Hail to a hundred,

hail to a thousand,

hail to ayuta [ten thousand],

hail to niyuta [hundred thousand],

hail to prayuta [million],

hail to arbuda [ten million],

hail to nyarbuda [hundred million],

hail to samudra [billion],

hail to madhya [ten billion],

hail to anta [hundred billion],

hail to parardha [trillion],

hail to the dawn,

hail to the daybreak,

hail to the world,

hail to all.

Although Sanskrit was primarily prized as a language of oral poetry, it gained a written form during the first millennium BCE. The climate of the region is not kind to perishable writing surfaces, but vernacular dialects were being inscribed on stone in the third century and there are inscriptions in Sanskrit itself by the first century BCE, with manuscripts surviving from around 150 CE onwards.

From the first half of the first millennium CE, there is also increasing evidence for the use of number and calculation in Sanskrit culture, in such applications as finding the number of possible verse metres, laying out the brick altars for fire sacrifices, or keeping track of the calendar for ritual purposes. It was astronomy (and astrology, and the study of calendars) that ultimately prompted the most mathematical work in Sanskrit. The discipline – one of the traditional ‘limbs’ of Vedic learning – flourished throughout the first millennium, producing a wealth of texts of increasing complexity and sophistication. The core of each text was in verse, but lengthy prose commentaries explained and amplified the rules of calculation and procedure.

One of the most striking features of Sanskrit astronomy is the model of space and time it used, which involved immense spans of space and cycles of time. Historian Kim Plofker explains:

The universe is created and destroyed during one kalpa or day-and-night period of Brahman, which lasts for 4,320,000,000 years. There is a shorter period called a mahayuga, or ‘great age’, of 4,320,000 years: it is divided into four smaller intervals in a 4 : 3 : 2 : 1 ratio, during the course of which the world decays from good to bad, as in the Golden, Silver, Bronze and Iron Ages of Greek legend. The last and worst of these sub-periods is the Kaliyuga, which is one-tenth of a mahayuga, or 432,000 years long.

These numbers of years may have been large, but the numbers denoting cosmic distances were still larger; one measure of the size of the heavens ran to 18,712,069,200,000,000 yojanas (a yojana being a unit of distance in the region of twelve kilometres). To insert such vast counts into verse in their verbal forms could hardly be satisfactory, creating a need for a special notation for numbers. Several different systems were tried, testimony to the difficulty of the problem and the urgency with which a solution was needed. Most proved to be brief experiments that hardly outlived their inventors, but one would become the world’s most widespread number notation.

One system, for instance, used by the author Aryabhata in the sixth century, assigned consonants to small numbers and vowels to powers of 10 in order to represent numbers as syllables: nonsense syllables, admittedly, but capable of being read aloud. The idea did not catch on, but another scholar – Haridatta – devised a variant in the following century, in which the thirty-three Sanskrit consonants were each assigned to one of the digits from 1 to 9. That meant there were several possible choices for each number; the four choices for ‘1’ were k, t, p and y, and the system is therefore known as katapayadi. This was a positional system: the first letter written denoted the units, the next the tens, the next the hundreds, and so on; and the position was the only indication of whether a given symbol represented units, tens or hundreds. This system became important in number magic and divination, and remained in use for centuries, although it never became the dominant system for writing numbers.

A different system again assigned one series of phonetic signs to the whole numbers up to 9, another to the multiples of ten up to 90, and sometimes also another to the multiples of 100: this system had the same structure as the Egyptian demotic and the later Greek alphabetic number symbols. It became the usual way to label the page numbers in books, surviving in some regions as late as the nineteenth century.

A different option, and one that avoided strings of nonsense syllables, was to create verbal synonyms for number words: such as ‘earth’ (meaning one); ‘eye’ (meaning two); ‘sense’ (meaning five); and so on. Like the katapayadi numbers, these could be strung together starting with the units, followed by the tens and the hundreds, as many times as was needed in order to express even the largest numbers in the Sanskrit cosmology. This had the advantage of involving actual words and being therefore much better suited to insertion into verse meant to be learned by heart. The earliest examples appeared in astrological texts in the third century CE, and this scheme, too, would remain popular for several centuries before fading away.

So much for the less successful experiments – there were more – illustrating what a way of writing numbers down could look like. But of course, the Brahmi script also possessed a set of dedicated number symbols, which made their first fateful appearance in inscriptions of the third century BCE and spread through the subcontinent during the following century or so. In this earliest form, there were distinct symbols for the numbers from 1 to 9 and for the multiples of ten from 10 to 90. Much has been written of the rival possibilities that they were independently invented in India or imported from elsewhere, or at least inspired by a foreign model. The structure of the system in this archaic form was once again the same as the Egyptian demotic one, and about a third of the signs show some resemblance to their Egyptian equivalents. Traders from the Ptolemaic kingdom certainly reached the Malabar coast in this period, and conversely Buddhist missionaries travelled to Alexandria in the third century BCE. So the possibility of influence is real.

The Brahmi number symbols were used in inscriptions, on coins, on land grants inscribed on copper plates, and to number the lines of verse, and they spread into central and south-east Asia by the sixth or seventh century. From those drier climates, manuscripts on wood, palm leaf and paper survive, in which the Brahmi numerals are used within texts in the Indo-European language called Tocharian. In southern India they continued in use until the eighteenth century; in Tamil, numbers with this structure continue in use today.

These symbols were not used for arithmetic or accounting, however; in mathematical contexts they did not displace the use of number words or the other systems of symbols described above. Not, that is, until an innovation occurred in their structure.

The idea, simple enough, was that instead of using the full set of symbols, just those for the numbers from 1 to 9 would be employed, with the convention that (starting from the right) the first number denoted units, the next tens, the next hundreds and so on. It was a positional system on the same lines as the katapayadi one. And if there were, for instance, no tens to record, the blank place was marked with a dot.

It is not certain when the innovation first occurred. Dates as early as the mid-first millennium BCE have been suggested on the basis of a possible link with the symbols used on Chinese counting boards. But the earliest undisputed direct evidence for the Indian place-value system occurs in inscriptions from the late seventh century CE, both in Southeast Asia and in India itself. The moment of innovation must lie somewhere in between.

The idea of a positional system, with a special sign for a blank column, spread throughout the region; by perhaps 800 CE, a preference for it was widespread. The actual shapes of the symbols continued to evolve in tandem with the complex family of different writing scripts used in South Asia: a process that continues to this day, when the major South Asian languages have their own alphabets and number symbols. But all structure their number symbols using the same decimal place-value system inherited from their Brahmi ancestor.

One of the tasks for which this system was well suited was written calculation, something that on the whole is unusual in the history of numbers and counting. Most cultures calculate using devices – like the counting board or the abacus – supplemented by mental arithmetic for simpler operations. But, as most readers of this book will know only too well, a place-value system of number symbols enables operations like addition, subtraction, multiplication and division to be done on paper using algorithms that a young child can learn, as long as the child is willing to memorise the addition and multiplication tables, each with a few dozen entries. It therefore tends to turn both counting and calculation into special kinds of writing, and to bring them into contact with a child’s early training in wielding writing implements and surfaces: to make learning and manipulating the number symbols a natural accompaniment to learning the letters and arranging them into words and sentences.

Oh Lilavati, intelligent girl, if you understand addition and subtraction, tell me the sum of the amounts two, five, thirty-two, one hundred ninety-three, eighteen, ten, and a hundred, as well as the remainder of those when subtracted from ten thousand.

The famous Indian mathematician Bhaskara II was one of the many heirs to the rich tradition of writing words and numbers in India. He lived in the twelfth century, in Vijjadavida, a city in the western region sometimes called the Great Escarpment of India or the Western Ghats, the north–south range that separates the Deccan Plateau from the coastal plain beside the Arabian Sea. A richly forested biome of exceptional biodiversity, it was at this time part of the Western Chalukya Empire, ruled at the time of Bhaskara’s birth (1114 CE) by the successful Vikramaditya VI; though Bhaskara’s lifetime saw the start of that empire’s breakup as Vikramaditya’s feudatories fought their rulers and each other. His was an agricultural society, with well-organised guilds and a complex taxation bureaucracy. Trade linked it with China, with Southeast Asia and with the Baghdad caliphates to the west.

Literature and scholarship flourished under the Chalukya kings, who patronised scholars writing in both the local Kannada language and in Sanskrit. Bhaskara’s ancestors were said to be court scholars: astronomers and astrologers. It is also said that he learned astronomy from his father, Mahesvara.

Bhaskara himself may have worked as a court astronomer, but very little is known in detail about his life. His writings, however, are among the best known of the whole Sanskrit mathematical tradition. By his day, the sections in astronomical texts dealing with arithmetic had become long and complex and had started to appear as independent treatises, called ganitas, setting out the art of calculation as a subject in its own right. Writing in this tradition, he produced his Lilavati – literally ‘the beautiful’ or ‘the playful’. It has long been rumoured that the book was addressed to his own daughter, though there is no positive proof one way or the other, and within the text the addressee is sometimes imagined as masculine, sometimes feminine.

The Lilavati discussed arithmetic comprehensively. Introducing the numbers themselves, Bhaskara named seventeen different powers of ten: 1, 10, 100, 1,000, ayuta, laksa, prayuta, kota, arbuda, abja, kharva, nikharva, mahapadma, sanku, jaladhi, antua, madhya, parardha. He explained how to write numbers down: in his verses, numbers were given in words, but in the prose sections they were written using the place-value system.

From the place-value system, the Lilavati continued through the operations of addition, subtraction, multiplication and division, and on to finding squares and square roots, cubes and cube roots, followed by operations involving fractions or zero. It concluded with algorithms to solve what would now be called algebra problems, rules for reasoning about quantities in proportion to each other, and some geometry. The arithmetic alone filled dozens of Sanskrit verses, and the prose sections which provided examples for each rule expanded its length many times over. The examples ranged from the pedestrian to the picturesque:

Tell me, quick-eyed girl, if you know the correct procedure for inversion, the number which, multiplied by three, added to three-fourths of the result, divided by seven, diminished by one-third of the result, multiplied by itself, decreased by fifty-two, having its square-root taken, increased by eight, and divided by ten, produces two.

A traveler on a pilgrimage gave one-half of his money at Prayaga, two-ninths of the rest at Kast, one-fourth of the remainder in toll fees, and six-tenths of the remainder at Gaya. Sixty-three gold coins were left over, and he returned with that to his own home. Tell me the initial amount of his money.

There is a hole at the foot of a pillar nine hastas high, and a pet peacock standing on top of it. Seeing a snake returning to the hole at a distance from the pillar equal to three times its height, the peacock descends upon it slantwise. Say quickly, at how many karas from the hole does the meeting of their two paths occur?

The ‘Peacock’ problem, in a manuscript of the Lilavati.

Columbia University Library, Smith Indic MS 3.

No very early manuscript of the Lilavati survives today, but chance survivals of other materials from the period show what it would have looked like. Palm leaves – or in northern India, birch bark – were used as a writing surface, and books traditionally had a rectangular format, broader than they were tall. The leaves were kept together not by binding along one edge but by passing a string through a hole in their centre. Boards and a cloth wrapper could be used to protect the whole volume, but even so it is rare for a manuscript of this kind to survive more than a few centuries at most.

Brahmi number symbols set off from the text in a box, in a manuscript of the Lilavati.

Wellcome collection, MS OR Indic beta 249. CC BY 4.0.

The writing itself formed a rectangular block of text in the centre of each page, without capital letters and with only a few spaces between words: there were no line breaks between verses, nor any page breaks between chapters of text, though sometimes red was used to highlight the starts of chapters. One of the few interruptions to the continuous stream of the text, in fact, was that the many number symbols were often set off with a box around them.

In 1207 Bhaskara’s grandson set up, with funds from a local ruler, a school for the study of his writings, which included as well as the Lilavati a book on algebra and two on astronomy. It is from the foundation inscription of that school, at a temple in Maharashtra, that most of the little information about Bhaskara’s life derives. His works became the standard and best-known mathematics textbooks in the Sanskrit tradition, with many thousands of manuscripts in circulation. He may justly be credited with doing more than anyone else to promote the decimal place-value system as a tool for writing numbers and for the practice of written calculation.