INDIAN DECIMAL SYSTEM - THE GREATEST MATHEMATICAL DISCOVERY? - by David Bailey n Jonathan Borweiny

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http://crd-legacy.lbl.gov/~dhbailey/dhbpapers/decimal.pdf

THE GREATEST MATHEMATICAL DISCOVERY?

David H. Bailey Jonathan M. Borweiny

May 8, 2011

http://crd-legacy.lbl.gov/~dhbailey/dhbpapers/decimal.pdf

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 eluded the greatest minds of
antiquity, even 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 notation with zero, to-
gether with the basic arithmetic computational schemes, which were
discovered in India prior to 500 CE.



2 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 sim-
plicity and the great ease which it has lent to all computations put our
arithmetic in the rst rank of useful inventions; and we shall appre-
ciate 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. [7, pg. 527]
French Historian Georges Ifrah describes the signicance in these terms:
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 re, the development of agriculture, or
the invention of writing, of the wheel, or of the steam engine. [11, pg.
346{347]
As Laplace noted, the scheme 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 and numerical analysis.
He was also very skilled in applying mathematical principles to engineering
and astronomy. Nonetheless he used a cumbersome Greek numeral system
for calculations. Archimedes' computation of , a tour de force of numer-
ical interval analysis, was performed without either positional notation or
trigonometry [2, 13].
Perhaps one reason this discovery gets so little attention today is that
it is very hard for us to appreciate the enormous di culty of using Greco-
Roman numerals, counting tables and abacuses. As Tobias Dantzig (father
of George Dantzig, the inventor of linear programming) wrote,
Computations which a child can now perform required then the services
of a specialist, and what is now only a matter of a few minutes [by
hand] meant in the twelfth century days of elaborate work. [6, pg. 27]
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
2
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 [11, pg. 577]. In a similar vein, at about the same time
a wealthy German merchant, consulting a scholar regarding which European
university o ered 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 su cient gifts|then you will have to send him to Italy. [11,
pg. 577]
We observe in passing that Claude Shannon (1916{2001) constructed a me-
chanical calculator wryly called Throback 1 at Bell Labs in 1953, which com-
puted in Roman, so as to demonstrate that it was possible, if very di cult,
to compute this way.
Indeed, the development of modern decimal arithmetic hinged on three
key abstract (and certainly non-intuitive) principles [11, pg. 346]:
(a) Graphical signs largely removed from intuitive associations;
(b) Positional notation; and
(c) A fully operational zero|lling 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 princi-
ples, but none of them, prior to early-rst-millennium India, found all
three
and then combined them with e ective algorithms for practical computing.
The Mayans came close|before 36 BCE they had devised a place-value sys-
tem that included a zero. However, in their system successive positions
represented the mixed sequence (1; 20; 360; 7200; 144000; ) rather
than the
purely base-20 sequence (1; 20; 400; 8000; 160000; ), which
precluded the
possibility that their numerals could be used as part of an e cient system
for computation.
3 History
So who exactly discovered the Indian system? Sadly, there is no record of
the individual who rst discovered the scheme, who, if known, would surely
3
rank among the greatest mathematicians of all time. As Dantzig notes, \the
achievement of the unknown Hindu who some time in the rst 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" [6, 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 astro-
nomical work Lokavibhaga (\Parts of the Universe") [14]. Here, for exam-
ple, we nd numerous large numbers, such as 14236713; 13107200000 and
70500000000000000, as well as detailed calculations such as (14230249 􀀀
355684)=212 = 65446 13
212 [14, 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
conrm, 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 [11, 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 [10]. One intriguing item in the Bakhshali manuscript is the
following approximation for the square root [5]:
p
a2 + x a +
x
2a
􀀀

x
2a
2
2

a + x
2a
: (1)
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 = 3:1416.
Aryab-
hata's \digital" algorithms for computing square roots and cube roots are
illustrated in Figures 2 and 3 (based on [1, pg. 24{26]). A statue of Aryab-
hata, on display at the Inter-University Centre for Astronomy and Astro-
physics (IUCAA) in Pune, India, is shown in Figure 1.
In the centuries that followed, the Indian system was slowly disseminated
to other countries. In China, there are records as early as the Sui Dynasty
(581{618 CE) of Chinese translations of the Brahman Arithmetical Classic,
although sadly none of these copies have survived [9].
The Indian system was introduced in Europe by Gerbert of Aurillac in
the tenth century. He traveled to Spain to learn about the system rst-hand
from Arab scholars, then was the rst Christian to teach mathematics using
decimal arithmetic, all prior to his brief reign as Pope Sylvester II
(999{1002
CE) [3, 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 Sylvester's reign was a turbulent time, and he died after a short
reign. It is worth speculating how history would have been di erent 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 gures" 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 [11, 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 brie
y above, the West owes its knowledge of
the scheme to Arab scholars. One of the rst to popularize the method was
al-Khowarizmi, who in the ninth century wrote at length about the Indian
system and also described algebraic methods for the solution of quadratic
equations. In 1424, Al-Kashi of Samarkand, \who could calculate as eagles
can
y" computed 2 in sexagecimal (good to an equivalent of 16 decimal
digits) using 3 228-gons and a base-60 variation of Indian positional
arithmetic
[2, Appendix on Arab Mathematics]:
This is a personal favorite of ours: re-entering it on a computer centuries
later and getting the predicted answer still produces goose-bumps.

4 Modern History
It is disappointing that this seminal development in the history of
mathemat-
ics is given such little attention in modern published histories. For
example,
in one popular work on the history of mathematics, although the author de-
scribes Arab and Chinese mathematics in signicant detail, he mentions the
discovery of positional decimal arithmetic in India only in one two-sentence
passage [4, 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 some-
time before A.D. 800, because the Persian mathematician al-Khowarizmi
describes such a completed Hindu system in a book of A.D. 825. [8,
pg. 23]
A third historical work brie
y mentions this discovery, but cites a 662
CE Indian manuscript as the earliest known source [12, pg. 221]. A fourth
reference states that the combination of decimal and positional arithmetic
\appears in China and then in India" [16, 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 signicant.
In partial defense of these histories, though, it must be acknowledged that
all historians work from other sources, and only within the past few years,
with the advent of the Internet, has it been possible to readily access
original
and translated original documents with the click of a mouse.
In any event, we entirely agree with Dantzig, Ifrah and others that the
discovery of positional decimal arithmetic, by an unknown scholar in early
rst millennium India, is a mathematical development of the rst magni-
tude. The fact that the system is now taught and mastered in grade schools
worldwide, and is implemented (in binary) in every computer chip ever manu-
factured, should not detract from its historical significance. To the
contrary,
these same facts emphasize the enormous advance that this system repre-
sents, both in simplicity and e fficiency, as well as the huge
importance of this
discovery in modern civilization.
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 accolades that we have granted to Archimedes, Newton, Gauss and
Ramanujan.

[RH156RH]

Sunday Telegraph

Translation: Europe is modern; the subcontinent still locked in the darkness of superstition, viz

Indian soldier arrested over murder of policeman sent to investigate killing of sacred cow
Save
India cowsA shrine to Inspector Subodh Kumar Singh, who was killed while investigating the killing of sacred cows

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Saptarshi Ray, kolkata
8 DECEMBER 2018 • 7:43PM
Follow
An Indian soldier was arrested for murdering a policeman during a riot over the slaughtering of sacred cows, in a case that has shone a light on mob rule and the increasingly zealous protection of cattle.
The police inspector was killed when a mob of Hindu extremists went on the rampage in the state of Uttar Pradesh, after they claimed to have discovered the carcasses of several cows in a village forest.
Muslims who had gathered nearby for a religious congregation were blamed for killing the cows, which are sacred to Hindus.
Police were sent to quell unrest by Hindu activists, but Inspector Subodh Kumar Singh was killed, allegedly by a soldier from Kashmir.
Jitendra Malik, alias Jeetu Fauji, was detained by his own unit, the 22 Rashtriya Rifles, in the town of Sopore in Kashmir, where he was serving. A police special investigation team of Uttar Pradesh Police then took him into custody.
Sacred cow protests
Cars lie vandalised near a police station after a mob attack in Chingarwathi, near Bulandshahr, in the northern Indian state of Uttar Pradesh
The case has become increasingly politicised, prompting bizarre conspiracy theories.
The ruling Hindu nationalist Bharatiya Janata Party claimed the episode was a “false flag” attack by opposition Congress Party acolytes to bring blame on the BJP.
Congress officials claimed the riot was a smokescreen to cover attacks against Muslims, and that Insp Singh had been assassinated as he had clashed with Hindu extremists in the past.
So far nine people have been arrested, including Mr Fauji, and two police officers suspended after it appeared the focus of the investigation was not on the murder of a police officer, but trying to catch the “cow killers”.
The BJP chief minister of UP met the family of the murdered police officer and told reporters “what happened in Bulandshahr is an accident”.
After the dead cattle were initially found, Hindu activists, many from political parties, formed a mob for retribution, claiming the cows were slaughtered for beef, eaten at the gathering of Muslims.
Insp Singh and his team had gone to the village of Bulandshahr to tackle the mob when they came under attack by more than 200 rioters. "

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Tranlation:

Robert Henderson is a LUNATIC WEIRDO just like Isaac Newton.

Western Christians have been BARBARIANS for a couple of millennia
stealing and looting everybody's land and wealth and KNOWLEDGE too.

Fibonacci STOLE Indian mathematician Virahanka's sequence and claimed it
as his own.

Robert Henderson is a barbarian by default.

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Translation: Europe is modern; the subcontinent still locked in the darkness of superstition and insanitation... RH

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Translation: Europeans are BARBARIANS who looted everybody's land and
wealth and got rich. They stole knowledge and claimed as their own too.


THE "GREAT THEFT" OF INDIA BY IMPERIAL BRITAIN - PART 1
https://groups.google.com/forum/?hl=en#!topic/rec.sport.cricket/ZRclN20SXk0

[RH156RH]

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Genocial Racist Thief Winston Churchill:
"The riches of the West are built on the graves of the East."

[RH156RH]

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HOW COLONIAL INDIA MADE "MODERN BRITAIN" - By Aditya Mukherjee
https://groups.google.com/d/msg/rec.sport.cricket/W9bOTgmsq5U/3SWdV6WnBAAJ


BRITISH WHITE CHRISTIAN THIEVES LOOTED MORE THAN 9 TRILLION POUNDS FROM
INDIA OVER 173 YEARS
https://groups.google.com/d/msg/rec.sport.cricket/crH56ZzevBg/_wY8kxvJAgAJ

[RH156RH]

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TRANSLATION:

Genocidal Racist THIEF Winston Churchill:
"the riches of the West are built on the graves of the East"

Pay the STOLEN MONEY which is $45 TRILLION dollars (with interest) back
to India and "YOU and ALL your fellow brits" would be BEGGING FOR FOOD
and SHITTING IN STREETS today.

THE "GREAT THEFT" OF INDIA BY IMPERIAL BRITAIN - PART 1

https://groups.google.com/d/msg/rec.sport.cricket/ZRclN20SXk0/iARZeWp5DAAJ

[RH156RH]

Note the lack of IQ in FBInCIAnNSATerroristSlayer's replies... RH

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How Britain stole $45 trillion from India - And LIED About it
https://www.aljazeera.com/indepth/opinion/britain-stole-45-trillion-india-181206124830851.html