Arabic numerals

• 1. Introduction

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  The Arabic numerals are ten symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. You used them today without a second thought. They sit on license plates and trademarks. They run through every character set a computer understands, from ASCII to Unicode to Morse code. Yet these shapes were not always universal, and they were not invented where their name suggests. They go by many other labels too: Western Arabic numerals, Western digits, European digits, Latin digits, even Ghubar numerals. The fuller name, Hindu-Arabic numerals, hints that something about this system began in India. So where did these ten symbols come from, and why did they take centuries to win over a continent that already had numbers? Why do scholars argue over a 2 and a 3 scratched onto a fragment from ninth-century Egypt? And how did a Pisan boy in an Algerian customs house help carry them into Europe?

• 2. Two Families Of Digits

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  Decimal was perfected using different symbols, and the Arabic numerals were a relatively recent development designed in North Africa. The immediate ancestors of these digits reached Europe in the 10th century, carried by Arabic speakers of Spain and North Africa. From Libya to Morocco, these forms were in wide use. To the east, a different set of shapes prevailed.

  From Egypt to Iraq and the Arabian Peninsula, the Arabs wrote with the Eastern Arabic numerals, also called Mashriqi numerals: the characters rendered as zero through nine in that script. Al-Nasawi observed in the early 11th century that mathematicians had not agreed on the form of the numerals. Most of them, he noted, had trained themselves on the forms now called Eastern Arabic.

  The oldest surviving specimens of the written numerals come from Egypt and date to 873-874 AD. They display three forms of the numeral 2 and two forms of the numeral 3. Those small variations mark the early split between what became the Eastern and the Western Arabic numerals. The Western forms took hold in the Maghreb and Al-Andalus from the 10th century onward.

  Some consistency in the Western forms held for centuries. It appears in a Latin manuscript of Isidore of Seville's Etymologiae from 976 and in the Gerbertian abacus. From there it carried into the 12th and 13th centuries, surviving in early manuscripts of translations from the city of Toledo.

• 3. Dust And Ink

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  Calculations were originally performed on a dust board, known as takht in Arabic and tabula in Latin. The method meant writing symbols with a stylus and then erasing them. This humble tool left a mark on language itself.

  The dust board split the terminology in two. In the east, Hindu reckoning was called hisab al-hindi. In the west it became hisab al-ghubar, meaning calculation with dust. The numerals were known there as ashkal al-ghubar, dust figures, or qalam al-ghubar, dust letters.

  Al-Uqlidisi later broke from the board entirely. He invented a system of calculations with ink and paper, described as without board and erasing, but with inkpot and sheet. That shift toward written-down arithmetic is why one of the system's many names is Ghubar numerals.

  A popular myth claims the symbols were shaped to reveal their value through the number of angles they contained. There is no contemporary evidence for the idea. The myth is hard to reconcile with any digit past 4.

• 4. From Sipos To Cipher

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  The 976 Codex Vigilanus holds the first mentions of the numerals from 1 to 9 in the West. It is an illuminated collection of historical documents spanning antiquity to the 10th century in Al-Andalus. Around those early digits, scribes left a clue about the missing tenth symbol.

  Numbers from 1 to 9 were sometimes supplemented by a placeholder called sipos, drawn as a circle or wheel. That shape foreshadowed the eventual symbol for zero. The Arabic word for zero is sifr, transliterated into Latin as cifra, which gave English the word cipher.

  From the 980s, Gerbert of Aurillac used his standing to spread knowledge of the numerals across Europe. He later became Pope Sylvester II. In his youth Gerbert had studied in Barcelona, and after returning to France he requested mathematical treatises on the astrolabe from Lupitus of Barcelona.

  Reception of the numerals stayed gradual and lukewarm, since older Roman numbers and other systems still circulated. Astronomers and astrologists were the first discipline to fold the new digits into their own writing. The evidence survives in manuscripts from mid-12th-century Bavaria. Reinher of Paderborn, who lived from 1140 to 1190, used the numerals in calendrical tables to fix the dates of Easter in his text Computus emendatus.

• 5. The Pisan And Bugia

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  Leonardo Fibonacci was a Pisan mathematician who studied in the trading colony of Bugia, the city now called Bejaia, in what is now Algeria. His 1202 book Liber Abaci set out to promote the numeral system in Europe. The book opens with how he first met the digits as a boy.

  Fibonacci recounted that his father, appointed public notary in the customs at Bugia for the Pisan merchants, summoned him there as a child. He wrote that he was introduced to the art of the Indians' nine symbols through remarkable teaching, and that knowledge of the art very soon pleased him above all else. From that schooling in accounting grew a lifelong project.

  The Liber Abaci analysis highlighting the advantages of positional notation proved widely influential. Its use of the Bejaia digits ultimately led to their widespread adoption in Europe. The timing helped, since the work coincided with the European commercial revolution of the 12th and 13th centuries centered in Italy.

  Positional notation made complex sums, such as currency conversion, faster than the Roman system allowed. It handled larger numbers, needed no separate reckoning tool, and let users check their work without redoing every step. Late medieval Italian merchants did not abandon Roman numerals or other tools. They simply added the new digits alongside the methods they already trusted.

• 6. An Italian Monopoly

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  By the late 14th century, only a few texts using Arabic numerals appeared outside Italy. The advantage the digits conferred in commerce remained a virtual Italian monopoly until the late 15th century. Part of the reason lay in language.

  Fibonacci's Liber Abaci was written in Latin, yet the Italian abacus traditions were written mostly in Italian vernaculars. Those texts circulated privately, held in the collections of abacus schools or individuals. The knowledge stayed local until a new technology pried it open.

  The printing press accelerated European acceptance, and the numerals became widely known during the 15th century. Their use grew in other centers of finance and trade, such as Lyon. Britain offers a trail of early inscriptions: an equal hour horary quadrant from 1396 in England, a 1445 inscription on the tower of Heathfield Church in Sussex, a 1448 inscription on a wooden lych-gate at Bray Church in Berkshire, and a 1487 inscription on the belfry door at Piddletrenthide church in Dorset. In Scotland, a 1470 inscription marks the tomb of the first Earl of Huntly in Elgin Cathedral.

  In central Europe, King Ladislaus the Posthumous of Hungary began the use of Arabic numerals, which first appear in a royal document of 1456. By the mid-16th century the digits had spread widely across Europe. By 1800 they had almost completely replaced counting boards and Roman numerals in accounting, leaving the older system for years and clock faces.

• 7. Russia And China

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  Cyrillic numerals, derived from the Cyrillic alphabet and Greek numerals, served South and East Slavs before the Arabic digits arrived. Russia used the system as late as the early 18th century. Peter the Great formally replaced it in official use in 1699.

  Historian Peter Brown argues that Peter's switch went beyond a wish to imitate the West, citing sociological, militaristic, and pedagogical reasons. Russian merchants, soldiers, and officials increasingly met Western counterparts who used Arabic numerals in common. Peter himself travelled covertly through Northern Europe from 1697 to 1698 during his Grand Embassy and was likely exposed to Western mathematics. The Cyrillic system proved inferior for ballistics, where Western mathematicians such as John Napier had been publishing since 1614.

  The Chinese Shang dynasty numerals from the 14th century BC predate the Indian Brahmi numerals by over 1,000 years and show substantial similarity to them. Like the modern Arabic numerals, the Shang system was decimal based and positional. China had its own positional traditions before the new digits came.

  The counting rod system and Suzhou numerals were already in use when the externally developed system reached medieval China through the Hui people. In the early 17th century, Spanish and Portuguese Jesuits introduced European-style Arabic numerals. That meeting of mathematical traditions reflects how far the ten symbols had travelled.

• 8. The Logic Of The Code

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  The numerals are encoded in virtually all character sets, including ASCII, Unicode, and even Morse code. Since Unicode includes ASCII, the same logic carries across both. The design hides a small piece of engineering elegance.

  In ASCII and therefore Unicode, masking all but the four least-significant binary digits yields the value of the decimal digit. That choice was made to ease the digitization of text. EBCDIC uses a different offset but supports a similar masking operation.

  The mapping is precise. The digit 0 sits at ASCII decimal 48, Unicode U+0030, and EBCDIC hex F0. The digit 9 sits at ASCII decimal 57, Unicode U+0039, and EBCDIC hex F9. Each step from one digit to the next moves the code point by exactly one. The ten symbols that began on a dust board in North Africa now run, in clean numeric order, through the machinery that stores nearly every written number.