Ancient Indian mathematics
David H. Bailey
Lawrence Berkeley National Laboratory (retired) andComputer Science Department, University of California, Davis

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Jon and Peter Borwein’s quartically convergent algorithm for πIn 1985, Jonathan and Peter Borwein published a “quartically convergent” algorithm
for π — each iteration approximately quadruples the number of correct digits(provided all iterations are performed with full precision).
√√
The algorithm is: Set a0 = 6 − 2, and y0 = 2 − 1. Iterate:yk+1 =

1 − (1 − yk4 )1/4
1 + (1 − yk4 )1/4

2ak+1 = ak (1 + yk+1 )4 − 22k+3 yk+1 (1 + yk+1 + yk+1)

Then ak converge quartically to 1/π.
Both this and the Salamin-Brent scheme are based on the arithmetic-geometric meanand some ideas due to Gauss, but evidently he (nor anyone else until 1976) ever saw
the connection to computation. Perhaps no one in the pre-computer age wasaccustomed to an “iterative” algorithm?
I J. M. Borwein and P. B. Borwein, Pi and the AGM: A Study in Analytic Number Theory andComputational Complexity, John Wiley, New York, 1987.

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A quartically convergent algorithm for square roots
To compute the square root of q, let x0 be the initial approximation. Then iterate:
an =q − xn2
2xnxn+1 = xn + an −Then xn converge quartically toIteration
012
3
4
567

√

an22 (xn + an )q. For example, for q = 336009, x0 = 579:Value
579.000000000000000000000000000000. . .
579.662833033259038411439488233421. . .
579.662833033134874975589542464552. . .
579.662833033134874975589542464552. . .
579.662833033134874975589542464552. . .
579.662833033134874975589542464552. . .
579.662833033134874975589542464552. . .
579.662833033134874975589542464552. . .

Relative error
1.143 × 10−3
2.142 × 10−13
2.631 × 10−52
5.993 × 10−208
1.612 × 10−830
8.449 × 10−3321
6.371 × 10−13282
2.060 × 10−53126

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Actual algorithm statement from the original manuscript“In the case of a number whose square root is to be found, divide [the difference fromthe square of the approximation] by the approximate root; multiply the denominator of
the resulting [fraction] by two; square [the result]; halve it; divide it by [the firstapproximation]; subtract [from the first approximation]; [the result is] the refinedroot.” [Translation].
Note that successive quantities are (in our notation):an =q − xn2,2xn

an2,
2 (xn + an )an22xn + an ,an2 ,xn + an −an22 (xn + an )4 / 26

One of numerous applications presented in the original manuscriptProblem: Find an accurate rational approximation to the solution of3/4x 2 + 3/4x = 7000 √(arising from analysis of progressions).
√
Solution: x = (−3 + 336009)/6. To calculate an accurate value for 336009, startwith the approximation 579. Note that q = 336009 = 5792 + 768. Then calculate:a0 = (qx02 )/(2x0 ) = 768/1158
x + a0 = 579 + 768/1158
2a0 /(2(x0 + a0 )) = 294912/777307500
Thus the refined root = 579 + 515225088 / 777307500 = 450576267588 /777307500. Note: This is 579.66283303325903841 . . . (correct to 13 digits).
The manuscript then performs a calculation to check that the original quadratic
equation is satisfied. It obtains, for the left-hand side of the quadratic equation:
50753383762746743271936 / 7250483394675000000
which, after subtracting the correction 21743271936 / [7250483394675000000], gives:
50753383762725000000000 / 7250483394675000000 = 7000 exactly.
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Proof that the scheme is quartically convergent
The scheme can seen to be equivalent to two consecutive iterations of theNewton-Raphson iteration for the solution of x 2 − q = 0:xn+1 = xn +q − xn22xn


2 2nq − xn + q−x2xn
xn+2 = xn+1 += xn ++2
q−x2xn+1
2xn
2 xn + 2xnn 
2q−xn2q−xn2
2 − 2xq−x−
2nn2xn2xn
q − xn= xn ++2
q−x2xn2 xn + 2xnn
2q−xn222xn
q − xn= xn +− 22xn
2 x + q−xn
2q − xn+1q − xn2n2xn

which is the same form as the manuscript iteration.
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Contents of the manuscript
I Solutions of systems of linear equations.
I Indeterminate (i.e., Diophantine) equations of the second degree.
I Arithmetic progressions of various types.
I Rational approximations of square roots.
I Additive series.
I Product series.
I Business applications.
I Geometry.

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Title? Author? Date?
Any guesses?

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The Bakhshali manuscript
The manuscript was found in 1881 in Bakhshali (near Peshawar, not far fromAfghanistan border), then in India but now in Pakistan. Here is the route to Bakhshali
from bin Laden’s home in Abbottabad:

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When was the Bakhshali manuscript written?
Scholarly opinion:
I Takao Hayashi: “Seventh century.”
I G. R. Kaye: “12th century.”
I B. Datta: “Towards the beginning of the Christian era.”
I M. N. Channabasappa: “200-400 CE.”I L. V. Gurjar: “Between 2nd century BCE and 2nd century CE.”I Rudolf Hoernle: “3rd or 4th century CE.”
Hoernle notes that the document is written in the “Shloka” style, which was replacedby 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, there are notational features (e.g., + is used for negative) that
do not appear in later documents.
Until very recently Hayashi’s “7th century” or Kaye’s “12th century” prevailed. A
reviewer of the Monthly paper by Jon Borwein and myself insisted that we use
Hayashi’s “7th century.”
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New radiocarbon dating study
A radiocarbon dating study commissioned by London’s Bodelian library, which has held
the manuscript for decades, has revealed that the oldest pages are from somewhere
between 224 CE and 383 CE.
The Bakhshali manuscript is thus the oldest artifact with a zero.

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Image of original Bakshali page with square root calculation above

The fraction
50753383762746743271936/7250483394675000000
is in right center box.
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Aryabhata’s mathematical treatise
In 510 CE, the Indian mathematician
Aryabhata presented schemes not only forvarious arithmetic operations, but also forsquare roots and cube roots. Additionally,Aryabhata gave a decimal value ofπ ≈ 3.1416.
A statue of Aryabhata, on display at theInter-University Centre for Astronomy andAstrophysics (IUCAA) in Pune, India, isshown here (no one really knows what helooked like):

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Aryabhata’s “digital” square root algorithm
4 53 69814 64404558 0496697344486
2
10
0

60
64
4

4
406
990

√b 45c = 662 = 36b94/(2 · 6)c = 77 · (2 · 6) = 8467774

72 = 49b578/(2 · 67)c = 44 · (2 · 67) = 536

442 = 16b4044/(2 · 674)c = 33 · (2 · 674) = 4044

332 = 9Finished; result = 6743
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Aryabhata’s “digital” cube root algorithm
4 52 71 81 3422224 9 9459271652 9 3395422755√
b 3 45c = 333 = 27
2b184/(3 · 3 )c = 6 (too high, so take 5)5 · (3 · 32 ) = 135

353 · 52 · 3 = 225954211257433395445753 = 125b26242/(3 · 352 )c = 7
7 · (3 · 352 ) = 25725
3 · 72 · 35 = 5145

33073 = 343
Finished; result = 357
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The Lokavibhaga manuscript
The earliest document that exhibits familiarity with positional decimal arithmetic, andwhich at the same time can be accurately and unambiguously dated, is the Indiancosmological work Lokavibhaga (“Parts of the Universe”).
Here, for example, we find numerous huge numbers, such as 14236713, 13107200000
and 70500000000000000, as well as many detailed calculations, e.g., (14230249 355684) / 212 = 65446 13/212.
Near the end of the Lokavibhaga, the author provides detailed astronomical data thatenable 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 French historian Georges Ifrah points out, this information not only allows us todate the document with precision, but also proves its authenticity.
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Even earlier square roots? Even earlier positional decimal arithmetic?
Datta notes√ that as early as 300–400 BCE, a scholar wrote, after erroneously assumingthat π = 10, that the “circumference” of a circle of diameter 100,000 yojana (a unitof distance, approx. 11.5 km) is (approximately):316227 yojana + 3 gavyuti + 128 dhanu + 13 1/2 angula, “and a little over.”According to several historical sources, these distance units are:4 gavyuti = 1 yojana; 2000 dhanu = 1 gavyuti; 96 angula = 1 dhanu.
Thus the “circumference” = 316227.766017578125 yojana.
√100000 · 10 = 316227.7660168379331 . . .
This is 12-digit agreement!
Questions:
I Did Indian scholars know numerical square root algorithms back then?
I Did Indian scholars know positional decimal arithmetic back then?
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Some forensic paleo-mathematics
What algorithm was used to compute√

10 to this accuracy?

Note that 13 1/2 angula, “and a little over” suggests 13 5/8. We can then compare as
follows:
Manuscript value, with 13 1/2
Manuscript value, except with 13 5/8
One Heron iteration (316227)
One Bakhshali iteration (316000)
One Bakhshali iteration √(316227)
Exact value of 100000 · 10316227.76601757812500 . . .
316227.76601774088541 . . .
316227.76601776571892 . . .
316227.76601684858667 . . .
316227.76601683793319 . . .
316227.76601683793319 . . .

From these results, it appears that one Newton-Raphson-Heron iteration wasperformed, with 316227 as the starting value.
Even this calculation would require substantial numerical computation.
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The significance of positional decimal arithmetic with zero
Pierre-Simon Laplace explained:
It is India that gave us the ingenious method of expressing all numbers bymeans 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 easewhich it has lent to all computations put our arithmetic in the first rank ofuseful inventions; and we shall appreciate the grandeur of this achievement
the more when we remember that it escaped the genius of Archimedes andApollonius, two of the greatest men produced by antiquity.
French historian Georges Ifrah describes the significance 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 fire, the development of agriculture, or theinvention of writing, of the wheel, or of the steam engine.
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The difficulty of pre-decimal arithmetic
Tobias Dantzig (father of George Dantzig, the inventor of linear programming):
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] meantin the twelfth century days of elaborate work.
In the 1500s (before widespread adoption of decimal arithmetic in commerce), 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, thenany 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.

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The scientist-Pope Sylvester II (Gerbert of Aurillac )Gerbert of Aurillac traveled to Spain to learn about theHindu-Arabic arithmetic system first-hand from Arabscholars, then was the first Christian to teach mathematics
using decimal arithmetic, all prior to his brief reign as PopeSylvester II (999-1002 CE).
Little progress was made at the time, in part because ofclerics who rumored that Sylvester II had sold his soul toLucifer during his travels to Islamic Spain.
These accusations persisted until 1648, when papal authorities
who reopened his tomb reported that Sylvesters body had notbeen dismembered in penance for Satanic practices.
Sylvester’s reign was a turbulent time, and he died after ashort reign. It is worth speculating how history would havebeen different had this remarkable scientist-Pope lived longer.
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Leonardo of Pisa
In 1202 CE, Leonardo of Pisa, also known as
Fibonacci, reintroduced the Hindu-Arabic system
into Europe with his book Liber Abaci, and asubsequent, simplified volume (now lost) that waswritten in vernacular Italian.
Keith Devlin, who has written extensively about
Leonardo, argues that Leonardo’s work launchedmuch of the subsequent scientific revolution inEurope. He notes that tutorials on Hindu-Arabicarithmetic exploded in number beginning in the
13th century, and were adopted by many tradersand scholars.
Read Devlin’s two books for further details.
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Kicking and screaming
Even though Leonardo published his Liber Abaci in 1202, usage of the system still
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 Arabworld (in spite of Fibonacci’s clear descriptions of the “nine Indian figures” plus zero).
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 modernusage, is likely a linguistic memory of the time when using decimal arithmetic wasdeemed evidence of involvement in the occult.
After much kicking and screaming, Europe finally adopted decimal arithmetic formathematical and scientific work (1300s and 1400s), but not universally for commerce
until nearly 1800.
In other words, Hindu-Arabic positional decimal arithmetic was not fully adopted inEurope until at least 1600 years (and probably more) after its discovery in India.
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Indian discovery of decimal arithmetic in present-day math history books
In one popular work on the history of mathematics, although the author describes Araband Chinese mathematics in significant detail, he mentions the discovery of positional
decimal arithmetic in India only in one two-sentence passage.
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.
A third historical work briefly mentions this discovery, but cites a 662 CE Indian
manuscript as the earliest known source.
A fourth reference states that the combination of decimal and positional arithmetic“appears in China and then in India.”
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.

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Eurocentric math versus experimental math
Math historian G. G. Joseph writes [The Crest of the Peakcock: Non-European Rootsof Mathematics, Princeton University Press, 2011, pg. xiii]:
A Eurocentric approach to the history of mathematics is intimately connected withthe dominant view of mathematics, both as a sociohistorical practice and as anintellectual activity. Despite evidence to the contrary, a number of earlier histories
viewed mathematics as a deductive system, ideally proceeding from axiomatic
foundations and revealing, by the “necessary” unfolding of its pure abstract forms,
the eternal/universal laws of the “mind.”
The concept of mathematics found outside the Graeco-European praxis was verydifferent. 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 workin Indian and Chinese mathematics ..., such as the summations of mathematicalseries, 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 withreference to any formal deductive system.
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For further reading
I D. H. Bailey and J. M. Borwein, “The greatest mathematical discovery?,”

available at http://crd.lbl.gov/~dhbailey/dhbpapers/decimal.pdf.
I D. H. Bailey and J. M. Borwein, “Ancient Indian square roots: An exercise inforensic paleo-mathematics,” American Mathematical Monthly, available at
http://crd.lbl.gov/~dhbailey/dhbpapers/india-sqrt.pdf.
I also have digital scans of numerous source documents relevant to this topic. Virtually
all of this material is out-of-print or otherwise very hard to obtain. One book isavailable only from five libraries worldwide. Some articles are from obscure and/ordefunct Indian math journals.
Thanks!
This talk is available here: http://www.davidhbailey.com/dhbtalks/dhb-india-2017.pdf
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