Indian Mathematics

History, politics and transmission of knowledge Recorded civilization in the Indus valley datesback to at least 2500 BC. During the first millenium BC, the Hindu religion developed as an amalga-mation of previous cultural and religious practices and beliefs. Buddhism and Jainism also began tospread in the later part of this period, primarily in the Ganges valley.

The Greeks, under Alexander the Great, conquered as fareast as the Indus in 326 BC, only three years before Alexan-der’s death. India was largely unified under the MauryaEmpire for the next 150 years, after which came 1000 yearsof shifting and changing empires ruling all or part of the In-dian peninsula. Protected in the north by the Himalayas andin the east by dense jungle, the Indus valley has been theprimary borderland of Indian culture for 3000 years with thearea changing control many times: the Persians expandedto the Indus and were pushed back on several occasionsover a 1000 year period. Eventually the expanding Muslimcaliphate conquered the valley and various Islamic dynas-ties made inroads before the greater part of India becamethe Islamic Mughal Empire in the 1500s.

The modern political situation reflects the complicated history. After the Mughal Empire declined,the British conquered India. After World War II, India won its independence from the British Empireand was partitioned according to religion. The Indus valley is now the central corridor of Pakistan(an Islamic state), whereas the country of India is nominally secular but majority Hindu. The upperIndus valley (Kashmir) remains contested and has been the site of several military conflicts betweenIndia and Pakistan.

Ancient India is important (for our purposes) not just for its mathematics, but as a crossroads.While some trade and knowledge passed north of the Himalayas between China and the MiddleEast/Europe, India’s location made it perfectly suited to absorb and synthesise ideas and technolo-gies from both east and west. The to-and-fro across the Indus also meant that India was exposedto ancient Greek and Babylonian learning, and gave back in kind. Far from being a backwater, it isestimated that India accounted for 25–30% of the world’s economy during the 1st millenium AD.

Hindu–Arabic Numerals

The Indian subcontinent is responsible for arguably the most important mathematical developmentin history, and certainly the most ubiquitous: our modern system of enumeration.

Brahmi Numerals Around the 3rd century BC, one of the earliest antecedents of our modern nu-merals appeared: the Brahmi numerals. The example below dates from around 100 BC and was usedin Mumbai/Bombay.

1 2 3 4 5 6 7 8 9 10

Additional symbols were used for multiples of 10, then 100, 1000, 10000, etc. The system was posi-tional in a similar style to Chinese characters, so that 800 would be written by prefixing the symbolfor 100 by that for 8. This isn’t a truly positional; the ‘8’ required a modifier to mean 800, and therewas symbol/placeholder for zero.

Sanskrit names for numbers The modern way that large numbers are named can also be linked tothe same time. Below is a table of old Sanskrit names for numbers.

1 2 3 4 5 6 7 8 9eka dvi tri catur pancha sat sapta asta nava

10 20 30 40 50 60 70 80 90dasa vimsati trimsati catvarimsat panchasat sasti saptati asiti navati

100 1000 10000 100000 1000000 107 108 109 1010

sata sahasra ayuta niyuta prayuta arbuda nyarbuda samudra madhya

• You should recognize some similarity to the numbers in some European languages.

• Sanskrit had distinct words for powers of 10 up to (at least!) 1062.

• For example, 3659 is a literal translation: “tri sashra sat sata panchasat nava.”

• Also had a version of pre-subtraction: “ekanna-niyuta” means ‘one less than 100000,’ or 99999.

Gwalior numerals During the first few centuries AD, a fully positional decimal place system cameinto being. The earliest evidence for this is in a manuscript recovered from Bakhshali (Pakistan) in1881 and which has now been carbon dated to the 3rd or 4th century AD. The manuscript containsthe earliest known symbol for the number zero: a circular dot. It is conjectured that the decimal placesystem was inspired by the Chinese counting-board method. Regardless, Chinese mathematicianswere copying the method by the 8th century. The examples below are better understood than thoseon the Bakhshali manuscript and come from Gwalior (northern India) around 876 AD.

0 1 2 3 4 5 6 7 8 9 10

270 = , 30984 =

• 0,1,2,3,4,7,9,10 are almost identical to modern numbers.

• The symbols for 2 and 3 are conjectured to have developed in an attempt to write earlier ver-sions (like the Brahmi numerals) cursively.

• Fully-developed decimal place-value system incorporating zero: e.g. 27, 207, and 270 are allclearly distinguishable.

• Sanskrit is written left-to-right, hence so are our modern numbers, with the leftmost digitsrepresenting the largest powers of 10.

• Zero has evolved to be a hollow circle.

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Zero On the right is a table of modern Sanskrit names and numerals: the digits and names arecertainly similar to their Gwalior counterparts.

The name of zero, ‘shuunya’ is interesting in and of it-self. The word literally means void or emptiness. It is re-lated to the Sanskrit word svi for ‘hollow,’ which in turnderives from an ancient word meaning ‘to grow.’ Thisreflects a major idea within Buddishm: the void is thesource of all things, of creation (and creativity). Con-templation of the void (the doctrine of Shunyata) is rec-ommended before composing music, creating art, etc.This is in marked contrast to western religion, wherethe void is something to be feared (the eternal absenceof god is an early conception of hell).

As the numerals travelled westwards, so too did the word for zero: here is a short version of thejourney.

• Shunya transliterated to Sifr (Arabic).

• Meaning split in Arabic: al-sifr (number zero) and safira (literally it was empty).

• Brought to Europe (Latin) in the 12th century (Fibonacci/Nemorarius) as cifra. Became blendedwith the word zephyrum which meant ‘West Wind’ (zephyr) which provided an alternate spelling.

• Cifra ultimately became the words cipher, chiffre and ziffer in English, French and German. Thesewords now mean a figure, digit, or code.

• Zephyrum became zefiro in Italian and zero in Venetian.

Our modern understanding of zero is really a fusion of several concepts:

Numerical positioning E.g., to distinguish 101 from 11.

Absence of a quantity E.g., 101 contains no 10’s.

A Symbol Began as a dot (bindu), then a circle (chidra/randhra meaning hole). Relationship betweenshunya and a symbol well-established by 2-300 AD. Here is a quote from c.400 AD (Vasavadatta)

The stars shone forth, like zero dots [shunya-bindu] scattered as if on a blue rug. The Creatorreckoned the total with a bit of the moon for chalk.

Math operations By time of Brahmagupta (7th C), mathematical texts often contained a sectioncalled shunya-gania: computations involving zero, including addition, multiplication, subtrac-tion, effect on ± signs, division and the relationship with ∞ (ananta). By the 12th century,Bhaskaracharya stated: if you were to divide by zero you would get a number that was “asinfinte as the god Vishnu.”

Other cultures had one or more of these aspects of zero, but the Indians were the first to put them alltogether. For instance:

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• The Egyptian hieroglyph nfr (meaning beautiful/complete) was used to indicate zero remain-der in calculations as early as 1700 BC. It was also used as a reference point/level in buildings.

• Very late in Babylonian times, a placeholder symbol was used to separate powers of 60 apart.It was never used as a number.

• With the Chinese counting board, one could simply leave an empty space as a placeholder.

‘Real’ Indian Mathematics

Of course Indian mathematicians made great progress on several fronts, not merely the decimal placesystem. Much ancient work was influenced by religion: the sulbasutras were written during pre-Hindu times and contain instructions for laying out altars using ruler-and-compass constructions.These could be quite complex, as the construction of the base of the Mahavedi (great altar) shows:

The center line is divided left-to-right in the ratio

1 : 7 : 12 : 11 : 5

See if you can spot all the following Pythagoreantriples in the picture:

(5, 12, 13), (12, 16, 20), (12, 35, 37),(15, 20, 25), (15, 8, 17)

30 pada 24 pada

36 pada

While developments were made in many other areas, it is the Indian work on trigonometry that mostlinks with our studies. Here are some of the highlights:

• In the early 5th century the text Paitamahasiddhanta appeared. It is assumed to be an extensionof Hipparchus’ work since the table of chords therin is based on a circle of radius 57,18; ratherthan Ptolemy’s 60.

• Instituted the use of half-chords which is in line with our modern understanding of ‘sine.’ Theword ‘sine’ is the end result of a long line of (mis)translations via Arabic and Latin from theSanskrit jya-ardha meaning ‘chord-half’ (sinus means bay/gulf/bosom!). Distinguished ‘basesine’ and ‘perpendicular sine’ (cosine).

• Created tables of sines/half-chords from 0 to 90 deg in steps of 3 34 deg: used linear interpolation

to find values in between.

• By 650, Bhramagupta had much better approximations, using quadratic polynomials to inter-polate.

• By 1530, Indian mathematicians had discovered cubic and higher approximations (essentiallyTaylor Polynomials 130 years before Newton) for even greater accuracy of sine, cosine andarctangent.

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