The greatest mathematical discovery?

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 recorded by George Dantzig’s father Tobias Dantzig, 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. [Dantzig2007, pg. 19]

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 [Netz2007,Marchant2008]. It is worth noting that 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]

We observe in passing that Claude Shannon (1916–2001) constructed a mechanical calculator wryly called
Throback 1 at Bell Labs in 1953, which computed in Roman, so as to demonstrate that it was possible if very difficult to compute this way.

To our knowledge, the best source currently available on the history of our modern number system is by French scholar Georges Ifrah [Ifrah2000]. He chronicles in encyclopedic detail the rise of modern numeration from its roots in primitive hand counting and tally schemes, to the Babylonian, Egyptian, Greek, Roman, Mayan, Indian and Chinese systems, and finally to the eventual discovery of full positional decimal arithmetic with zero in India, and its belated, kicking-and-screaming adoption in the West. Ifrah emphasizes more than once that the discovery of this system was by no means obvious or inevitable:

The measure of the genius of Indian civilization, to which we owe our modern system, is all the greater in that it was the only one in all history to have achieved this triumph. … Some cultures succeeded, earlier than the Indian, in discovering one or at best two of the characteristics of this intellectual feat. But none of them managed to bring together into a complete and coherent system the necessary and sufficient conditions for a number-system with the same potential as our own. [Ifrah2000, pg. 346]

Indeed, the development of this system hinged on three key abstract (and certainly non-intuitive) principles [Ifrah2000, pg. 346]: (a) the idea of attaching to each basic figure graphical signs which were removed from all intuitive associations, and did not visually evoke the units they represented; (b) the idea of adopting the principle according to which the basic figures have a value which depends on the position they occupy in the representation of a number; and (c) the idea of a fully operational zero, filling the empty spaces of missing units and at the same time having the meaning of a null number. Ifrah describes the significance of this discovery in these terms:

This fundamental realization therefore profoundly changed human existence, by bringing a simple and perfectly coherent notation for all numbers and allowing anyone, even those most resistant to elementary arithmetic, the means to easily perform all sorts of calculations; also by henceforth making it possible to carry out operations which previously, since the dawn of time, had been inconceivable; and opening up thereby the path which led to the development of mathematics, science and technology. … Now that we can stand back from the story, the birth of our modern number-system seems a colossal event in the history of humanity, as momentous as the mastery of fire, the development of agriculture, or the invention of writing, of the wheel, or of the steam engine. [Ifrah2000, pg. 346-347]

It is astonishing how many years passed before this system finally gained full acceptance in the rest of the world. There are indications that Indian numerals reached southern Europe perhaps as early as 500 CE, but with Europe mired in the Dark Ages, few paid any attention. Similarly, there is mention in Sui Dynasty (581-618 CE) records of Chinese translations of the “Brahman Arithmetical Classic,” although sadly none of these copies survived [Gupta1983].

The Indian system (also known as the Indo-Arabic system) was introduced to Europeans by Gerbert of Aurillac in the tenth century. He traveled to Spain to learn about the system first-hand from Arab scholars, prior to being named Pope Sylvester II in 999 CE. However, his advocacy of the system encountered stiff resistance, in part from accountants who did not want their craft rendered obsolete, to clerics who were aghast to hear that he had traveled to Islamic lands to study the method. As a result, it was widely rumored that he was a sorcerer and that he must have sold his soul to Lucifer during his travels, an accusation that persisted until 1648, when papal authorities reopened his tomb to make sure that his body had not been infested by Satan! [Ifrah2000,pg. 579].

The system was later reintroduced to Europe by Leonardo of Pisa, also known as Fibonacci, in his 1202 CE book Liber Abaci. However, usage of the system still remained limited for centuries, in part because the scheme continued to be considered “diabolical,” due 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) [Ifrah2000, pg. 361-362]. 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 very likely a linguistic memory of the time when using decimal arithmetic was deemed evidence of dabbling in the occult, which was potentially punishable by death at the hands of the Inquisition [Ifrah2000, pg. 588-589]. Decimal arithmetic began to be widely used by scientists beginning in the 1500s, and was employed, for instance, by Copernicus, Galileo, Kepler and Newton, but it was not universally used in European commerce until after the French Revolution in 1793 [Ifrah2000, pg. 590]. In limited defense of the 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 2²⁸-gons and a base-60 variation of Indian positional arithmetic [Berggren2004, Appendix on Arab Mathematics]:

2 pi ~ 6 + 16/60¹ + 59/60² + 28/60³ + 1/60⁴ + 34/60⁵ + 51/60⁶ + 46/60⁷ + 14/60⁸ + 50/60⁹

This is a personal favourite of ours: re-entering it on a computer centuries later and getting the predicted answer still produces goose-bumps.

So who exactly discovered the Indian system? Sadly, there is no record of the individual that first discovered the scheme, who, if known, would surely rank among the greatest mathematicians of all time. The very earliest document clearly describing positional decimal arithmetic with zero is an Indian astronomical work entitled Lokavibhaga (“Parts of the Universe”). Here, for example, the number 13,107,200,000 is written as

panchabhyah khalu shunyebhyah param dve sapta chambaram ekam trini cha rupam cha

(“five voids, then two and seven, the sky, one and three and the form”), i.e., 00000 2 7 0 1 3 1, which, when written in reverse order, is 13,107,200,000. One section of this same work gives detailed astronomical observations that confirm to modern scholars that this was written on the date it claimed to be written: 25 August 458 CE (Julian calendar). As Ifrah points out, this information not only allows us to date the document with precision, but it also proves its authenticity.

Fifty-two years later, in 510 CE, Indian mathematician Aryabhata described schemes for various arithmetic operations, even including square roots and cube roots, which most likely were known even earlier than this date. Aryabhata’s actual scheme for computing square roots, as described in greater detail in a 628 CE manuscript by a faithful disciple named Bhaskara I, is presented in full in Ifrah’s book [Ifrah2000, pg. 361-362]. Aryabhata even gave a decimal value of pi = 3.1416 [Ifrah2000, pg. 361-362]. From these and other sources there can be no doubt that our modern system of arithmetic — differing only in the symbols used for the digits — originated in India at least by the year 458 CE and probably several decades earlier.

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 that 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.

References

1. [Berggren2004] L. Berggren, J. M. Borwein and P. B. Borwein, Pi: a Source Book, Springer-Verlag, New York, third edition, 2004.
2. [Burton2003] David M. Burton, The History of Mathematics: An Introduction, McGraw-Hill, New York, 2003.
3. [Dantzig2007] Tobias Dantzig and Joseph Mazur, Number: The Language of Science, Plume, New York, 2007.
4. [Eves1990} Howard Eves, An Introduction to the History of Mathematics, Holt, Rinehart and Winston, New York, 1990.
5. [Gupta1983] R. C. Gupta, “Spread and triumph of Indian numerals,” Indian Journal of Historical Science, vol. 18 (1983), pg. 23-38, available at Online article.
6. [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.
7. [Katz1998] Victor J. Katz, A History of Mathematics: An Introduction, Addison Wesley, New York, 1998.
8. [Netz2007] Reviel Netz and William Noel, The Archimedes Codex, Da Capo Press, 2007.
9. [Marchant2008] Josephine Marchant, Decoding the Heavens: Solving the Mystery of the World’s First Computer, Arrow Books, New York, 2008.
10. [Stillwell2002] John Stillwell, Mathematics and Its History, Springer, New York, 2002.
11. [Struik1987] Dirk J. Struik, A Concise History of Mathematics, Dover, New York, 1987.