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Mathematics & Computer Science

Decimal place-value system

by 5th c. CEIndian mathematicians (Aryabhata, Bakhshali tradition)Aryabhatiya; Bakhshali manuscriptPan-Indian / unspecified

401 CE

A base-10 positional notation where a digit's value depends on its column, enabling compact numbers and easy computation.

Dated: 401–500 CE
As scholars put it: by 5th c. CE
Originator: Indian mathematicians (Aryabhata, Bakhshali tradition)
Source text: Aryabhatiya; Bakhshali manuscript
Region: Pan-Indian / unspecified
Discipline: Mathematics & Computer Science
Evidence: Documented
Confidence: High
Verification: Verified-web

Nine signs and an empty column

The oldest Indian numerals, the Brahmi numerals of the 3rd century BCE, were not positional. They had one sign for each of 1 to 9, further signs for 10, 20 and so on up to 90, and more for 100, 200 and 1000, so a number was built by stringing signs together, much as Roman numerals are ⁷. To write a bigger number you needed a new symbol.

Place value removes that limit. Only nine signs are needed, plus a mark for an empty position, and the column a sign sits in tells you whether it means sevens, seventies or seven hundreds. The Babylonians had used position as early as the 19th century BCE, but in base 60 and without a true zero; the Indian innovation was a base-ten positional system ⁷. Pierre-Simon Laplace later called it "the ingenious method of expressing every possible number using a set of ten symbols" ¹².

Counting in words and syllables

Indian texts had long been at ease with very large numbers. In the Lalitavistara, a Buddhist work of roughly the 1st or 2nd century CE, the young Gautama is asked to name the ranks of numbers beyond the koti, ten million, and runs on to ten raised to the 53rd power and beyond ⁷. A naming scheme in which each rank is ten times the last is already a place-value idea, even before it is written with digits.

Aryabhata, who finished his Aryabhatiya in 499 at the age of 23 near Kusumapura (Patna), packed numbers into verse with a code of his own: consonants of the Sanskrit alphabet stood for values from 1 to 100, and vowels shifted them up by powers of ten, reaching 10 to the 18th ⁹. The historian Georges Ifrah judged that "it is extremely likely that Aryabhata knew the sign for zero and the numerals of the place value system" ⁹. Five centuries later the Persian scholar al-Biruni, who lived in India in the 1020s, reported that "the Indians do not use letters at all for arithmetic", meaning they calculated with digits ⁷.

Dating the first written digits

Pinning down when the digits first appear in writing is hard. A copper-plate charter of the ruler Dadda III from Sankheda in Gujarat carries a date equal to 594 CE written in place-value numerals, and similar charters follow from the 8th and 9th centuries, but historians have argued that some of these dates were added later or that the plates are forgeries ⁷. The first dated use that nobody disputes is the inscription of 876 CE in a temple at Gwalior ⁷.

The Bakhshali manuscript, 70 leaves of birch bark dug up in 1881 near Peshawar, uses a dot as a placeholder throughout ¹⁰. The Bodleian Library announced in 2017 that radiocarbon dating put parts of it as early as the 3rd or 4th century ¹⁰. Specialists in Indian mathematics replied that the treatise reads as one continuous work, so it was probably written when its youngest leaf was made, around the 8th to 10th century ¹¹. Either way, the written system was in everyday use well before the year 1000.

From dust boards to Pisa

The earliest hint of the digits outside India comes from 662 CE, when Severus Sebokht, a Syrian bishop on the Euphrates, praised Indian computation "done by means of nine signs" ⁸. In the 770s an Indian scholar reached the court of the caliph al-Mansur with an astronomical work that was translated into Arabic ⁸. Arab and Persian calculators worked the new numerals on a board spread with dust, which is why the western forms were called ghubar, from the Arabic for dust; in the 10th century al-Uqlidisi adapted the methods to pen and paper and wrote the earliest surviving book on what he called Indian arithmetic ⁸.

The digits first appear in a European manuscript in the Codex Vigilanus, copied in Spain in 976 ⁸. They spread slowly until Leonardo of Pisa, who had learned the "art of the Indians' nine symbols" while his father worked at Bugia in North Africa, published Liber abaci in 1202 ⁸. Merchants took them up first, and within a few centuries they had replaced Roman numerals across Europe and, later, the world ¹.

Background

Early Indian inscriptions wrote numbers with Brahmi numerals, which had separate signs for the units, the tens and the hundreds, so they worked by adding symbols rather than by position. In the first centuries CE, Indian calculators came to use only nine signs plus a mark for an empty place, letting a digit's column set its worth. Aryabhata's Aryabhatiya of 499 CE assumes a decimal place-value scheme, and dated inscriptions using it survive from the late 6th century onward.

Significance

With nine digits and a zero, any number can be written in a few marks and calculated with the same short routines: carrying, borrowing, long multiplication and division. Carried west through Baghdad and North Africa, the system displaced Roman numerals and the counting board in Europe and is now the way the whole world writes numbers.

Meanwhile in the West

Adopted in Europe as the 'Hindu-Arabic' numerals, replacing Roman numerals.

Fair to say

Positional ideas appear in other cultures; the fully decimal, symbol-based system is Indian.

Path to modern innovation

The basis of virtually all written arithmetic today.

401 CEDecimal place-value system

Builds on

1,000 BCENames for very large powers of ten

Transmission

Carried by Arab scholars to Europe; Fibonacci popularised it in 1202.

Kusumapura (Patna), 499 CE. Aryabhata
Aryabhata's Aryabhatiya, written here, works with a base-ten place-value system, where a digit's position sets its worth.
Bhillamala (Bhinmal), 628 CE. Brahmagupta
Brahmagupta's Brahmasphutasiddhanta gives the first rules for calculating with zero and with negative numbers.
Baghdad, c. 773 CE. An Indian astronomer at the court of Caliph al-Mansur
Indian astronomical tables reach the new Abbasid capital and are translated into Arabic. About fifty years later al-Khwarizmi writes his book on calculating with Hindu numerals here.
Albelda, La Rioja, 976 CE. Vigila and the monks of Albelda
The Codex Vigilanus, copied in this monastery, is the oldest known European manuscript to show the nine Hindu-Arabic digits.
Pisa, 1202 CE. Leonardo of Pisa (Fibonacci)
Having learned 'the method of the Indians' in North Africa, Fibonacci publishes Liber Abaci, which spreads the numerals among Italian merchants and on to all of Europe.

Play it in Then → Now

References

1Plofker (2009)
2G. G. Joseph, The Crest of the Peacock