[Earlier version posted 6 Feb 2010]

### Introduction

**Question**: What mathematical discovery more than 1500 years ago:

- Is one of the greatest, if not the greatest, single discovery in the field of mathematics?
- Involved three subtle ideas that evaded the greatest minds of antiquity, even including geniuses such as Archimedes?
- Was fiercely resisted in Europe for hundreds of years after its discovery?
- Even today, in historical treatments of mathematics, is often dismissed with scant mention, or else is ascribed to the wrong source?

**Answer**: Our modern system of positional decimal arithmetic with zero, which was discovered in India in the fourth or fifth century.

### Why?

As the 19th century mathematician Pierre-Simon Laplace explained:

*It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity and the great ease which it has lent to all computations put our arithmetic in the first rank of useful inventions; and we shall appreciate the grandeur of this achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity*. [Durant1954, pg. 527]

As Laplace noted, today we often take this scheme for granted as “trivial,” but it is anything but trivial (as any youngster in grade school will attest), since it eluded the best minds of the ancient world, even the genius Archimedes. Note that although Archimedes saw far beyond the mathematics of his time, even anticipating numerous key ideas of modern calculus and numerical analysis, and also even though he was very skilled in applying these mathematical principles to engineering and astronomy, nonetheless he used the traditional Greek-Roman numeral system for calculations. Archimedes’ computation of pi was a *tour de force* of numerical interval analysis performed without either positional notation or trigonometry [Berggren2004; Netz2007].

Perhaps one reason this discovery gets so little attention today is that it is very hard for us to appreciate the enormous difficulty of using Roman numerals, counting tables and abacuses. Michel de Montaigne, Mayor of Bordeaux and one of the most learned men of his day, confessed in 1588 (prior to the widespread adoption of decimal arithmetic in Europe) that in spite of his great education and erudition, “*I cannot yet cast account either with penne or Counters.*” That is, he could not do basic arithmetic [Ifrah2000, pg. 577]. In a similar vein, at about the same time a wealthy German merchant, consulting a scholar regarding which European university offered the best education for his son, was told the following:

*If you only want him to be able to cope with addition and subtraction, then any French or German university will do. But if you are intent on your son going on to multiplication and division—assuming that he has sufficient gifts—then you will have to send him to Italy.* [Ifrah2000, pg. 577]

Indeed, the development of this system hinged on three key abstract (and certainly non-intuitive) principles [Ifrah2000, pg. 346]: (a) graphical signs largely removed from intuitive associations; (b) positional notation; and (c) a fully operational zero—filling the empty spaces of missing units and at the same time representing a null value. Some civilizations succeeded in discovering one or two of these principles, but none of them, prior to early-first-millennium India, found all three and then combined them with effective algorithms for practical computing.

### History

So who exactly discovered the Indian system? Sadly, there is no record of the individual who first discovered the scheme, who, if known, would surely rank among the greatest mathematicians of all time. As Dantzig notes, “the achievement of the unknown Hindu who some time in the first centuries of our era discovered [positional decimal arithmetic] assumes the proportions of a world-event. Not only did this principle constitute a radical departure in method, but we know now that without it no progress in arithmetic was possible” [Dantzig2007, pg. 29-30].

The earliest document that exhibits familiarity with decimal arithmetic, and which at the same time can be accurately dated, is the Indian astronomical work *Lokavibhaga* (“Parts of the Universe”) [Lokavibhaga]. Here, for example, we find numerous large numbers, such as 14236713, 13107200000 and 70500000000000000, as well as detailed calculations such as (14230249 – 355684) / 212 = 65446 13/212 [Lokavighaga, pg. 70, 79, 131, 69]. Methods for computation were not presented in this work — the author evidently presumed that the reader understood decimal arithmetic. Near the end of the *Lokavibhaga*, the author provides detailed astronomical data that enable modern scholars to confirm, in two independent ways, that this text was written on 25 August 458 CE (Julian calendar). The text also mentions that it was written in the 22nd year of the reign of Simhavarman, which corresponds to 458 CE. As Ifrah points out, this information not only allows us to date the document with precision, but also proves its authenticity [Ifrah2000, pg. 417].

An even more ancient source employing positional decimal arithmetic is the Bakhshali manuscript, a copy of a ancient mathematical treatise that gives rules for computing with fractions. British scholar Rudolf Hoernle, for instance, has noted that the document was written in the “Shloka” style, which was replaced by the “Arya” style prior to 500 CE, and furthermore that it was written in the “Gatha” dialect, which was largely replaced by Sanskrit, at least in secular writings, prior to 300 CE. Also, unlike later documents, it used a plus sign for negative numbers and did not use a dot for zero (although it used a dot for empty position). For these and other reasons scholars have concluded that the original document most likely was written in the third or fourth century [Hoernle1887].

In 510 CE, the Indian mathematician Aryabhata presented schemes not only for various arithmetic operations, but also for square roots and cube roots. Additionally, Aryabhata gave a decimal value of Pi = 3.1416. Aryabhata’s “digital” algorithms for computing square roots and cube roots are given in [Decimal]. A statue of Aryabhata, on display at the Inter-University Centre for Astronomy and Astrophysics (IUCAA) in Pune, India, is shown here:

The Indian system was introduced in Europe by Gerbert of Aurillac in the tenth century. He traveled to Spain to learn about the system first-hand from Arab scholars, then was the first Christian to teach mathematics using decimal arithmetic, all prior to his brief reign as Pope Sylvester II (999–1002 CE) [Brown2010, pg. 5]. Little progress was made at the time, though, in part because of clerics who, in the wake of the crusades, rumored that Sylvester II had been a sorcerer, and that he had sold his soul to Lucifer during his travels to Islamic Spain. These accusations persisted until 1648, when papal authorities who reopened his tomb reported that Sylvester’s body had not, as suggested in historical accounts, been dismembered in penance for Satanic practices [Brown2010, pg. 236]. Sylvester’s reign was a turbulent time, and he died after a short reign. It is worth speculating how history would have been different had this remarkable scientist-Pope lived longer.

In 1202 CE, Leonardo of Pisa, also known as Fibonacci, reintroduced the Indian system into Europe with his book *Liber Abaci*. However, usage of the system remained limited for many years, in part because the scheme was considered “diabolical,” due in part to the mistaken impression that it originated in the Arab world (in spite of Fibonacci’s clear descriptions of the “nine Indian figures” plus zero). Indeed, our modern English word “cipher” or “cypher,” which is derived from the Arabic *zephirum* for zero, and which alternately means “zero” or “secret code” in modern usage, is likely a linguistic memory of the time when using decimal arithmetic was deemed evidence of involvement in the occult [Ifrah2000, pg. 588-589].

Decimal arithmetic began to be widely used by scientists beginning in the 1400s, and was employed, for instance, by Copernicus, Galileo, Kepler and Newton, but it was not universally used in European commerce until 1800, at least 1300 years after its discovery. In limited defense of the Greco-Roman system, it is harder to alter Roman entries in an account book or the sum payable in a cheque, but this does not excuse the continuing practice of performing arithmetic using Roman numerals and counting tables.

The Arabic world, by comparison, was much more accepting of the Indian system — in fact, as mentioned briefly above, the West owes its knowledge of the scheme to Arab scholars. One of the first to popularize the method was al-Khuwarizmi, who in the ninth century wrote at length about the Indian place-value system and also described algebraic methods for the solution of quadratic equations. In 1424, Al-Kashi of Samarkand, “*who could calculate as eagles can fly*” computed 2 pi in *sexagecimal* (good to an equivalent of 16 decimal digits) using 3 x 2^{28}-gons and a base-60 variation of Indian positional arithmetic [Berggren2004, Appendix on Arab Mathematics]:

2 pi ~ 6 + 16/60^{1} + 59/60^{2} + 28/60^{3} + 1/60^{4} + 34/60^{5} + 51/60^{6} + 46/60^{7} + 14/60^{8} + 50/60^{9}

### Modern History

It is both disappointing and perplexing that this seminal development in the history of mathematics is given such little attention in contemporary published histories. For example, in one popular work on the history of mathematics, although the author describes Arab and Chinese mathematics in significant detail, he mentions the discovery of decimal arithmetic in India only in one two-sentence passage [Burton2003, pg. 253]. Another popular history of mathematics mentions the discovery of the “Hindu-Arabic Numeral System,” but says only that

*Positional value and a zero must have been introduced in India sometime before A.D. 800, because the Persian mathematician al-Khowarizmi describes such a completed Hindu system in a book of A.D. 825.* [Eves1990, pg. 23]

A third historical work briefly mentions this discovery, but cites a 662 CE Indian manuscript as the earliest known source [Katz1998, pg. 221]. A fourth reference states that the combination of decimal and positional arithmetic “*appears in China and then in India*” [Struik1987, pg. 67]. None of these authors devotes more than a few sentences to the subject, and, more importantly, none suggests that this discovery is regarded as particularly significant.

We entirely agree with Ifrah, Dantzig and othersthat this discovery is of the first magnitude. The mere fact that the system is now taught in grade schools worldwide, and is implemented (in binary) in every computer ever manufactured, should not detract from its historical significance — to the contrary, these same facts emphasize the enormous advance that this system represented over earlier systems, both in simplicity and efficiency, as well as the huge importance of this discovery in modern life.

Perhaps some day we will finally learn the identity of this mysterious Indian mathematician. If we do, we surely must accord him or her the same honors that we have granted to Archimedes, Newton, Gauss and Ramanujan.

For additional references and full details, see [Decimal].

### References

- [Berggren2004] L. Berggren, J. M. Borwein and P. B. Borwein,
*Pi: a Source Book*, Springer-Verlag, New York, third edition, 2004. - [Burton2003] David M. Burton,
*The History of Mathematics: An Introduction*, McGraw-Hill, New York, 2003. - [Dantzig2007] Tobias Dantzig and Joseph Mazur,
*Number: The Language of Science*, Plume, New York, 2007. - [Decimal] David H. Bailey and Jonathan M. Borwein, “The Greatest Mathematical Discovery,” available at Online article.
- [Durant1954] Will Durant,
*Our Oriental Heritage*, vol. 1 of*The Story of Civilization*, 11 vols., Simon and Schuster, New York, 1954 (date of vol. 1). - [Eves1990} Howard Eves,
*An Introduction to the History of Mathematics*, Holt, Rinehart and Winston, New York, 1990. - [Hoernle1887} Rudolf Hoernle,
*On the Bakhshali Manuscript*, Alfred Holder, Vienna, 1887. - [Ifrah2000] Georges Ifrah,
*The Universal History of Numbers: From Prehistory to the Invention of the Computer*, translated from French by David Vellos, E. F. Harding, Sophie Wood and Ian Monk, John Wiley and Sons, New York, 2000. - [Lokavibhaga] Balachandra Siddhanta-Shastri, trans.,
*Loka-vibhaga*, Gulabchand Hirachand Doshi, Sholapur, India, 1962. - [Katz1998] Victor J. Katz,
*A History of Mathematics: An Introduction*, Addison Wesley, New York, 1998. - [Netz2007] Reviel Netz and William Noel,
*The Archimedes Codex*, Da Capo Press, 2007. - [Marchant2008] Josephine Marchant,
*Decoding the Heavens: Solving the Mystery of the World’s First Computer*, Arrow Books, New York, 2008. - [Stillwell2002] John Stillwell,
*Mathematics and Its History*, Springer, New York, 2002. - [Struik1987] Dirk J. Struik,
*A Concise History of Mathematics*, Dover, New York, 1987.