The ten Arabic numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 are the most commonly used symbols for writing numbers. The term often also implies a positional notation using the numerals, as well as the use of a decimal base, in particular when contrasted with other systems such as Roman numerals. However, the symbols are also used to write numbers in other bases such as octal, as well as for writing non-numerical information such as trademarks or license plate identifiers.
They are also called Western Arabic numerals, Ghubār numerals, Hindu–Arabic numerals,[1]Western digits, Latin digits, or European digits.[2] The Oxford English Dictionary differentiates them with the fully capitalized Arabic Numerals to refer to the Eastern digits.[3] The term numbers or numerals or digits often implies only these symbols, however this can only be inferred from context.
Europeans first learned of Arabic numerals about the 10th century, though their spread was a gradual process. Two centuries later, in the Algerian city of Béjaïa, the Italian scholar Fibonacci encountered the numerals; his 13th century work Liber Abaci was crucial in making them known throughout Europe. Until the evolution of the printing press in the 15th century, use of Arabic numerals in Europe was mainly confined to Northern Italy.[4] European trade, books, and colonialism subsequently helped popularize the adoption of Arabic numerals around the world. The numerals have found worldwide use significantly beyond the contemporary spread of the Latin alphabet, and have become common in the writing systems where other numeral systems existed previously, such as Chinese and Japanese numerals.
History
Origin
Evolution of Indian numerals into Arabic numerals and their adoption in Europe
Positional decimal notation including a zero symbol was developed in India, using symbols visually distinct from those that would eventually enter into international use. As the concept spread, the sets of symbols used in different regions diverged over time.
The immediate ancestors of the digits now commonly called "Arabic numerals" were introduced to Europe in the 10th century by Arabic speakers of Spain and North Africa, with digits at the time in wide use from Libya to Morocco. In the eastern part of the Arabian Peninsula, the Arabs were using the Eastern Arabic numerals or "Mashriki" numerals: ٠, ١, ٢, ٣, ٤, ٥, ٦, ٧, ٨, ٩.[5]
Al-Nasawi wrote in the early 11th century that mathematicians had not agreed on the form of the numerals, but most of them had agreed to train themselves with the forms now known as Eastern Arabic numerals.[6] The oldest specimens of the written numerals available are from Egypt and date to 873–874 AD. They show three forms of the numeral "2" and two forms of the numeral "3", and these variations indicate the divergence between what later became known as the Eastern Arabic numerals and the Western Arabic numerals.[7] The Western Arabic numerals came to be used in the Maghreb and Al-Andalus from the 10th century onward.[8] Some amount of consistency in the Western Arabic numeral forms endured from the 10th century, found in a Latin manuscript of Isidore of Seville's Etymologiae from 976 and the Gerbertian abacus, into the 12th and 13th centuries, in early manuscripts of translations from the city of Toledo.[5]
Calculations were originally performed using a dust board (takht, Latin: tabula), which involved writing symbols with a stylus and erasing them. The use of the dust board appears to have introduced a divergence in terminology as well: whereas the Hindu reckoning was called ḥisāb al-hindī in the east, it was called ḥisāb al-ghubār in the west (literally, "calculation with dust").[9] The numerals themselves were referred to in the west as ashkāl al‐ghubār ("dust figures") or qalam al-ghubår ("dust letters").[10]Al-Uqlidisi later invented a system of calculations with ink and paper "without board and erasing" (bi-ghayr takht wa-lā maḥw bal bi-dawāt wa-qirṭās).[11]
A popular myth claims that the symbols were designed to indicate their numeric value through the number of angles they contained, but there is no contemporary evidence of this, and the myth is difficult to reconcile with any digits past 4.[12]
Adoption and spread
The first Arabic numerals in the West appeared in the Codex Albeldensis in Spain.
The first mentions of the numerals from 1 to 9 in the West are found in the 976 Codex Vigilanus, an illuminated collection of various historical documents covering a period from antiquity to the 10th century in Hispania.[13] Other texts show that numbers from 1 to 9 were occasionally supplemented by a placeholder known as sipos, represented as a circle or wheel, reminiscent of the eventual symbol for zero. The Arabic term for zero is sifr (صفر), transliterated into Latin as cifra, and the origin of the English word cipher.
From the 980s, Gerbert of Aurillac (later Pope Sylvester II) used his position to spread knowledge of the numerals in Europe. Gerbert studied in Barcelona in his youth. He was known to have requested mathematical treatises concerning the astrolabe from Lupitus of Barcelona after he had returned to France.[13]
The reception of Arabic numerals in the West was gradual and lukewarm, as other numeral systems circulated in addition to the older Roman numbers. As a discipline, the first to adopt Arabic numerals as part of their own writings were astronomers and astrologists, evidenced from manuscripts surviving from mid-12th-century Bavaria. Reinher of Paderborn (1140–1190) used the numerals in his calendrical tables to calculate the dates of Easter more easily in his text Compotus emendatus.[14]
Leonardo Fibonacci was a Pisan mathematician who had studied in the Pisan trading colony of Bugia, in what is now Algeria,[15] and he endeavored to promote the numeral system in Europe with his 1202 book Liber Abaci:
When my father, who had been appointed by his country as public notary in the customs at Bugia acting for the Pisan merchants going there, was in charge, he summoned me to him while I was still a child, and having an eye to usefulness and future convenience, desired me to stay there and receive instruction in the school of accounting. There, when I had been introduced to the art of the Indians' nine symbols through remarkable teaching, knowledge of the art very soon pleased me above all else and I came to understand it.
The Liber Abaci introduced the huge advantages of a positional numeric system, and was widely influential. As Fibonacci used the symbols from Béjaïa for the digits, these symbols were also introduced in the same instruction, ultimately leading to their widespread adoption.[16]
Fibonacci's introduction coincided with Europe's commercial revolution of the 12th and 13th centuries, centered in Italy. Positional notation could be used for quicker and more complex mathematical operations (such as currency conversion) than Roman and other numeric systems could. They could also handle larger numbers, did not require a separate reckoning tool, and allowed the user to check a calculation without repeating the entire procedure.[16] Although positional notation opened possibilities that were hampered by previous systems, late medieval Italian merchants did not stop using Roman numerals (or other reckoning tools). Rather, Arabic numerals became an additional tool that could be used alongside others.[16]
Europe
A German manuscript page teaching use of Arabic numerals (Talhoffer Thott, 1459). At this time, knowledge of the numerals was still widely seen as esoteric, and Talhoffer presents them with the Hebrew alphabet and astrology.Table of numerals in many variants, 1757, by Jean-Étienne Montucla
By the late 14th century, only a few texts using Arabic numerals appeared outside of Italy. This suggests that the use of Arabic numerals in commercial practice, and the significant advantage they conferred, remained a virtual Italian monopoly until the late 15th century.[16] This may in part have been due to language barriers: although Fibonacci's Liber Abaci was written in Latin, the Italian abacus traditions was predominantly written in Italian vernaculars that circulated in the private collections of abacus schools or individuals. It was likely difficult for non-Italian merchant bankers to access comprehensive information.
The European acceptance of the numerals was accelerated by the invention of the printing press, and they became widely known during the 15th century. Their use grew steadily in other centers of finance and trade such as Lyon.[17] Early evidence of their use in Britain includes: an equal hour horary quadrant from 1396,[18] in England, a 1445 inscription on the tower of Heathfield Church, Sussex; a 1448 inscription on a wooden lych-gate of Bray Church, Berkshire; and a 1487 inscription on the belfry door at Piddletrenthide church, Dorset; and in Scotland a 1470 inscription on the tomb of the first Earl of Huntly in Elgin Cathedral.[19] In central Europe, the King of HungaryLadislaus the Posthumous, started the use of Arabic numerals, which appear for the first time in a royal document of 1456.[20]
By the mid-16th century, they were in common use in most of Europe. Roman numerals remained in use mostly for the notation of Anno Domini (“A.D.”) years, and for numbers on clock faces.[citation needed] Other digits (such as Eastern Arabic) were virtually unknown.[citation needed]
Russia
Prior to the introduction of Arabic numerals, Cyrillic numerals, derived from the Cyrillic alphabet, were used by South and East Slavs. The system was used in Russia as late as the early 18th century, although it was formally replaced in official use by Peter the Great in 1699.[21] Reasons for Peter's switch from the alphanumerical system are believed to go beyond a surface-level desire to imitate the West. Historian Peter Brown makes arguments for sociological, militaristic, and pedagogical reasons for the change. At a broad, societal level, Russian merchants, soldiers, and officials increasingly came into contact with counterparts from the West and became familiar with the communal use of Arabic numerals. Peter also covertly travelled throughout Northern Europe from 1697 to 1698 during his Grand Embassy and was likely informally exposed to Western mathematics during this time.[22] The Cyrillic system was found to be inferior for calculating practical kinematic values, such as the trajectories and parabolic flight patterns of artillery. With its use, it was difficult to keep pace with Arabic numerals in the growing field of ballistics, whereas Western mathematicians such as John Napier had been publishing on the topic since 1614.[23]
China
Chinese Shang dynasty oracle bone numerals of 14th century B.C.
The Chinese Shang dynasty numerals from the 14th century B.C. predates the Indian Brahmi numerals by over 1000 years and shows substantial similarity to the Brahmi numerals. Similar to the modern Arabic numerals, the Shang dynasty numeral system was also decimal based and positional.[26][27]
While positional Chinese numeral systems such as the counting rod system and Suzhou numerals had been in use prior to the introduction of modern Arabic numerals,[28][29] the externally-developed system was eventually introduced to medieval China by the Hui people. In the early 17th century, European-style Arabic numerals were introduced by Spanish and Portuguese Jesuits.[30][31][32]
Encoding
The ten Arabic numerals are encoded in virtually every character set designed for electric, radio, and digital communication, such as Morse code. They are encoded in ASCII (and therefore in Unicode encodings[33]) at positions 0x30 to 0x39. Masking all but the four least-significant binary digits gives the value of the decimal digit, a design decision facilitating the digitization of text onto early computers. EBCDIC used a different offset, but also possessed the aforementioned masking property.
↑ Kunitzsch 2003, p.7: "Les personnes qui se sont occupées de la science du calcul n'ont pas été d'accord sur une partie des formes de ces neuf signes; mais la plupart d'entre elles sont convenues de les former comme il suit."
↑ Kunitzsch 2003, pp.12–13: "While specimens of Western Arabic numerals from the early period—the tenth to thirteenth centuries—are still not available, we know at least that Hindu reckoning (called ḥisāb al-ghubār) was known in the West from the 10th century onward..."
↑ Ifrah, Georges (1998). The universal history of numbers: from prehistory to the invention of the computer. Translated by David Bellos (from the French). London: Harvill Press. pp.356–357. ISBN9781860463242.
↑ Danna, Raffaele; Iori, Martina; Mina, Andrea (22 June 2022). "A Numerical Revolution: The Diffusion of Practical Mathematics and the Growth of Pre-modern European Economies". SSRN4143442.
↑ The Shorter Science & Civilisation in China Vol 2, An abridgement by Colin Ronan of Joseph Needham's original text, Table 20, p. 6, Cambridge University Press ISBN0-521-23582-0
↑ The Shorter Science & Civilisation in China Vol 2, An abridgement by Colin Ronan of Joseph Needham's original text, Table 20, p. 6, Cambridge University Press ISBN0-521-23582-0
Burnett, Charles (2006). "The Semantics of Indian Numerals in Arabic, Greek and Latin". Journal of Indian Philosophy. 34 (1–2). Springer-Netherlands: 15–30. doi:10.1007/s10781-005-8153-z. S2CID170783929.
Hayashi, Takao (1995). The Bakhshālī Manuscript: An Ancient Indian Mathematical Treatise. Groningen, Netherlands: Egbert Forsten. ISBN906980087X.
Ifrah, Georges (2000). A Universal History of Numbers: From Prehistory to Computers. New York: Wiley. ISBN0471393401.
Katz, Victor J., ed. (20 July 2007). The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton, New Jersey: Princeton University Press. ISBN978-0691114859.
The decimal numeral system is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral system. The way of denoting numbers in the decimal system is often referred to as decimal notation.
Fibonacci, also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano, was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western mathematician of the Middle Ages".
A numeral system is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner.
A number is a mathematical object used to count, measure, and label. The most basic examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can be represented by symbols, called numerals; for example, "5" is a numeral that represents the number five. As only a relatively small number of symbols can be memorized, basic numerals are commonly organized in a numeral system, which is an organized way to represent any number. The most common numeral system is the Hindu–Arabic numeral system, which allows for the representation of any non-negative integer using a combination of ten fundamental numeric symbols, called digits. In addition to their use in counting and measuring, numerals are often used for labels, for ordering, and for codes. In common usage, a numeral is not clearly distinguished from the number that it represents.
1 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. In conventions of sign where zero is considered neither positive nor negative, 1 is the first and smallest positive integer. It is also sometimes considered the first of the infinite sequence of natural numbers, followed by 2, although by other definitions 1 is the second natural number, following 0.
0 (zero) is a number representing an empty quantity. Adding 0 to any number leaves that number unchanged. In mathematical terminology, 0 is the additive identity of the integers, rational numbers, real numbers, and complex numbers, as well as other algebraic structures. Multiplying any number by 0 has the result 0, and consequently, division by zero has no meaning in arithmetic.
A decimal separator is a symbol used to separate the integer part from the fractional part of a number written in decimal form. Different countries officially designate different symbols for use as the separator. The choice of symbol also affects the choice of symbol for the thousands separator used in digit grouping.
Liber Abaci is a historic 1202 Latin manuscript on arithmetic by Leonardo of Pisa, posthumously known as Fibonacci.
A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method of mathematical expression which uses only two symbols: typically "0" (zero) and "1" (one).
A numerical digit or numeral is a single symbol used alone or in combinations, to represent numbers in a positional numeral system. The name "digit" comes from the fact that the ten digits of the hands correspond to the ten symbols of the common base 10 numeral system, i.e. the decimal digits.
Algorism is the technique of performing basic arithmetic by writing numbers in place value form and applying a set of memorized rules and facts to the digits. One who practices algorism is known as an algorist. This positional notation system has largely superseded earlier calculation systems that used a different set of symbols for each numerical magnitude, such as Roman numerals, and in some cases required a device such as an abacus.
Positional notation usually denotes the extension to any base of the Hindu–Arabic numeral system. More generally, a positional system is a numeral system in which the contribution of a digit to the value of a number is the value of the digit multiplied by a factor determined by the position of the digit. In early numeral systems, such as Roman numerals, a digit has only one value: I means one, X means ten and C a hundred. In modern positional systems, such as the decimal system, the position of the digit means that its value must be multiplied by some value: in 555, the three identical symbols represent five hundreds, five tens, and five units, respectively, due to their different positions in the digit string.
Lattice multiplication, also known as the Italian method, Chinese method, Chinese lattice, gelosia multiplication, sieve multiplication, shabakh, diagonally or Venetian squares, is a method of multiplication that uses a lattice to multiply two multi-digit numbers. It is mathematically identical to the more commonly used long multiplication algorithm, but it breaks the process into smaller steps, which some practitioners find easier to use.
The Hindu–Arabic numeral system is a decimal place-value numeral system that uses a zero glyph as in "205".
The Hindu–Arabic numeral system is a positional base ten numeral system for representing integers; its extension to non-integers is the decimal numeral system, which is presently the most common numeral system.
The Eastern Arabic numerals, also called Indo-Arabic numerals, are the symbols used to represent numerical digits in conjunction with the Arabic alphabet in the countries of the Mashriq, the Arabian Peninsula, and its variant in other countries that use the Persian numerals on the Iranian plateau and in Asia.
A timeline of numerals and arithmetic.
Principles of Hindu Reckoning is a mathematics book written by the 10th- and 11th-century Persian mathematician Kushyar ibn Labban. It is the second-oldest book extant in Arabic about Hindu arithmetic using Hindu-Arabic numerals, preceded by Kibab al-Fusul fi al-Hisub al-Hindi by Abul al-Hassan Ahmad ibn Ibrahim al-Uglidis, written in 952.
An alphabetic numeral system is a type of numeral system. Developed in classical antiquity, it flourished during the early Middle Ages. In alphabetic numeral systems, numbers are written using the characters of an alphabet, syllabary, or another writing system. Unlike acrophonic numeral systems, where a numeral is represented by the first letter of the lexical name of the numeral, alphabetic numeral systems can arbitrarily assign letters to numerical values. Some systems, including the Arabic, Georgian and Hebrew systems, use an already established alphabetical order. Alphabetic numeral systems originated with Greek numerals around 600 BC and became largely extinct by the 16th century. After the development of positional numeral systems like Hindu–Arabic numerals, the use of alphabetic numeral systems dwindled to predominantly ordered lists, pagination, religious functions, and divinatory magic.
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