Ancient Indian square roots

As we have argued in an earlier blog, our modern system of positional decimal notation with zero, together with efficient algorithms for computation, which were discovered in India some time prior to 500 CE, certainly must rank among the most significant achievements of all time. As Pierre-Simon Laplace explained:

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.


So who exactly discovered the Indian system? Sadly, there is no record of who first discovered positional decimal notation with zero, nor is there any clear record of the development of efficient algorithms for computation. One person who deserves at least some credit for the proliferation of decimal arithmetic calculation is the Indian mathematician Aryabhata, who in 500 CE presented algorithms not only for various arithmetic operations but also for square roots and cube roots. His ingenious digit-by-digit algorithms for computing square roots and cube roots, which were presented in his work Aryabhatiya, are illustrated in [BaileyBorwein2011].

However, it is abundantly clear that the underlying system of positional decimal arithmetic with zero, together with some reasonably efficient algorithms for various arithmetic operations, were known even earlier. One piece of evidence is the Indian astronomical work Lokavibhaga (“Parts of the Universe”). Here, for example, we find numerous large numbers and detailed calculations, such as (14230249 – 355684) / 212 = 65446 +13/212. Near the end of the document, the author provides detailed astronomical data that enable modern scholars to confirm that this text was written on 25 August 458 CE (Julian calendar).

The Bakhshali manuscript

Another ancient source that clearly exhibits considerable familiarity with decimal arithmetic in general and square roots in particular is the Bakhshali manuscript. This document, an ancient mathematical treatise, was found in 1881 in the village of Bakhshali, approximately 80 kilometers northeast of Peshawar (then in India, now in Pakistan). Among the topics covered in this document, at least in the fragments that have been recovered, are solutions of systems of linear equations, indeterminate (Diophantine) equations of the second degree, arithmetic progressions of various types, and rational approximations of square roots.

The manuscript appears to be a 10-11th century copy and commentary of an even earlier work. While one or two scholars disagree, the present consensus is that the original Bakhshali manuscript was probably written 200-400 CE, earlier than either the Aryabhatiya or the Lokavibhaga.

The Bakhshali square root

One particularly intriguing item in the Bakhshali manuscript is the following algorithm for computing square roots:

In the case of a number whose square root is to be found, divide it by the by the approximate root [the root of the nearest square number]; multiply the denominator of the resulting [ratio of the remainder to the divisor] by two; square it [the fraction just obtained]; halve it; divide it by composite fraction [the first approximation]; subtract [from the composite fraction]; [the result is] the refined root. [Translation due to M. N. Channabasappa]

In modern notation, this algorithm is as follows. To obtain the square root of a number q, start with an approximation x0 and then calculate, for n >= 0,
an = (q – xn2) / (2 xn)
xn+1 = xn + an – an2 / (2 (xn + an))

In the examples presented in the Bakhshali manuscript, this formula is used to obtain rational approximations to square roots only for integer arguments q, only for integer-valued starting values x0, and is only applied once in case (even though the result after one iteration is described as the “refined root,” possibly suggesting it could be repeated). But from a modern perspective, the scheme clearly can be repeated, and in fact converges very rapidly to sqrt(q), as we shall see in the next section.

Several explicit applications of this scheme are presented in the Bakshshali manuscript. One example is to find an accurate rational approximation to the solution of the quadratic equation 3 x2 / 4 + 3 x / 4 = 7000. The manuscript notes that x = (sqrt(336009) – 3) / 6, and then calculates an accurate value for sqrt(336009), starting with the approximation 579. The result obtained is

579 + 515225088 / 777307500 = 450576267588 / 777307500

This is 579.66283303325903841…, which agrees with sqrt(336009) = 579.66283303313487498… to 12-significant-digit accuracy. From a modern perspective, this happens because the Bakhshali square root algorithm is quartically convergent — each iteration approximately quadruples the number of correct digits in the result, provided that either exact rational arithmetic or sufficiently high precision floating-point arithmetic is used.

An even more ancient square root

There are instances of highly accurate square roots in Indian sources that are even more ancient than the Bakhshali manuscript. For example, Srinivasiengar noted that the ancient Jain work Jambudvipa Prajnapti (~400 BCE), after erroneously assuming that pi = sqrt(10), asserts that the “circumference” of a circle of diameter 100,000 yojana is 316227 yojana + 3 gavyuti + 128 dhanu + (13+1/2) angula, “and a little over.” Other scholars have noted other instances of this reckoning in manuscripts dated 0 CE or older.

According to one commonly used ancient convention these units are: 1 yojana = 14.6 kilometers (approximately); 4 gavyuti = 1 yojana; 2000 dhanu = 1 gavyuti; and 96 angula = 1 dhanu. Converting these units to yojana, we conclude that the “circumference” is 316227.766017578125… yojana. This agrees with 100000 sqrt(10) = 316227.766016837933… to 12-significant-digit accuracy!

As we note in our recent paper [BaileyBorwein2011], it is most likely that this value was obtained by applying Heron’s formula, namely xn+1 = (xn + q / xn)/2, with x0 = 316227. Heron’s formula is equivalent to one Newton-Raphson iteration to solve the equation f(x) = x2 – q = 0. Note that just to perform one Newton-Raphson-Heron iteration, with starting value 316227, one would need to perform at least the following rather demanding calculation:

(316227 + 100000000000 / 316227) / 2 = (3162272 + 100000000000 / 316227) / 2
= (99999515529 + 100000000000) / 632454 = 316227 + 484471 / 632454,

followed by several additional steps to convert the result to the given units. By any reasonable standard, this is a rather high-precision computation, which we were surprised to find evidence for in manuscripts of this ancient vintage (300-400 BCE). Numerous other examples of prodigious computations have been noted in the Jambudvipa Prajnapti and other ancient Indian sources.

In any event, this analysis leads to the inescapable conclusion that ancient Indian mathematicians, roughly contemporaneous with Greeks such as Euclid and Archimedes, had command of a rather powerful system of arithmetic, very likely some variation of positional decimal arithmetic. How exactly did they do these calculations? We can only hope that further study of ancient Indian mathematics will shed light on this intriguing question.


Aside from historical interest, does any of this matter? In his latest book, mathematical historian George G. Joseph explains as follows:

A Eurocentric approach to the history of mathematics is intimately connected with the dominant view of mathematics, both as a sociohistorical practice and as an intellectual 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 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.

So this is why it matters: The Greek heritage that underlies much of Western mathematics, as valuable as it is, may have unduly predisposed many of us against experimental approaches that are now facilitated by the availability of powerful computer technology. In addition, more and more documents are now accessible for careful study — from Chinese, Babylonian, Mayan and other sources as well. Thus a renewed exposure to non-Western traditions may lead to new insights and results, and may clarify the age-old issue of the relationship between mathematics as a language of science and technology, and mathematics as a supreme human intellectual discipline.


For brevity, we have not included references here, but these together will full details of the above may be found in our recent paper:

[BaileyBorwein2011] David H. Bailey and Jonathan M. Borwein, “Ancient Indian Square Roots: An Exercise in Forensic Paleo-Mathematics,” manuscript, 13 Jun 2011, available at Online article.

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