Positional decimal system and zero
A way of writing numbers in which the value of a digit depends on its place, and an 'empty place' is marked by a separate sign: without it 12, 102 and 1002 cannot be told apart. In India this idea joined the decimal count, and zero became not only a sign but a number.
Verbal place-value notation is already present in the Lokavibhaga (458 CE) and in Aryabhata (499). In 628 Brahmagupta in the Brahmasphutasiddhanta gave rules for operating with zero and negative numbers: here zero is already a number, not an empty place.
Through Arabic arithmetic and the 12th-century Latin translations the notation reached Europe and made written column arithmetic a uniform procedure. One and the same algorithm works for thousands and for units: from this point on you can compute without thinking about what exactly you are counting.
Why this date
era, not a date. Axis position follows Brahmagupta's Brahmasphutasiddhanta (628), the first datable text with rules for operating with zero. Positional decimal notation appears in Indian texts earlier: Lokavibhaga (458), Aryabhata (499); the oldest dated inscription with positional numerals is 595, with a zero symbol 683 (Cambodia, K-127, as dated by Swetz; Britannica is more cautious and admits no inscription with such a symbol before the 9th century) and 876 (Gwalior). The dating of the Bakhshali manuscript is disputed: the Bodleian radiocarbon results (2017) gave 224-383, 680-779 and 885-993 for three folios, while Plofker et al. (2017) regard the text as a single, later work
What it grew from
-
Writing
influenced · Confidence: low
Positional decimal notation exists as a written notation: Indian numerals are signs of the Brahmi script, and the zero dot and positional numbers are known from inscriptions and manuscripts; but no source asserting this link directly has been found.
Dispute: Chrisomalis (Numerical Notation, 2010) treats numerical notations as a separate class of sign systems rather than a part of writing, and stresses that before modern times arithmetic was done on a counting board while notation only recorded results; the positional principle is possible on a board as well (Chinese rod numerals), so writing is not a physical prerequisite but the medium in which place value became a notation. Britannica ignores the Cambodian inscription of 683 with a zero symbol (Coedès 1931; Swetz).
Link sources
- Britannica, «Numerals and numeral systems», «The Hindu-Arabic system» (2024) Verified 24 Sep 2026
None of these early Indian inscriptions gives evidence of place value or of a zero that would make modern place value possible. Hindu literature gives evidence that the zero may have been known earlier, but there is no inscription with such a symbol before the 9th century.
- Chrisomalis S. «Re-counting the Cognitive History of Numerals», The MIT Press Reader (2021)
arithmetic was an embodied and material practice (e.g., on a counting board or abacus) whereas numerical notation was used for writing and recording.
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.
- Britannica, «Numerals and numeral systems», «The Hindu-Arabic system» (2024) Verified 24 Sep 2026
What grew out of it
-
Decimal fractions in computation
prerequisite for · Confidence: high
Decimal fractions extend positional decimal notation to the right of the units place: Stevin reads 3.27 as 327 units of 0.01, and the earliest direct treatment of decimal fractions (al-Uqlidisi, 952/953) appears in a book on Indian arithmetic; without positional notation such a fraction is impossible.
Link sources
- Minnaert M. G. J. «Stevin, Simon», Dictionary of Scientific Biography (1976) Verified 20 Sep 2026
327 items of the unit 0.01
- O'Connor J. J., Robertson E. F. «Al-Uqlidisi», MacTutor History of Mathematics (1999) Verified 20 Sep 2026
the earliest known text offering a direct treatment of decimal fractions
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.
- Minnaert M. G. J. «Stevin, Simon», Dictionary of Scientific Biography (1976) Verified 20 Sep 2026
-
Liber abaci: Indian numerals and Arabic algebra in a Latin book
prerequisite for · Confidence: high
The Liber abaci introduced the Hindu-Arabic place-valued decimal system and Arabic numerals into Europe; without them the book would not exist.
Link sources
- O'Connor J. J., Robertson E. F. «Leonardo Pisano Fibonacci», MacTutor History of Mathematics Verified 4 Oct 2026
The book, which went on to be widely copied and imitated, introduced the Hindu-Arabic place-valued decimal system and the use of Arabic numerals into Europe.
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.
- O'Connor J. J., Robertson E. F. «Leonardo Pisano Fibonacci», MacTutor History of Mathematics Verified 4 Oct 2026
Node sources
- O'Connor J. J., Robertson E. F. «Brahmagupta», MacTutor History of Mathematics (2000) Verified 20 Sep 2026
In particular he wrote Brahmasphutasiddhanta, in 628.
- O'Connor J. J., Robertson E. F. «Brahmagupta», MacTutor History of Mathematics (2000) Verified 20 Sep 2026
In the Brahmasphutasiddhanta he defined zero as the result of subtracting a number from itself. He gave some properties as follows:- When zero is added to a number or subtracted from a number, the number remains unchanged; and a number multiplied by zero becomes zero. He also gives arithmetical rules in terms of fortunes (positive numbers) and debts (negative numbers).
- Bailey D. H., Borwein J. M. «Ancient Indian Square Roots: An Exercise in Forensic Paleo-Mathematics», American Mathematical Monthly 119(8), 646-657 (2012)
But one manuscript whose ancient authenticity cannot be denied is the Lokavibhaga ("Parts of the Universe"). This has numerous large numbers in positional decimal notation (using Sanskrit names or word-symbols for the digits), such as 14236713, 13107200000 and 70500000000000000, and detailed calculations. Near the end of the Lokavibhaga, the author provides some astronomical observations that enable modern scholars to determine, in two independent ways, that this text was written on 25 August 458 CE (Julian calendar).
- Bailey D. H., Borwein J. M. «Ancient Indian Square Roots: An Exercise in Forensic Paleo-Mathematics», American Mathematical Monthly 119(8), 646-657 (2012)
Similarly, in 499 CE the Indian mathematician Aryabhata wrote, "The numbers one, ten, hundred, thousand, ten thousand, hundred thousand, million, ten million, hundred million, and billion are from place to place each ten times the preceding".
- Britannica, «Numerals and numeral systems», «The Hindu-Arabic system» (2024)
Modern numeral systems are place-value systems. That is, the value of the symbol depends upon the position or place of the symbol in the representation; for example, the 2 in 20 and 200 represents two tens and two hundreds, respectively.
- Plofker K., Keller A., Hayashi T., Montelle C., Wujastyk D. «The Bakhshālī Manuscript: A Response to the Bodleian Library's Radiocarbon Dating», History of Science in South Asia 5(1) (2017) Verified 20 Sep 2026, quotation assembled from fragments
In this paper, we disagree with the interpretation of the findings announced by the Bodleian team. In particular, we argue that the earliest dated folio of this manuscript is unlikely to be the date of the whole text. Rather, the latest dateable folio is logically the date of the scribal activity. ... And we argue that the Bakhshālī Manuscript does include written zeros that function as arithmetical operators, i.e., as numbers in their own right, and not merely as place-holders, as asserted by the Bodleian team.
- Howell D., Evison G., Formigatto C., Llado-Buisan V., Chivall D. «Bodleian Library Bakhshali Research Statement 2017» (2017)
It is actually composed of material from at least three dates, with some pages dating from as early as the 3rd to 4th century and others dating from the 8th and 10th centuries.
- Swetz F. J. «Mathematical Treasure: The Cambodian Zero», MAA Convergence. DOI 10.4169/Convergence20151010 404, MAA (2015) Verified 25 Sep 2026
Amir Aczel (1950–2015) located and documented the existence of this zero, a Khmer zero used in a stone inscription denoting the Khmer year 604 (682–683 CE). The chronological inscription is contained on a stone stele first discovered in 1891 and designated in archaeological coding as K-127.
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong. "Assembled from fragments" means the words do not stand together on the page: we found each fragment separately and joined them with an ellipsis.