunits; they are integers represented by repeated instances of the same digit. For example, 333 is a repdigit. The primacy
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
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 numberLychrel number
A Lychrel number is a natural number which cannot form a palindrome through the iterative process of repeatedly reversing its base 10 digits and adding the resulting numbers. This process is called the 196-algorithm. The name "Lychrel" was coined by Wade VanLandingham—a rough anagram of his...
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
Recreational mathematics
Recreational mathematics is an umbrella term, referring to mathematical puzzles and mathematical games.Not all problems in this field require a knowledge of advanced mathematics, and thus, recreational mathematics often attracts the curiosity of non-mathematicians, and inspires their further study...
; the smallest candidate is 196
196 (number)
196 is the natural number following 195 and preceding 197.-In mathematics:* 196 is an even number* 196 is an abundant number, as 203 is greater than 196* 196 is a composite number...
.
History
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 10 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
Tally marks
Tally marks, or hash marks, are a unary numeral system. They are a form of numeral used for counting. They allow updating written intermediate results without erasing or discarding anything written down...
carved in wood, bone, and stone have been used since prehistoric times. Stone age cultures, including ancient American Indian 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 BCE. This was done with small clay tokens of various shapes that were strung like beads on a string. Beginning about 3500 BCE 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 BCE written numbers were dissociated from the things being counted and became abstract numerals.
Between 2700 BCE and 2000 BCE 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
Sign-value notation
A sign-value notation represents numbers by a series of numeric signs that added together equal the number represented. In Roman numerals for example, X means ten and L means fifty. Hence LXXX means eighty . There is no need for zero in 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. This sexagesimal number system was fully developed at the beginning of the Old Babylonia period (about 1950 BC) and became standard in Babylonia.
Sexagesimal numerals were a mixed radix
Mixed radix
Mixed radix numeral systems are non-standard positional numeral systems in which the numerical base varies from position to position. Such numerical representation applies when a quantity is expressed using a sequence of units that are each a multiple of the next smaller one, but not by the same...
system that retained the alternating base 10 and base 6 in a sequence of cuneiform vertical wedges and chevrons. By 1950 BCE this was a positional notation
Positional notation
Positional notation or place-value notation is a method of representing or encoding numbers. Positional notation is distinguished from other notations for its use of the same symbol for the different orders of magnitude...
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
Time
Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....
(minutes per hour) and angle
Angle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...
s (degrees).
In China
China
Chinese civilization may refer to:* China for more general discussion of the country.* Chinese culture* Greater China, the transnational community of ethnic Chinese.* History of China* Sinosphere, the area historically affected by Chinese culture...
, armies and provisions were counted using modular tallies of prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
s. Unique numbers of troops and measures of rice appear as unique combinations of these tallies. A great convenience of modular arithmetic
Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
is that it is easy to multiply, though quite difficult to add. 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
Digital signal processing
Digital signal processing is concerned with the representation of discrete time signals by a sequence of numbers or symbols and the processing of these signals. Digital signal processing and analog signal processing are subfields of signal processing...
.
The oldest Greek system was the that of the Attic numerals
Attic numerals
Attic numerals were used by the ancient Greeks, possibly from the 7th century BC. They were also known as Herodianic numerals because they were first described in a 2nd century manuscript by Herodian...
, but in the 4th century BC they began to use a quasidecimal alphabetic system (see Greek numerals
Greek numerals
Greek numerals are a system of representing numbers using letters of the Greek alphabet. They are also known by the names Ionian numerals, Milesian numerals , Alexandrian numerals, or alphabetic numerals...
). Jews began using a similar system (Hebrew numerals
Hebrew numerals
The system of Hebrew numerals is a quasi-decimal alphabetic numeral system using the letters of the Hebrew alphabet.In this system, there is no notation for zero, and the numeric values for individual letters are added together...
), 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
Roman numerals
The numeral system of ancient Rome, or Roman numerals, uses combinations of letters from the Latin alphabet to signify values. The numbers 1 to 10 can be expressed in Roman numerals as:...
remained in common use in Europe until positional notation
Positional notation
Positional notation or place-value notation is a method of representing or encoding numbers. Positional notation is distinguished from other notations for its use of the same symbol for the different orders of magnitude...
came into common use in the 16th century.
The Maya
Maya numerals
Maya Numerals were a vigesimal numeral system used by the Pre-Columbian Maya civilization.The numerals are made up of three symbols; zero , one and five...
of Central America used a mixed base 18 and base 20 system, possibly inherited from the Olmec
Olmec
The Olmec were the first major Pre-Columbian civilization in Mexico. They lived in the tropical lowlands of south-central Mexico, in the modern-day states of Veracruz and Tabasco....
, including advanced features such as positional notation and a zero
0 (number)
0 is both a numberand the numerical digit used to represent that number in numerals.It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems...
. 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
Venus
Venus is the second planet from the Sun, orbiting it every 224.7 Earth days. The planet is named after Venus, the Roman goddess of love and beauty. After the Moon, it is the brightest natural object in the night sky, reaching an apparent magnitude of −4.6, bright enough to cast shadows...
.
The Incan Empire ran a large command economy using quipu
Quipu
Quipus or khipus were recording devices used in the Inca Empire and its predecessor societies in the Andean region. A quipu usually consisted of colored, spun, and plied thread or strings from llama or alpaca hair. It could also be made of cotton cords...
, tallies made by knotting colored fibers. Knowledge of the encodings of the knots and colors was suppressed by the Spanish
Spain
Spain , officially the Kingdom of Spain languages]] under the European Charter for Regional or Minority Languages. In each of these, Spain's official name is as follows:;;;;;;), is a country and member state of the European Union located in southwestern Europe on the Iberian Peninsula...
conquistador
Conquistador
Conquistadors were Spanish soldiers, explorers, and adventurers who brought much of the Americas under the control of Spain in the 15th to 16th centuries, following Europe's discovery of the New World by Christopher Columbus in 1492...
s in the 16th century, and has not survived although simple quipu-like recording devices are still used in the Andean
Andes
The Andes is the world's longest continental mountain range. It is a continual range of highlands along the western coast of South America. This range is about long, about to wide , and of an average height of about .Along its length, the Andes is split into several ranges, which are separated...
region.
Some authorities believe that positional arithmetic began with the wide use of counting rods
Counting rods
Counting rods are small bars, typically 3–14 cm long, used by mathematicians for calculation in China, Japan, Korea, and Vietnam. They are placed either horizontally or vertically to represent any number and any fraction....
in China. The earliest written positional records seem to be rod calculus
Rod calculus
Rod calculus or rod calculation is the method of mathematical computation with counting rods in China from the Warring States to Ming dynasty before the counting rods were replaced by the more convenient and faster abacus.-Hardware:...
results in China around 400. In particular, zero was correctly described by Chinese mathematicians around 932.
The modern positional Arabic numeral system was developed by mathematicians in India
Indian mathematics
Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics , important contributions were made by scholars like Aryabhata, Brahmagupta, and Bhaskara II. The decimal number system in use today was first...
, and passed on to Muslim mathematicians
Islamic mathematics
In the history of mathematics, mathematics in medieval Islam, often termed Islamic mathematics or Arabic mathematics, covers the body of mathematics preserved and developed under the Islamic civilization between circa 622 and 1600...
, along with astronomical tables brought to Baghdad
Baghdad
Baghdad is the capital of Iraq, as well as the coterminous Baghdad Governorate. The population of Baghdad in 2011 is approximately 7,216,040...
by an Indian ambassador around 773.
From India
India
India , officially the Republic of India , is a country in South Asia. It is the seventh-largest country by geographical area, the second-most populous country with over 1.2 billion people, and the most populous democracy in the world...
, the thriving trade between Islamic sultans and Africa carried the concept to Cairo
Cairo
Cairo , is the capital of Egypt and the largest city in the Arab world and Africa, and the 16th largest metropolitan area in the world. Nicknamed "The City of a Thousand Minarets" for its preponderance of Islamic architecture, Cairo has long been a centre of the region's political and cultural life...
. Arabic mathematicians extended the system to include decimal fractions
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....
, and wrote an important work about it in the 9th century. The modern Arabic numerals
Arabic numerals
Arabic numerals or Hindu numerals or Hindu-Arabic numerals or Indo-Arabic numerals are the ten digits . They are descended from the Hindu-Arabic numeral system developed by Indian mathematicians, in which a sequence of digits such as "975" is read as a numeral...
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
Binary numeral system
The binary numeral system, or base-2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2...
(base 2), was propagated in the 17th century by Gottfried Leibniz
Gottfried Leibniz
Gottfried Wilhelm Leibniz was a German philosopher and mathematician. He wrote in different languages, primarily in Latin , French and German ....
. Leibniz had developed the concept early in his career, and had revisited it when he reviewed a copy of the I ching
I Ching
The I Ching or "Yì Jīng" , also known as the Classic of Changes, Book of Changes and Zhouyi, is one of the oldest of the Chinese classic texts...
from China. Binary numbers came into common use in the 20th century because of computer applications.
Numerals in most popular systems
| West Arabic | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|---|
| East Arabic | ٠ | ١ | ٢ | ٣ | ٤ | ٥ | ٦ | ٧ | ٨ | ٩ |
| Persian | ٠ | ١ | ٢ | ٣ | ۴ | ۵ | ۶ | ٧ | ٨ | ٩ |
| Asomiya (Assamese) | ০ | ১ | ২ | ৩ | ৪ | ৫ | ৬ | ৭ | ৮ | ৯ |
| Bengali | ০ | ১ | ২ | ৩ | ৪ | ৫ | ৬ | ৭ | ৮ | ৯ |
| Chinese (everyday) |
零 | 一 | 二 | 三 | 四 | 五 | 六 | 七 | 八 | 九 |
| Chinese (formal) |
零 | 壹 | 贰/貳 | 叁/叄 | 肆 | 伍 | 陆/陸 | 柒 | 捌 | 玖 |
| Chinese (Suzhou) |
〇 | 〡 | 〢 | 〣 | 〤 | 〥 | 〦 | 〧 | 〨 | 〩 |
| Devanagari | ० | १ | २ | ३ | ४ | ५ | ६ | ७ | ८ | ९ |
| Ge'ez (Ethiopic) |
፩ | ፪ | ፫ | ፬ | ፭ | ፮ | ፯ | ፰ | ፱ | |
| Gujarati | ૦ | ૧ | ૨ | ૩ | ૪ | ૫ | ૬ | ૭ | ૮ | ૯ |
| Gurmukhi | ੦ | ੧ | ੨ | ੩ | ੪ | ੫ | ੬ | ੭ | ੮ | ੯ |
| Hieroglyphic Egyptian | 𓏺 | 𓏻 | 𓏼 | 𓏽 | 𓏾 | 𓏿 | 𓐀 | 𓐁 | 𓐂 | |
| Kannada | ೦ | ೧ | ೨ | ೩ | ೪ | ೫ | ೬ | ೭ | ೮ | ೯ |
| Khmer | 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 | |
| Tamil Tamil language Tamil is a Dravidian language spoken predominantly by Tamil people of the Indian subcontinent. It has official status in the Indian state of Tamil Nadu and in the Indian union territory of Pondicherry. Tamil is also an official language of Sri Lanka and Singapore... |
௦ | ௧ | ௨ | ௩ | ௪ | ௫ | ௬ | ௭ | ௮ | ௯ |
| Telugu Telugu language Telugu is a Central Dravidian language primarily spoken in the state of Andhra Pradesh, India, where it is an official language. It is also spoken in the neighbouring states of Chattisgarh, Karnataka, Maharashtra, Orissa and Tamil Nadu... |
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 |
| Urdu Urdu Urdu is a register of the Hindustani language that is identified with Muslims in South Asia. It belongs to the Indo-European family. Urdu is the national language and lingua franca of Pakistan. It is also widely spoken in some regions of India, where it is one of the 22 scheduled languages and an... |
۰ | ۱ | ۲ | ۳ | ۴ | ۵ | ۶ | ۷ | ۸ | ۹ |
Additional numerals
| 10 | 20 | 30 | 40 | 100 | 1000 | 10000 | 108 | 1012 | |
|---|---|---|---|---|---|---|---|---|---|
| Chinese (simple) |
十 | 廿 | 卅 | 卌 | 百 | 千 | 万 | 亿 | 兆 |
| Chinese (complex) |
拾 | 佰 | 仟 | 萬 | 億 | 兆 |
| 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 | 10000 | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Ge'ez (Ethiopic) |
፲ | ፳ | ፴ | ፵ | ፶ | ፷ | ፸ | ፹ | ፺ | ፻ | ፼ |
| 10 | 50 | 100 | 500 | 1000 | 5000 | |
|---|---|---|---|---|---|---|
| Roman |
See also
- HexadecimalHexadecimalIn mathematics and computer science, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F to represent values ten to fifteen...
- BitBitA bit is the basic unit of information in computing and telecommunications; it is the amount of information stored by a digital device or other physical system that exists in one of two possible distinct states...
- Significant digit
- Large numbersLarge numbersThis article is about large numbers in the sense of numbers that are significantly larger than those ordinarily used in everyday life, for instance in simple counting or in monetary transactions...
- AbacusAbacusThe abacus, also called a counting frame, is a calculating tool used primarily in parts of Asia for performing arithmetic processes. Today, abaci are often constructed as a bamboo frame with beads sliding on wires, but originally they were beans or stones moved in grooves in sand or on tablets of...
- History of large numbersHistory of large numbersDifferent cultures used different traditional numeral systems for naming large numbers. The extent of large numbers used varied in each culture....
- List of numeral system topics
Numeral notation in various scripts
- Arabic numeralsArabic numeralsArabic numerals or Hindu numerals or Hindu-Arabic numerals or Indo-Arabic numerals are the ten digits . They are descended from the Hindu-Arabic numeral system developed by Indian mathematicians, in which a sequence of digits such as "975" is read as a numeral...
- Armenian numeralsArmenian numeralsThe system of Armenian numerals is a historic numeral system created using the majuscules of the Armenian alphabet.There was no notation for zero in the old system, and the numeric values for individual letters were added together. The principles behind this system are the same as for the Ancient...
- Babylonian numeralsBabylonian numeralsBabylonian numerals were written in cuneiform, using a wedge-tipped reed stylus to make a mark on a soft clay tablet which would be exposed in the sun to harden to create a permanent record....
- Burmese numeralsBurmese numeralsBurmese numerals are a set of numerals traditionally used in the Burmese language, although the Arabic numerals are also used. Burmese numerals follow the Hindu-Arabic numeral system commonly used in the rest of the world.-Main numbers:-Zero to nine:...
- Chinese numeralsChinese numeralsChinese numerals are characters for writing numbers in Chinese. Today speakers of Chinese use three numeral systems:the ubiquitous Arabic numerals and two indigenous systems....
- Greek numeralsGreek numeralsGreek numerals are a system of representing numbers using letters of the Greek alphabet. They are also known by the names Ionian numerals, Milesian numerals , Alexandrian numerals, or alphabetic numerals...
- Hebrew numeralsHebrew numeralsThe system of Hebrew numerals is a quasi-decimal alphabetic numeral system using the letters of the Hebrew alphabet.In this system, there is no notation for zero, and the numeric values for individual letters are added together...
- Indian numeralsIndian numeralsMost of the positional base 10 numeral systems in the world have originated from India, where the concept of positional numeration was first developed...
- Japanese numeralsJapanese numeralsThe system of Japanese numerals is the system of number names used in the Japanese language. The Japanese numerals in writing are entirely based on the Chinese numerals and the grouping of large numbers follow the Chinese tradition of grouping by 10,000...
- Korean numeralsKorean numeralsThe Korean language has two regularly used sets of numerals, a native Korean system and Sino-Korean system.-Construction:For both native and Sino- Korean numerals, the teens are represented by a combination of tens and the ones places...
- Mayan numerals
- QuipuQuipuQuipus or khipus were recording devices used in the Inca Empire and its predecessor societies in the Andean region. A quipu usually consisted of colored, spun, and plied thread or strings from llama or alpaca hair. It could also be made of cotton cords...
- Rod numerals
- Roman numeralsRoman numeralsThe numeral system of ancient Rome, or Roman numerals, uses combinations of letters from the Latin alphabet to signify values. The numbers 1 to 10 can be expressed in Roman numerals as:...
- Suzhou numeralsSuzhou numeralsThe Suzhou numerals or huama is a numeral system used in China before the introduction of Arabic numerals.-History:The Suzhou numeral system is the only surviving variation of the rod numeral system. The rod numeral system is a positional numeral system used by the Chinese in mathematics...