Wednesday, October 18, 2017

What if base-10 arithmetic had been discovered earlier?

 We take our arithmetic for granted, but it was never trivial.  It obviously circulated among scholars only from India through to Europe when i presume it got picked up by Italian bankers in the Renaissance in the fourteenth century.

Without the arithmetic, the zero is not too obvious an improvement and was easily dispensed with.  If not easily, it was still possible to get by.  Worse, calculation is something that once learned becomes conservative.  For example, no one studies Mayan calculation and it is much easier.

The usefulness of our arithmetic in preparing the mathematical mind itself is also not easily understood. It is just there allowing easy leaps forward.

In practical terms, arithmetic using the zero was a European cultural discovery and made the nascent art of science work to say nothing of double entry book keeping which delights in the application of zero..

What if base-10 arithmetic had been discovered earlier?
[Note: A condensed and revised version of this article was published here in The Conversation, an online forum of academic research headquartered in Melbourne, Australia.]


Monumental inventions of history can be grouped into three categories: (a) those whose origin is well known and well appreciated; (b) those whose origin is completely lost to history; and (c) those who origin may be known, at least in general terms, but which are not very well appreciated in modern society. Among those in the first category are efficient steam engines (by James Watt in 1765), movable-type printing (by the Chinese inventor Bi Sheng in 1040 CE, although often credited to Gutenberg), and of course Newton’s apple — even if it is in part apocryphal. Among those in the second category are the invention of writing and the wheel, both of which predate recorded history. Among those in the third category is the invention of paper by the Chinese court official Cai Lun in 105 CE. While this invention has had enormous impact through history, up to and certainly including our information age, it has been granted only scant attention in western histories. One exception is [Hart1978], where Cai Lun is ranked as the seventh most influential person in history.

Here we wish to consider another item in the under-appreciated third category: the discovery of our modern system of positional decimal arithmetic with zero (including calculation schemes similar to those that we learned in grade school), which was developed in India at least by 500 CE and probably earlier.

Let us recall exactly what we mean. The counting numbers 1,2,3, … were found by all civilizations, but zero (nothing) seems to have originated independently in India and central America (by the Mayans). Positional arithmetic can be in decimal (for us) or binary (for computers). The main idea is that you do not need new symbols for tens, hundreds, thousands etc such as X, C, M in Roman notation. The Mayans got close but not close enough to allow for modern arithmetic.

Perhaps because we all learn decimal arithmetic at an early age, and thus presume it to be “trivial,” this discovery is given disappointingly brief mention in most western histories of mathematics. Yet is indisputably one of the most important discoveries of all time. As 19th century mathematician Pierre-Simon Laplace explained it all started in India:
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]
French Historian Georges Ifrah  compares its significance to writing or the wheel:
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]
As Laplace noted, decimal arithmetic is anything but “trivial,” since it eluded the best minds of the ancient world, even superhuman geniuses such as Archimedes. Archimedes saw far beyond the mathematics of his time, even anticipating numerous key ideas of modern calculus. Nonetheless he used a cumbersome Greek numeral system for calculations. Archimedes’ computation of pi to two decimal place accuracy, a tour de force of ancient mathematics, was performed without either positional notation or trigonometry [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 Greco-Roman numerals, counting tables and abacuses. Along this line, in the 16th century, 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]
Discovery and proliferation of decimal arithmetic

So who exactly discovered decimal arithmetic? One person who deserves at least some credit is the Indian mathematician Aryabhata, who in 499 CE presented schemes not only for various arithmetic operations, but also for square roots and cube roots. Additionally, he gave a decimal value of pi = 3.1416, correct to four digits. A statue of Aryabhata, on display at the Inter-University Centre for Astronomy and Astrophysics (IUCAA) in Pune, India, is shown in Figure 1.

An earlier document that exhibits familiarity with full decimal arithmetic, including zero and positional notation, is the Lokavibhaga (“Parts of the Universe”), which provides detailed astronomical data that enable modern scholars to confirm that it was written on 25 August 458 CE (Julian calendar). Another ancient source is the Bakhshali manuscript. Although there is some disagreement among scholars, it appears to be a 7-8th century copy and commentary of a slightly older mathematical treatise.

Statue of the Indian mathematician Aryabhata
Decimal arithmetic likely originated even earlier in ancient India. Even by 300 BCE, Indian scholars exhibited remarkable facility with computation, based on extant writings. For instance, a documented dated roughly 300 BCE, after erroneously assuming that pi is the square root of ten, gives a value of the square root of ten that is accurate to 12 digit precision [Bailey2011]. In any event, it is clear that India was the home to this pivotal event of mathematical history (although it may have based on some earlier developments in China).

The Indian system of decimal arithmetic was first 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 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, later rumored that Sylvester II had been a sorcerer and 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].

In 1202 CE, Leonardo of Pisa, also known as Fibonacci, reintroduced the Indian system into Europe with his book Liber Abaci [Devlin2011]. 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. 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.

What if…?

Many readers may be familiar with the growing genre of “alternate history.” These semi-historical, semi-fictional works explore various possibilities that events took different branches at certain critical junctures. What would have happened if Islamic forces, instead of the Franks, had prevailed in the pivotal Battle of Tours in 732 CE? Or if Harold II, instead of William of Normandy, had prevailed in the Battle of Hastings in 1066 CE? Or if Adolf Hitler had not diverted his forces to Russia in 1940, but had instead pursued the Battle of Britain until he prevailed? Much of this literature is highly speculative, but there is a valid premise here: human affairs and human history are highly contingent, and a slight turn of events at one epoch can have enormous impact in the years and centuries to come.

So it is with the history of science and technology. Historians now widely recognize that the emergence of European nations as leaders of science and technology during the past 400 years was hardly inevitable. Indeed, the underlying factors remain matters of controversy. After all, China was the most advanced nation, politically and technologically, from about 500 CE until 1400 CE, when they (for reasons still not fully understood) turned inwards and began to discourage exploration and scientific advancement. Similarly, the Islamic nations of the Middle East were the principal centers of scientific and mathematical progress during the period 900-1200 CE. They deserve special credit for preserving and transmitting the Greek classics to the West, particularly since we now know that the Graeco-Roman world was capable of great technological sophistication.

So let us indulge in some alternate history with regards to the history of mathematics in general and decimal arithmetic in particular:

1. What if a cultural connection had been made between pre-Christian-era Indian mathematicians and, say, Archimedes and his colleagues? Very likely Archimedes would have adopted with glee the slick computational schemes then being developed by the Indian scholars, and very likely the Indian scholars would have studied with great interest the impressive logical edifice of Greek geometry. Such an exchange would doubtless have enhanced both worlds, resulting in a synthesis that combined the deductive foundation of Greek mathematics with the unlimited exploratory power of Indian computation. The contemporary scholar George Joseph makes this very clear.
The concept of [ancient] mathematics found outside the Graeco-European praxis was very different. The aim was not to build an imposing edifice on a few self-evident axioms but to validate a result by any suitable method. Some of the most impressive work in Indian and Chinese mathematics, … such as the summations of mathematical series, or the use of Pascal’s triangle in solving higher-order numerical equations or the derivations of infinite series, or “proofs” of the so-called Pythagorean theorem, involve computations and visual demonstrations that were not formulated with reference to any formal deductive system. [Joseph2010, pg. xiii]
2. Failing contingency #1, what if the writings of Indian mathematicians Aryabhata and/or his brilliant successor Bhaskara I had reached Europe in 500 CE or 600 CE, instead of much later? If this had happened, and if scholars had used this as a springboard not only for developing science but also further cultural exchanges between East and West, it is likely that much of the turmoil of the “dark ages” in Europe would have been avoided. Indeed, potentially the development of the modern era could have been moved forward by nearly a full millennium.

3. Failing #1 and #2, what if Pope Sylvester II had lived another 10 or 20 years, instead of dying in 1002 after a brief and turbulent three-year reign (some say he was poisoned [Durant1954, pg. 538-541])? As contemporary historian Nancy Brown has noted, Sylvester II was, quite literally, the Pope who brought the light of science to the dark ages [Brown2010]. After all, for at least 200 years prior to his reign, and for at least 200 years afterwards, European history in general and the Papacy in particular were marked by continual warfare and intrigue, with little opportunity for scholarship and science to flourish.

As we mentioned above, Sylvester II studied decimal arithmetic first-hand from Islamic scholars in Spain, and was the first scholar to teach decimal arithmetic in Europe. If this remarkable mathematician-Pope had lived longer, he may well have been an agent of major historical change, incorporating not only Indian arithmetic but also many aspects of Greek and Islamic culture into European society, thus accelerating the Renaissance and the development of modern science and technology by several centuries. At the least, such a cultural exchange may well have forestalled the crusades and the resulting animosity between Christian and Islamic peoples that remains to the present day.

One conclusion that is clear from history is that mathematics matters. Without advanced mathematics (and arithmetic), progress in science and technology is not possible. Yet it is also clear that mathematics is not sufficient. The prodigious computational abilities of ancient Indian mathematicians were not translated into technology. Why? No one knows for sure. Similarly, while nearly as advanced mathematics was developed in China at the time, and China certainly did advance during medieval times, nonetheless at some point the Chinese nation decided not to further pursue science or technology, and as a result fell into a doldrums of sorts from which it did not fully extricate itself until the 20th century.

The lesson for our time is clear — we ignore or downplay mathematics and science at our peril. Let us hope that in the current worldwide economic downturn, that governmental bodies will not forsake research funding for mathematics and science, nor turn their backs on technology, but instead will recognize that mathematics, science and technology are the pathways to the future.


[Bailey2011] David H. Bailey and Jonathan M. Borwein, “Ancient Indian square roots: An exercise in forensic paleo-mathematics,” July 2011, available at Online article.
[Brown2010] Nancy M. Brown, The Abacus and the Cross: The Story of the Pope Who Brought the Light of Science to the Dark Ages, Basic Books, New York, 2010.
[Dantzig2007] Tobias Dantzig and Joseph Mazur, Number: The Language of Science, Plume, New York, 2007. This is a reprint, with Preface by Mazur, of Dantzig’s book as originally published by MacMillan in 1930.
[Devlin2011] Keith Devlin, The Man of Numbers: Fibonacci’s Arithmetic Revolution, Walker and Company, New York, 2011.
[Durant1935] Will Durant, Our Oriental Heritage, vol. 1 of The Story of Civilization, 11 vols., Simon and Schuster, New York, 1935.
[Durant1950] Will Durant, The Age of Faith, vol. 4 of The Story of Civilization, Simon and Schuster, New York, 1950.
[Hart1978] Michael H. Hart, The 100: A Ranking of the Most Influential Persons in History, Hart Publishing Company, 1978.
[Ifrah2000] Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer, translated by David Vellos, E. F. Harding, Sophie Wood and Ian Monk, John Wiley and Sons, New York, 2000.
[Joseph2010] George G. Joseph, The Crest of the Peacock: Non-European Roots
of Mathematics, Princeton University Press, Princeton, NJ, 2010.
[Netz2007] Reviel Netz and William Noel, The Archimedes Codex, Da Capo Press, 2007.

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