Mathematics in India

The picture, adapted from Ifrah's book [16], shows how the numeral 2 in various Indian and other languages has evolved from it's Brahmī genesis; in particular, the Brahmī–Gupta–Nagari–western Arabic–European route taken by the numeral to reach the modern universally used 2 is also seen. — The picture, adapted from Ifrah’s book [16], shows how the numeral 2 in various Indian and other languages has evolved from it’s Brahmī genesis; in particular, the Brahmī–Gupta–Nagari–western Arabic–European route taken by the numeral to reach the modern universally used 2 is also seen. Sanjana N. Murthy

In the previous two parts of the article in this series, we had presented glimpses of the mathematics present in Vedic and Sūtra Literature, and highlighted the chief source materials, especially the two major direct sources of mathematical results and techniques: the Śulba-sūtras of Baudhāyana, Mānava, Āpastamba and Kātyāyana (c. 800–500 BCE), and the last section of the Chandaḥ-Sūtra of Piṅgalācārya (c. 300 BCE). These texts, being a part of the Vedāṅga literature, have been carefully preserved by successive generations for around 25 centuries or more. The Vedāṅga body of literature was considered essential for a proper study of the sacred Vedas and the performance of the Vedic rituals.

Introduction

Treatises were written in ancient India on materials like palm leaves or barks of trees which usually do not last long. Thus a treatise could be preserved only through memorisation, or through repeated copying. Therefore the more concise the work, the better its chance of survival through faithful memorisation or copying. Further, scholars had to be selective regarding the texts to be preserved, especially during the earlier periods. It would be natural under the circumstances not to make the tedious efforts to preserve a mathematics treatise of a certain period if there arises a later mathematics treatise by a stalwart containing the essence of the knowledge of the earlier period alongside new original results.

To cite a concrete example, the intensely concise Āryabhaṭīya (c. 499 CE) of Āryabhaṭa is considered a landmark in Indian astronomy and mathematics. But, thanks to this treatise, most of the preceding mathematics texts became redundant and hence became lost to posterity. Āryabhaṭa acknowledges at the beginning (verse 1) of the mathematics chapter Gaṇita of Āryabhaṭīya that he is recording ancient knowledge “honoured at Kusumapura” (probably near modern Patna). His commentator Bhāskara I (c. 600 CE) makes an incidental reference to names like Maskarī, Pūraṇa, Mudgala, Pūtana, referring to them as ācāryas (masters); however, nothing is known to us about such revered mathematicians. Evidently, they had composed voluminous mathematics texts, for Bhāskara I explains why Āryabhaṭa records only a “bit of mathematics” ([21], pp. liv–lv):

This has not been composed as a treatise by ācāryas Maskarī, Pūraṇa, and others, with even one lakh verses for each [topic]. How can the ācārya [Āryabhaṭa] manage [to state all of] it within such a short treatise?

As a consequence, there is a gap of nearly a thousand years between the Vedāṅga treatises like the Śulba-sūtras and the extant post-Vedic texts on mathematics. Therefore, although there was a long tradition of computational mathematics in ancient India, there arises a serious difficulty in reconstructing its precise history.

The most ancient Indian mathematics manuscript discovered so far is the “Bakhshali manuscript”, an incomplete manuscript comprising 70 leaves of birch bark that was discovered in 1881 at the village of Bakhshali near Peshawar. It is preserved at the Bodleian Library at the University of Oxford. There is a wide divergence of views regarding the date of the manuscript. Its mathematical content suggests that it predates Āryabhaṭīya of Āryabhaṭa (see [8]). B. Datta’s estimate (third century CE) is in tune with the recent radiocarbon dating (224–383 CE) of one of the folios at Oxford.

In Part 2 of the article (cf. [12]), we had highlighted the decimal number-vocabulary in Vedic literature and discussed the mathematical significance of the decimal system (applicable to both the decimal number-vocabulary and the decimal notation). In this part, we shall discuss some of the salient features of decimal notation and the early records of its occurrence in India. We shall also discuss some historical and mathematical aspects of the zero. We shall then demonstrate a few examples of ancient Indian methods for performing the fundamental arithmetic operations using decimal notation, highlighting the algebraic ideas involved. Finally, we shall indicate how the decimal notation and arithmetic got transmitted to Europe via Arabia. We shall also have an appendix describing Āryabhaṭa’s coding of large numbers.

The Decimal Notation: Some Features

Recall that, at the heart of the decimal notation lies the mathematical principle that every natural number $N$ can be expressed as a “decimal expansion”

\[\begin{equation}\label{decimalexpansion}
N=10^na_n+ \cdots + 100a_2 + 10a_1 + a_0,
\end{equation}\]where $a_0, a_1, \dots, a_n$ are integers between 0 and 9.

In the decimal notation, the number $N$ with the above expansion is written as $a_n \dots a_0$ .

We highlight three key ideas which are responsible for the efficiency of the standard decimal notation.

Three key ideas in the decimal notation

1. Place-value. Recall that the “place-value” (or “positional value”) principle assigns the values $1, 10, 10^2, \dots 10^n$ respectively to the first, second, third, $\dots$ $(n+1)^{\rm th}$ position from the right in a number, so that a digit $d$ in the $r$ th position (place) from the right is imparted the place-value $d \times 10^{r-1}$ . For instance, when we write 2019 in the decimal notation, the symbol 2 acquires the value $2 \times 10^3$ (i.e., “2 thousand”) by virtue of it being in the fourth “place” (position) from the right. It is the place-value principle that enables the concise representation of a number with the expansion described in expression (\ref{decimalexpansion}) as $a_n \dots a_0$ .

2. Zero-symbol. In the decimal notation for a number $N$ , a symbol is assigned to the concept of zero and this zero-symbol is used as a “place-holder” whenever there is no positive contribution of a place (or power of ten) to the number $N$ . For instance, in 2019 ( $=2 \times 10^3+1 \times 10+9$ ), where there is no hundred ( $10^2$ ), the symbol 0 holds the place for $10^2$ . Without this strategic placing of the zero-symbol at the third place from the right to indicate “no hundred”, the number “two thousand and nineteen” might be mistaken for “two hundred and nineteen”. By manning the $10^2$ -place, the symbol 0 enables 2 to get unambiguously stationed in the $10^3$ -place.

3. One symbol for one digit. The decimal notation stipulates a separate single symbol for each of the ten primary numbers: $0,1, \dots, 9$ . For instance, the symbol for “three” is a distinct figure 3, and not three copies of the symbol for 1 (i.e., 3 is not denoted in the decimal notation as $1 1 1$ ). The principle “one digit one symbol” is reminiscent of the phonetic principle “one sound one symbol” for the Sanskrit alphabet. Further, an interesting feature of the ten Sanskrit numerals in decimal notation (for instance, in the Nāgarī script) is that each of them can be drawn in just one stroke (without lifting the pen/pencil from the paper), thereby imparting speed to the writing process.

Though the principle “one symbol for one digit” is not highlighted to the same extent as the place-value and the zero, its importance can be seen by comparing number-representation in ancient Babylonia with the decimal notation.

The Indian decimal notation and the Babylonian sexagesimal system

In Part 2 of the article (cf. [12]), our mathematical discussions mostly pertained to the polynomial-like expansion of numbers displayed in expression (\ref{decimalexpansion}). It would then appear that the adjective “decimal” has only a peripheral role. After all, in expression (\ref{decimalexpansion}), 10 could have been replaced by any natural number $M$ and the coefficients $a_i$ ‘s could have been allowed to vary between $0$ and $M-1$ .

Indeed, for the writing of numbers, the Babylonians used a “sexagesimal” place-value system with base $M=60$ from as early as 1700 BCE. The system was imperfect since there was no symbol for zero in the early Babylonian system. As a result, the interpretation of the place-value of a digit would not always be unambiguous from sheer notation—it would have to be determined from the context. For example, the symbols for 1 and 60 are identical. In a later phase, the Babylonians used a placeholder to represent zero, but only in the medial positions and not on the right-hand side of the number. So the scope for ambiguity remained.

But the crucial difference between the Indian decimal notation and the Babylonian system is that while the former comprises ten distinct figures for the ten digits, and each of these ten figures can be drawn in just one stroke, the Babylonians did not use 60 distinct symbols for their digits. As is well-known, they had only two basic symbols, for ten and one respectively, for representing any digit. Consequently, the Babylonian sexagesimal system could not attain the economy, compactness and elegance of the Indian decimal notation. Though based on profound ideas, the Babylonian system was quite cumbersome in practice. Figure 1 shows the complications involved in depicting even the basic 59 digits.

Thus, along with the place-value idea and the zero, the principle “one symbol for one digit” too contributes crucially to the effectiveness of the Indian decimal notation.

The number 60 has remarkable divisibility properties. It is the smallest number which is divisible by every number from 1 to 6. It is a “highly composite number”,¹ that is, 60 has more divisors (factors) than any number less than it: 60 has twelve divisors 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60; the next highest among lower numbers is 48 which has ten divisors. Consequently, in the sexagesimal system, many fractions get simplified. For example, one hour (i.e., 60 minutes) can be conveniently divided evenly into sections of 30 minutes, 20 minutes, 15 minutes, 12 minutes, 10 minutes, 6 minutes, 5 minutes, 4 minutes, 3 minutes, 2 minutes, and 1 minute.

Figure 1: Babylonian numerals from 1 to 59 **courtesy:** MacTutor History of Mathematics Archive

Therefore, a partial blending of the sexagesimal and decimal systems is now used for recording time, angles and geographic coordinates. For instance, we know that 1 hour is divided into 60 minutes, and 1 minute is divided into 60 seconds. A watch displaying a time of 6:54:23 (6 hours, 54 minutes, 23 seconds) can be interpreted as the sexagesimal number

6 \times 60^2 + 54 \times 60^1 + 23

of seconds that have elapsed since midnight. Note that, in the modern system, each of the three sexagesimal digits (6, 54 and 23) is written in decimal notation, thus overcoming the cumbersomeness of the original Babylonian system.

Appreciation of the decimal notation and the zero

The decimal notation is celebrated both for its sheer brilliance of abstract thought as well as its utility as a practical invention. Thinkers on mathematics and history have expressed profound admiration for decimal notation, especially the two innovations: the place-value and the zero. We quote a few examples.

The great French mathematician Pierre-Simon Laplace writes (1814 CE):²

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 simplicity, the great ease which it has lent to all computations, puts 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.

The power of the place-value of zero has been beautifully highlighted by the American mathematician G.B. Halsted³

The importance of the creation of the zero mark can never be exaggerated. This giving to airy nothing, not merely a local habitation and a name, a picture, a symbol, but helpful power, is the characteristic of the Hindu race whence it sprang. It is like coining the Nirvana into dynamos. No single mathematical creation has been more potent for the general on-go of intelligence and power.

The above passage contains an allusion to the following verses in Shakespeare’s A Midsummer Night’s Dream (Act V, Scene I):

And, as imagination bodies forth
The form of things unknown, the poet’s pen
Turns them to shapes, and gives to airy nothing
A local habitation and a name.

The view of Halsted on the zero-symbol has been reiterated by several authors. The British scientist L. Hogben writes⁴

In the whole history of mathematics, there has been no more revolutionary step than the one which the Hindus made when they invented the sign `0′ …

The eminent Dutch mathematician Bartel Leendert van der Waerden calls the idea of zero “a stroke of genius” ⁵

The zero is the most important digit. It is a stroke of genius, to make something out of nothing by giving it a name and inventing a symbol for it.

Finally, we recall the remarks of the British historian A.L. Basham
on the decimal notation (cf. [2], pp. 495–496):

Most of the great discoveries and inventions of which Europe is so proud would have been impossible without a developed system of mathematics, and this in turn would have been impossible if Europe had been shackled by the unwieldy system of Roman numerals. The unknown man who devised the new system was from the world’s point of view, after the Buddha, the most important son of India. His achievement, though easily taken for granted, was the work of an analytic mind of the first order, and he deserves much more honour than he has so far received.

Indians were aware of the greatness of the invention—it appears to have had a status similar to that of an inspired revelation. The Arab historian Abul Hasan Al-Masūdī, who visited India during the tenth century, writes (943 CE):

A congress of sages at the command of the Creator Brahmā invented the nine figures … ([7], p. 97).

In Līlāvatī (1150 CE), Bhāskarācārya makes a special mention that the decimal system of place-values was conceived by the ancients to simplify practical computations.

Appearance of zero and decimal place-value notation

One does not know when the idea of zero and the other nine numerals occurred in the context of number-representation in symbols, and when the decimal place-value notation was invented in India. It has been suggested, on the basis of Indus valley metrology and apparent number-symbols, that a decimal system was in use in the Indus valley civilisation. In the epic Mahābhārata (3.134.16), there is a phrase nava yogo gaṇanāmeti śaśvat, “A combination of nine [digits] always [suffices] for any count [or calculation]”. This incidental reference to the decimal place-value notation occurs during the narration of a tale (3.132–134) involving names like Uddālaka, Śvetaketu, Aṣṭāvakra, Janaka, et al, who belong to the Brāhmaṇa phase of the Vedic era. In fact, the word śaśvat (perpetual) has the nuance of “from immemorial time”.

Some Sanskrit scholars see in the term lopa (elision, disappearance, absence) of Pāṇini’s grammar treatise Aṣṭādhyāyī a concept analogous to zero as a marker for a non-occupied position, and have wondered whether lopa led to the idea of zero in mathematics, or the other way.⁶ Indeed, in a text Jainendra Vyākaraṇa of Pūjyapāda (c. 450 CE), the term “lopa” is replaced by khaṃ, a standard Sanskrit term for the mathematical zero. Unfortunately, mathematics texts of the time of Pāṇini have not survived.

We give below a few instances of (i) a verbal mention of the chief ingredients of the decimal notation: places, nine numerals, zero-symbol (or zero), etc., and (ii) actual occurrence of decimal notation in written documents. We shall also discuss the treatment of zero as a number in its own right.

Mention of zero in Chandaḥ-Sūtra of Piṅgalācārya

In Part 2 of the article (cf. [12]), we had seen that in his prosody-text Chandaḥ-Sūtra (c. 300 BCE), Piṅgalācārya mentions an algorithm to compute the total number of metres containing $n$ syllables (i.e., the number $2^n$ ). His instructions involve the use of dvi (two) and śūnya (zero) as distinct labels; the word śūnya occurs in two consecutive terse aphorisms: rūpe śūnyam and dviḥ śūnye. The choice of the labels suggests the prevalence of the mathematical zero and a zero-symbol.

Mention of zero and decimal places in Jaina texts

B. Datta (cf. [6]) has pointed out that the concept of decimal “place” (sthāna) is explicitly mentioned in the Jaina text Anuyogadvāra-Sūtra (c. 100 BCE). The text mentions that the number $2^{96}$ “occupies 29 places (sthāna)”. Indeed, the number $2^{96}(=79228162514264337593543950336)$ requires precisely 29 digits in the decimal notation. The 0 occurs in the thousand’s place in this 29-place decimal expansion. This reference to 29 places for the number $2^{96}$ indicates that the concept of zero as a placeholder had emerged in some form.

Zero is explicitly mentioned in 6th-century Jaina texts. Jinabhadra Gaṇi (529–589 CE) abridges the verbal description of the number 3200400000000 as “thirty-two, two zeros, four, eight zeros”.

The Jaina schools adopted the approximation formula $\sqrt{N}=a+\frac{r}{2a}$ , where $N$ is a positive integer, $a$ the largest integer for which $a^2<N$ and $r=N-a^2$ . Jinabhadra Gaṇi describes the (approximate) square root $\sqrt{58545048750}$ (which is $241960\frac{407150}{483920}$ by the above formula) as “two hundred thousand forty one thousand nine hundred and sixty, removing the zero, the numerator is four-zero-seven-one-five and the denominator four-eight-three-nine-two”. The reference to removal of zero (i.e., to cancel the common factor 10) suggests the use of a zero-symbol. Similar references occur in the work of Siddhasena Gaṇi.

The Brāhmī numerals

The present scripts of various Indian languages evolved from the ancient Indian Brāhmī script. The figures for the modern nine numerals 1 2, 3, …, 9, are all derived from the corresponding Brāhmī numerals. The evolution of the numeral 2 is shown in the opening image of the article.

In the Brāhmī script, the numbers are represented in base 10, but not in “place-value” notation and the numerals do not satisfy the “one symbol for one digit” criterion. Apart from symbols for the nine numerals, the Brāhmī system has separate symbols for multiples of 10 (up to 90) and multiples of 100 (up to 900). A number is obtained by adding the numerical values of the symbols representing the number. Thus zero is not required in the Brāhmī system.

There are inscriptions from the 3rd century BCE in the Brāhmī script, including the rock edicts of Emperor Aśoka.

In the present scripts of Indian languages including Sanskrit, numbers are represented in decimal place-value notation using modifications of the nine Brāhmī numerals and a zero-symbol. The most widely used symbols in India for the ten numerals are those belonging to the Nāgarī (or Devanāgarī) script.

Occurrence of written decimal notation in the Bakhshali manuscript

Numbers are written in decimal place-value notation, with a dot as the zero-symbol, in the “Bakhshali Manuscript”. Figures 2 and 3 display a few examples of large numbers written in the decimal notation in a page of the Bakhshali Manuscript; for instance the number 7,250,483,394,675,000,000.

Exposition of decimal place-value by Āryabhaṭa

The idea of visualizing each power of ten as a sthāna (denominational place) occurs explictly in the Gaṇita (mathematics) chapter of the treatise Āryabhaṭīya (499 CE) of Āryabhaṭa. After an invocation in the first verse, the chapter narrates (in Verse 2) the first ten notational places: eka, daśa, śata (hundred), sahasra (thousand), ayuta (ten thousand), niyuta (hundred thousand), prayuta (million), koṭi (ten million), arbuda (hundred million) and vṛnda (thousand million) and declares them as being sthānāt sthānaṁ daśaguṇaṁ syāt}, “from place to place, it is ten times (the preceding)”. Subsequent authors list more places (usually 18, sometimes 24). Āryabhaṭa’s verse is only an exposition and not an announcement of a new concept. As mentioned in the Introduction, Āryabhaṭa acknowledges his predecessors in the preceding verse.

Figure 2: Page 46-recto of the Bakhshali manuscript, in the Bodleian library copy, on which numerals are dense. **courtesy:** Bill Casselman

Figure 3: Same page, annotated with modern numerals by Bill Casselman. The fully visible Bakhshali digits are overlaid in yellow colour. Those one can see parts of are in green, and those in red are conjectural.8 — Figure 3: Same page, annotated with modern numerals by Bill Casselman. The fully visible Bakhshali digits are overlaid in yellow colour. Those one can see parts of are in green, and those in red are conjectural.⁷ **courtesy:** Bill Casselman

Āryabhaṭa’s methods in Gaṇita for extraction of the square-root and the cube-root of a number, which are slight variants of the modern methods, make intricate uses of the concepts of place-value and zero. The fact that these methods are briefly stated without elaboration, indicates that the decimal place-value notation with zero had already become firmly established in India. Āryabhaṭa’s alphabetical coding of large numbers, described in the Appendix of this part, reiterates how strong his mastery was over all the ideas encapsulated in the decimal notation.

Occurrence of written decimal notation in inscriptions

A copper plate from Gujarat of 595 CE is, so far, the oldest discovered inscription depicting a number (346) in decimal place-value notation. For numbers in decimal notation with zero as a digit, the earliest known examples are certain Sanskrit inscriptions dated 683–687 CE from places in modern Cambodia and Indonesia (then a part of Greater India) which give their Śaka dates 604, 606 and 608 in decimal notation.

Among early inscriptions within present-day India, the most famous depiction of a number in decimal notation with a zero digit is the clear engraving of the number 270 in the Nāgarī script in the Gwalior inscription of King Bhojadeva, dated 876 CE. But older inscriptions with zero in decimal notation have also been reported, e.g., the number 30 in Trilingi plates (690 CE) and Ragholi plates (8th century); the number 201 in the Khandela inscription (807 CE); (cf. [7] and [22] for more details). A list of inscriptions from 578 CE which depict the zero symbol, not necessarily as part of decimal notation, is presented in [22].

Kaṭapayādi

A decimal place-value notation called kaṭapayādi makes use of letters of the Sanskrit alphabet in place of numerical figures.

In one of the versions, the nine consonants from k to jh, as also the nine consonants from ṭ to dh, denote (in usual order) the digits from 1 to 9; the five consonants from p to m denote the digits 1 to 5 while the eight unclassed consonants from y to h denote 1 to 8. Thus 1 could be denoted by any of the letters ka, ṭa, pa, ya; hence the scheme is named kaṭapayādi.

The consonants ñ and n and all pure vowels not preceded by a consonant (i.e., occurring at the beginning) denote 0. A vowel joined to a consonant, or a consonant not joined to a vowel, did not carry numerical value. In a conjoined consonant, only the last one denotes a digit.

The letters are usually written in ascending powers of 10 (the digit in the unit’s place written first followed by the digit in the ten’s place to its right, and so on).

For instance, the kaṭapayādi word tatvāloke denotes the number $1346$ (ta=6, tvā=4, lo=3, ke=1).

Although the kaṭapayādi has the rigidity of position imposed by its place-value character, it actually attains considerable flexibility from the near-absolute freedom in the choice of vowels and partial choices in the use of consonants. The kaṭapayādi provides scope for skilled authors to compose brief but pleasant-sounding chronograms, often with connected meanings.

Terms are sometimes coined in ancient treatises in such a manner that the kaṭapayādi value of the chosen word encodes some numerical feature of the defined concept.

An interesting example is the word anantapura—an Indian name for the lunar cycle. The word ananta means “endless”, “boundless”, “eternal”, “infinite’, “sky”, “atmosphere”; while the word pura means “abode”, “towards (or from) the east”. Apart from the literary nuance of the Moon’s endless eastward (relative to the fixed stars) orbit in the boundless sky, the kaṭapayādi value of anantapura is 21600 (a=0, na=0, ta=6, pu=1, ra=2) the number of minutes in a pakṣa (lunar half-month): $15 \times 24 \times 60$ .

It is said that the great philosopher Śankara was so named since the kaṭapayādi value of the name (215; śa=5, ka=1, ra=2) indicates his birth-date—the fifth day of the first fortnight of the second month in the Indian lunar calendar.

The kaṭapayādi was used in Kerala to compose vākyas (sentences) recording planetary positions at regular intervals. The earliest known treatise of this type is the Candra-vākyāni of Vararuci (fourth century CE).

The kaṭapayādi system can be seen in several subsequent treatises of South India, like the Grahacāranibandhana of Haridatta (c. 600 CE).

The kaṭapayādi scheme has been popular in South India, especially in Kerala. It is applied in Karnatic (South Indian) music where the 72 Janaka (root) rāgas are classified into 12 groups of 6 rāgas each and numbered systematically, according to their notes, in such a way that the notes of a rāga can be quickly determined from the rāga number. The 72 rāgas are named in such a way that the kaṭapayādi value of the first two syllables of its name gives the serial number of the rāga and hence its notes.⁸

Bhūtasaṅkhyā

As most ancient Indian scientific treatises are composed in verses, there has been a tradition of expressing numbers verbally by the decimal place-value principle including zero using word-numerals instead of symbols.

The Bakhshali Manuscript has both word-numerals and representations in symbols, the word-numerals in this treatise being simply usual number-names. Another early instance, cited in [16], is a Jaina cosmology text Lokavibhāga dated 458 CE.

In a popular system of word-numerals, which emerged by the third century CE, the word-name for a digit (or even a number) $n$ is chosen to be a well-known object or idea which usually occurs with frequency $n$ (or has $n$ components). Such a word-numeral is called bhūtasaṅkhyā, i.e., a number (saṅkhyā) using an object (bhūta) illustrating the number.

For instance, candra (Moon) or mahī (Earth) stands for one, netra (eyes) for two, guṇa (sattva, rajas, tamas) or kāla (time: past-present-future) for three, yuga (satya, tretā, dvāpara, kali) for four, indriya (sense-organs) for five, rasa (taste: sweet, sour, salty, bitter, pungent, and astringent) for six and so on. For zero, its Sanskrit synonyms (most of them being synonyms for the sky) are used; examples are given later in this article. The place-values of word numerals in a bhūtasaṅkhyā increases from left to right.

The bhūtasaṅkhyā system can be seen in the Yavanajātaka of Sphujidhvaja (270 CE), in the Sūryasiddhānta and Puliśasiddhānta (original versions composed within fourth century CE) and in later astronomy and mathematics treatises, and in the Agni-Purāṇa (portions of which were composed within fourth century CE). Word-numerals are also found in certain Sanskrit inscriptions of the seventh-century CE in the Far East.

The numerous choices for a digit, often with profound nuances, not only helps in maintaining the rhythms of the verses but also infuses a poetic charm in the technical presentations. For instance, Bhāskarācārya describes his year of birth 1036 Śaka Era (1114 CE) as rasa-guṇa-pūrṇa-mahī. The phrase rasa (taste; delight) – guṇa (quality) – pūrṇa (full) – mahī (Earth) for the number 1036 has a sweet meaning: “The Earth is full of delight”.

B. Datta feels that decimal place-value notation had been in common use long before the idea of applying the place-value principle to a system of word-names was conceived. In the beginning, number-names are used as word-numerals. Later, Bhūtasaṅkhyā becomes popular due to its literary advantages.

The algebraic role of zero

A distinction is often made between two of the important features of zero in mathematics:

The notational innovation of the zero-symbol as a place-holder in a place-value numeral system.
The abstract conception of the number zero as an integral part of a system where one can perform fundamental binary operations like addition, subtraction and multiplication.

Roughly speaking, (i) and (ii) correspond respectively to the roles played by zero in basic arithmetic and algebra. The two aspects are mathematically equivalent, for the decimal notation $a_n \dots a_1a_0$ corresponds to the polynomial expression $10^na_n+ \cdots + 10a_1 + a_0$ whose coefficients $a_n, \dots, a_0$ are regarded as integers between 0 and 9; a placeholder zero in the representation of a number through the decimal notation corresponds to the vanishing of a coefficient in the polynomial-type representation. Aspect (ii) is central in modern mathematics with its emphasis on algebraic structures.

Though the distinction between (i) and (ii) might appear artificial to a modern mathematician, it is indeed convenient to make this distinction while surveying the history of ideas. It helps us appreciate the extent to which the potential of the zero had been harnessed in different mathematical cultures and at different epochs, even in the same mathematical culture. That step (ii) needs a conceptual leap from (i) can be seen from the fact that although the decimal notation requires ten symbols, ancient and medieval scholars often refer to it as a system with “nine figures” indicating that the place-holder zero-symbol did not always receive the dignity accorded to the other nine numerals.

The profound step (ii) of elevating zero to an algebraic number, saṅkhyā}, and not treating it as a mere place-holder, was taken in India. Indian mathematicians were perhaps the only ancient mathematicians to explicitly define mathematical operations with zero. This point, which often gets lost in discourses on zero, is made in passing by N. Bourbaki⁹ in (cf. [3], p. 46):

Our actual decimal system, which (by the agency of the Arabs) is derived from Hindu mathematics, where its use is attested already from the first centuries of our era. It must be noted moreover that the conception of zero as a number (and not as a simple symbol of separation) and its introduction into calculations, also count amongst the original contributions of the Hindus.

The earliest extant text in the world which gives a detailed exposition on operations with zero is the treatise Brāhma-Sphuṭa-Siddhānta (BSS in short) of Brahmagupta, unambiguously dated 628 CE. In a section of Chapter 18 (verses 30–35) of this treatise, Brahmagupta states rules for the fundamental operations (addition, subtraction, multiplication, division, involution, evolution) involving numbers—positive, negative and zero—treating all three on the same footing. The BSS became a trend-setter: the mathematics of zero is elucidated in subsequent extant works of major Indian mathematicians like Śridhara (c. 750), Mahāvīra (850), Āryabhaṭa II (950), Śrīpati (1039), Bhāskara II (1150) and Nārāyaṇa (1350). To cite a few examples, Chapter 18 of BSS explicitly states rules like $x+(-x)=x-x=0$ , $0+0=0$ (verse 30), $0-0=0$ (verse 32) $x \times 0=0$ for all $x$ positive, negative or zero (verse 33) and Triśatikā of Śridhara states the rule $x\pm 0=x$ (cf. [7, [p. 240]]). A striking aspect of some of these rules is that they amount to the sophisticated idea of giving integers a ring structure with “zero as the additive identity’’.

The use or mention of zero as a number in arithmetic can be seen scattered in the Bakhshali Manuscript (probably third century CE), in the sixth-century treatise Pañca-Siddhāntikā of Varāhamihira and in the early seventh-century commentary on Āryabhaṭīya by Bhāskara I. The reference to zero by Piṅgala suggests a strong possibility that zero was already considered a number by his time (c. 300 BCE). But it is from the BSS onwards that we see a systematic formal presentation of the mathematics of zero either as a topic by itself or as a part of the topic: rules of operations with numbers (positive, negative and zero).

In a personal email correspondence with the present author, David Mumford remarked (on 8 February 2013),

It seems clear to me that Brahmagupta is the key person in the creation of algebra as we know it.

The history of the conceptualization of zero as a number amenable to the fundamental operations is closely, and symbiotically, related to the emergence and recognition in India of algebra as a “distinct branch of mathematics’’. In his magnum opus BSS, Brahmagupta compiles a separate chapter (Chapter 18) on certain topics that we now recognize to be algebra, and emphasises its importance. In the process, he makes a conscious formulation of the science of algebra. After a statement on the indispensability and power of the subject matter of this algebra-chapter (verse 1), Brahmagupta lists the core topics of the algebra-chapter in verse 2. The very second item in Brahmagupta’s list is “the zero, negative and positive quantities”. In retrospect, one may say that the emergence of the zero as a central object (pun intended) in a number system was the outcome of the algebraic mentality of the Indian mathematicians (sometimes dormant, sometimes in the open) and conversely, the comprehensive treatment of zero and negative numbers was an important pillar in the foundations of algebra laid by Brahmagupta in the above-mentioned chapter.

The introduction of binary operations with zero (and negative numbers) reminds us of one of the high points of post-Vedic Indian mathematics: Brahmagupta’s formulation of his “composition law” that was later named bhāvanā in Indian mathematics. The binary operation bhāvanā has momentous consequences in algebra and number theory. Brahmagupta’s laws would be generalised by Gauss in the 19th century (one of Gauss’s major works) and a further generalisation has been achieved in the present century by Manjul Bhargava. A brief account of the significance of Brahmagupta’s Composition Law has been given in [10]. For someone who could conceive of such an abstract principle of binary composition in the seventh century in the context of solving an equation, it would have been natural to foresee the importance of systematic promulgation of binary operations with zero (and negative numbers), which is what he did during his “creation of algebra as we know it”.

Brahmagupta had also taken the bold step of considering numbers with zero in the denominator (which he called kha-cheda), and such numbers were discussed by subsequent Indian algebraists too, especially by Bhāskara II (he called them khahara). Modern scholars, used to a different convention, have often criticised ancient Indian algebraists for their statements around division by zero. But, viewed with an open mind, one can see that the ancients had simply followed a different convention from ours and there is nothing mathematically unsound in their approach; in fact, some of their approaches have analogues in modern higher mathematics. We plan to discuss this theme in a future article.

The etymology of zero

The śūnya of Sanskrit mathematics was translated to Arabic (820 CE) as ṣifr, the Arabic word for “empty”. This eventually became sifra in Hebrew (c. 1146), cifra in Spanish and Italian, cifre in old French, ziffer in German, chiffre in French and cipher in English.

In another direction, ṣifr, the Arabic translation of Sanskrit śūnya, became zephyr or zephyrus, a Latin word for the “west wind”. (Zephyrus is the Roman name of Zephyros, the Greek god of the west wind.) These words came to mean a “light breeze”, an “almost nothing”.

Fibonacci (1202), who advocated the use of the Indian decimal system, coined the term zephirum for śūnya. Subsequently, these words became zefiro in Italian. There were other variants: zeuero, zepiro, zefiro, zefro, zeviro, zevro, zeiro. Eventually, these got contracted to zero in Italian. The word “zero” came into the English language via French zéro from Italian zero.

The first known occurrence of the term “zero” is in the Italian book De Arithmetica Opusculum by Philippi Calandri, published in Florence in 1491.

Significance of the Sanskrit terms for zero

Most Sanskrit terms for “zero” are also the Sanskrit words for the sky—which is both empty and huge—and words closely associated with the sky, like cloud, vastness, and void. A frequently used term for zero in Sanskrit mathematics is “kha”. While in Ṛgveda, the word kha refers to the hole in the nave of a wheel through which the axle runs, the meanings of kha include, apart from “cavity”, “hollow”, “aperture”, “vacuity” and “empty space”, the “sky” and the “great Void”—something apparently empty but containing all! The Bṛhadāraṇyaka Upaniṣad identifies kham with Brahman, the Supreme Spirit.¹⁰

The most well-known Sanskrit term for zero is śūnya whose meanings include the sky and the void, apart from nothing. Other Sanskrit words for zero include ākāśa, antarikṣa, abhra, ambara, gagana, diva, nabha, puṣkara, jaladharapatha, viṣṇupada, vihāyas, vyoma (all are Sanskrit words for the sky), ambuda, jalada, jaladhara, payoda, payodhara, megha (Sanskrit words for the cloud), ananta (infinite), pūrṇa (whole, full, complete), vṛhat (vast), pṛthu (great, large, ample, numerous, extensive), pṛthula (large), viyat (sky, being dissolved, vanishing), randhra, chidra (small hole), bindu (dot), vindu (drop), nagna (bare), asat (non-existent). Zero has been denoted in India sometimes by a dot and sometimes by a small circle.

The kha (or śūnya) is not a mere void, it is a sublime state with immense possibilities—it carries the essence of all that is yet to be formed or created. It has a subtle nuance of a potential for growth from a deficient state (a zero state), to a state of fullness. The first landmark in Buddha’s spiritual quest is śūnyatā (vacuity). Under his first teacher Ārāḍa Kālāpa (or Āḷāra Kālāma), Buddha learnt meditation, especially a dhyānic state: the sphere of nothingness.

The literary nuance of śūnya (or kha) is not exactly zero (or the dismissive “cipher”); it is a “fathomless zero”. In the Canto “The Symbol Dawn” in his epic Savitri, while describing the condition just before the beginning of Creation, Sri Aurobindo writes:

A fathomless zero occupied the world.

Place-value and zero as metaphors in philosophy and literature

The place-value notation had become so well-established among the cultured elite in India, that the idea could be used as a metaphor in works on philosophy. A reference to place-value occurs in a work of the Buddhist philosopher Vasumitra (probably 1st century CE):

When [the same] clay counting-piece is in the place of units, it is denoted as one, when in hundreds, one hundred. (Quoted in [18], p. 46.)

Both the Vyāsa-Bhāṣya (probably 5th century CE) on the Yoga-Sūtra of Patañjali and Sārīraka-Bhāṣya of Śaṅkarācārya (c. 750 CE) use the following simile:¹¹

Just as the same stroke (or figure) conveys different ideas such as a unit, a ten, a hundred, a thousand, etc., according to the place in which it is set down, $\dots$

The romantic literary work Vāsavadattā of Subandhu¹² uses the simile of a zero-symbol śūnya-bindu to describe the stars.

The zero of the decimal system has also been used to explain spiritual principles. Indian philosophy regards the universe as a manifestation of `The One Reality’, the Pūrṇa, but emphasises that without the realisation of (or at least a conscious link with) the `One Brahman’, the transient universe is an unreal vacant nought, a śūnya; just as many zeros form a large number if preceded by 1; but without that 1, a large collection of zeros amounts to nothing. In the utterance of Sri Ramakrishna (1884):¹³

First realise God, then think of the creation and other things. … If you put fifty zeros after one, you have a large sum; but erase the one and nothing remains. It is the one that makes the many. First one, then many.

We end this subsection with a witty remark made nearly a hundred years ago by the Bengali author Pramatha Chaudhuri (pen-name Birbal) which blends two nuances of the word zero:

The mathematical zero as a place-holder.
The cipher, a person of no value or importance who is used by others for their own purposes.

In a certain session of the Indian National Congress in September 1920, M.K. Gandhi brought a resolution which was strongly opposed by several stalwarts. However, thanks to his followers, the resolution got passed. Birbal wrote¹⁴

There was only one vote in favour of Mahatma Gandhi’s resolution and nine hundred and ninety nine against it. Yet, the one triumphed as the one was accompanied by several zeros.

A brief history of the decimal notation is presented in [9] and a detailed history in [7] (also see [4], [13] and [16]). For a concise illustrated history of zero, see [17].

Examples of Ancient Indian Arithmetic Algorithms

In this section we shall describe examples of ancient Indian algorithms for the fundamental arithmetic operations: addition, subtraction, multiplication, division, taking squares and cubes, and extractions of square roots and cube roots. Our present algorithms for these operations are slight variants of some of the ancient Indian methods and follow the same mathematical principles.

Calculations in ancient India were usually performed on sand spread on a board called pāṭī (hence the Sanskrit term pāṭīgaṇita for basic arithmetic). Not many lines of figures could be written on the sand-board. On the other hand, it was easy to erase figures. The ancient Indian methods were therefore planned in such a way that figures not required for subsequent work could be erased. When ink and paper got introduced, the compulsion for involving a minimal number of lines decreased; on the other hand, erasing became inconvenient. Hence the slight modifications from the ancient Indian to our present methods.

A great merit of both the decimal system, as well as the ancient Indian methods (or our present methods) for the arithmetic operations emanating out of the decimal notation, is their incredible simplicity. We have seen that there have been well-articulated statements on the greatness of the decimal notation, and some of its cardinal features like the place-value principle and the zero-symbol as a place-holder. This has ensured that the decimal notation is reckoned among the great mathematical achievements. There is, however, a dearth of comparable statements for the fundamental operations. Ironically, it is due to the extreme ease achieved by these arithmetic methods, that the mathematical acumen involved in these methods does not receive due appreciation.

Often, we do not realise that a distinction has to be made between the “learning” and the “invention” of these methods. While it does not require any serious mathematical talent or maturity to learn the prevalent methods, the invention of these methods needed algebraic insights and sophistication.

We had seen in the Part 2 of the article (cf. [12]), that our standard multiplication algorithm implicitly involves the knowledge and skilful application of the “distributive property” of numbers

$(a+b)c=ac+bc,$ an algebraic principle important enough to be conferred a distinct name; and of course the polynomial-like expansion of numbers in the decimal system, which is an algebraic concept, together with adjustments for “carrying over”. The reader can similarly analyse the algebraic aspects of the standard algorithms for addition and subtraction.

An ancient Indian multiplication algorithm

Indians describe several multiplication algorithms like gomūtrikā, sthāna-khaṇḍa and kapāṭa-sandhi with the same underlying principles but slightly different “arrangements”, or algorithmic layouts. The precise details of various multiplication algorithms in ancient India are narrated lucidly in (cf. [7] pp. 134–149). We illustrate below the gomūtrikā (zigzag) method described by Brahmagupta (cf. [7], pp. 147–148).

To multiply 1223 by 235, the numbers are arranged in the gomūtrikā plan as:

\[\begin{array}{llllllllllllllll}
2&&&&&&&&&&1&2&2&3&&\\
3&&&&&&&&&&&1&2&2&3&\\
5&&&&&&&&&&&&1&2&2&3
\end{array}\]Thus, the multiplication $235 \times 1223$ is broken up into three parts: $200 \times 1223= (2 \times 1223) \times 100$ , $30 \times 1223= (3 \times 1223) \times 10$ and $5 \times 1223$ to be performed in three horizontal rows. The number 1223 is written in each of the three horizontal rows; but the first and second rows are shifted to the left by two and one places respectively, corresponding to the multiplications by 100 and 10. The first horizontal line (of 1223) is then multiplied by 2, beginning at the unit’s place, and the original digits are replaced by the product. For instance, $2 \times 3=6$ ; so the 3 on the right-hand corner is erased and 6 substituted in its place. Then the second horizontal line is multiplied by 3 and, as before, the original digits are replaced by the product. Recall that the digits in the second row are placed one place to the right of the digits in the first row. After all the original horizontal lines have thus been multiplied and replaced, the resultant horizontal lines are added as in our present method. After all steps are performed, the numbers on the pāṭī will finally appear as:

\[\begin{array}{llllll}
2&4&4&6& & \\
&3&6&6&9& \\
& &6&1&1&5 \\
\hline
2&8&7&4&0&5
\end{array}\]A little reflection will show that only a brilliant algebraic mind could have conceived of the original algorithms and that the standard modern method closely follows the original plans.

The “long division” algorithm

A variant of the modern algorithm for “long division” is described in the arithmetic treatises of Śrīdhara (750 CE), Mahāvīra (850 CE) and later writers (cf. [7], pp. 150–154).

We first describe the algorithm for long division through an example: division of 1620 by 12. The divisor is placed below the dividend:\[\begin{array}{llllllllllllll}
1&6&2&0&&&&&&&&&& \\
1&2& & &&&&&&&&&&
\end{array}\]The process begins from the left with the figure 16 which, when divided by 12, gives the quotient 1 and the remainder 4. The quotient 1 is placed in a separate line:

\[\begin{array}{llllllllllllll}
1&6&2&0&&&&&&&&&&1 \\
1&2& & &&&&&&&&&&
\end{array}\]The figure 16 is erased and replaced by the remainder 4 as follows:

\[\begin{array}{llllllllllllll}
&4&2&0&&&&&&&&&&1 \\
1&2& & &&&&&&&&&&
\end{array}\]The divisor 12 is shifted one place to the right:

\[\begin{array}{llllllllllllll}
&4&2&0&&&&&&&&&&1 \\
&1&2& &&&&&&&&&&
\end{array}\]The process is repeated with the new dividend 42 and divisor 12, giving the new quotient 3 and new remainder 6. The new quotient 3 is placed in the line of the quotient, next to the earlier quotient 1 while the new remainder 6 replaces 42:

\[\begin{array}{llllllllllllllll}
&&&6&0&&&&&&&&&&1&3 \\
&&1&2& &&&&&&&&&&&
\end{array}\]Proceeding as above, we obtain

\[\begin{array}{lllllllllllllllll}
&&&&6&0&&&&&&&&&&1&3 \\
&&&&1&2&&&&&&&&&&&
\end{array}\]Now the divisor 12 is a factor of the new dividend 60 with quotient 5 (and no remainder). The process therefore terminates and we obtain the quotient 135.

\[\begin{array}{lllllllll}
&&1&3&5 \\
\end{array}\]The algorithm involves, apart from the subtleties of the decimal place-value system and the distributive property of numbers, the following algebraic principle of “division with remainder” mentioned in Part 2 (cf. [12]):

Remainder Principle: Given two positive integers $a, b$ with $a \ge b$ , there exist unique non-negative integers $q, r$ such that $a=bq+r$ with $0 \le r < b$ .

The Indian method for long division cleverly uses the decimal expansion of 1620 to break up the number 1620 into separate pieces so that the concrete realisation of the $q$ and $r$ of the Remainder Principle has to be made, perhaps repeatedly, for small (here two-digit) numbers $a$ (here $b=12$ ). For $a=16$ , we have

16=12 \times 1+4 \quad \mathrm{and\quad hence}\quad 1600 = 12 \times 100 +400

so that 1620 gets split into:
$1620=1600+20=12 \times 100 +400+20 =12 \times 100 +420.$

For $a=42$ , we have

\[\begin{align*}42=12 \times 3+6 \quad\mathrm{ and \quad hence}\quad 420&= 12 \times 30+ 60\\
&=12 \times 30+ 12 \times 5.\end{align*}\]

Now the distributive law is used to get

\[\begin{align*}
1620 (=12 \times 100 +420)&= 12 \times 100+12 \times 30+ 12 \times 5\\
&=12 \times (100+30+5)\\
&=12 \times 135.\end{align*}\]

One can see that the modern method follows the same steps. In the modern presentation, the figures 16 (in 1620) and 42 (in 420) are not erased. Instead, the table is extended below for writing the replacements 420 and 60 at appropriate places below 1620. Also, the constant divisor 12 is written separately at a top left position.

The antiquity of the method of long division is not known. We have seen in Part 2 of the article (cf. [12]) that division was known even during the Vedic era. The Śulba-sūtras show clear use of division; they even give examples of division by fractions. But their methods are not known.

The method for long division must have become firmly established in India well before the fifth century CE, for Āryabhaṭa’s ingenious algorithms for computing square roots and cube roots (which are again slight variants of the present methods) involve arrangements akin to that of long division, described earlier. Evidently, Āryabhaṭa (499 CE) and Brahmagupta (628 CE) consider the method to be too well-known and elementary for their treatises.

It is not known how Greeks performed division during Euclid’s time. D.E. Smith writes (cf. [23], p. 133.):

We are quite ignorant as to the way in which the Greeks and Romans performed the operation of division before the Christian Era.

Let $a$ and $b$ be two real numbers (or lengths of two line segments in ancient Greek terminology). If $a>b$ , then one subtracts $b$ from $a$ , then $b$ from $a-b$ if $a-b > b$ , then $b$ from $a-2b$ if $a-2b$ still exceeds $b$ , and so on, till one reaches a stage, say after $q_1$ subtractions, when the remainder $r_1=a-q_1b$ does not exceed $b$ . Now, if $b>r_1$ , then, as before, one keeps on subtracting $r_1$ from $b$ as often as possible, say $q_2$ times, till one reaches the remainder $r_2=b-q_2r_1$ which is $\le r_1$ . The subtractive process is then continued with the lengths $r_1$ and $r_2$ to get $q_3, r_3$ , then with $r_2$ and $r_3$ , and so on. This process was called Antanairesis (reciprocal subtraction) in ancient Greek mathematics.

For integers, a part of the principle of Antanairesis may be abbreviated to the Remainder Principle. This principle is implicit in the proof of Propositions 1–2 of Book VII of Euclid’s Elements.¹⁵ But, as mentioned earlier, there is no algorithm in extant Greek literature for computing the quotient and remainder, other than the method of repeated subtraction described above (which is not practicable if the dividend $a$ in the Remainder Principle is large).

The Indian version of long division was transmitted to the Arabs and occurs in Arabic works from the ninth century onwards. From the Arabs, the method went to Europe where it came to be known as the galley (galea, batello) method (also called the “scratch method”); see ([7], pp. 153–154). The method, very convenient in the earlier system where erasing of figures was easy and desirable, was popular in Europe during the 15th–18th centuries.

To appreciate how much admiration the early Indian invention of the long division method deserves, we point out that the operation of division was regarded as a difficult operation by European scholars till as late as the fifteenth century. L. Pacioli remarked in 1494 CE (cf. [23], p. 132):

if a man can divide well, everything else is easy, for all the rest is involved therein.

An algorithm for squaring

Squaring is a special case of multiplication; thus any multiplication algorithm can be used to compute a square as well. But Indic mathematicians make repeated and judicious use of the basic algebra formula

$(a+b)^2=a^2+2ab+b^2,$ and some of the features of the decimal place value system with zero, to construct ingenious algorithms for squaring a number which are “more efficient’’ than the standard multiplication algorithm applied to two equal numbers.

We highlight one ancient method of squaring, which is described by Brahmagupta, Bhāskara I and later authors. Though Āryabhaṭa does not describe the method, Datta and Singh have pointed out that his square-root algorithm shows that this method too must have been known to him (cf. [7], p. 156). In his commentary on Āryabhaṭīya, Bhāskara I cites a sūtra describing the method (see [21], p. 49.) For other arrangements, see ([7], pp. 155–162).

We explain the method through the squaring of the three-digit number 125. Note that the usual multiplication algorithm will involve nine small multiplications. In the method for squaring, the number of operations is minimised through the implicit use of the identity

\[\begin{align*}
&(100a+10b+c)^2\\
&\quad =c^2+2c(10a+b) \times 10+ b^2 \times 10^2+ 2ab \times 10^3\\
&\hskip 1.75in +a^2 \times 10^4,\end{align*}\]which requires only the doubling of two multiplications and three squares of digits. Thus, $125^2$ is split as

\[\begin{align*}
125^2=(5+120)^2&=25+ (2\times 5)\times (12 \times 10)+120^2 \\
&=1225+(2+10)^2 \times 10^2\\
&= 1225 + 4 \times 10^2+4 \times 10^3+1 \times 10^4 \\
&=1625+4 \times 10^3+1 \times 10^4\\
&= 5625+1 \times 10^4=15625.\end{align*}\]The above remainder principle is efficiently arranged algorithmically, using the decimal place-value system and the erasing facility then in vogue, into the following steps. To start with, the square of the last digit ( $5^2$ ) is placed above the given number.

\[\begin{array}{lll}
&2&5\\
1&2&5
\end{array}\]The last digit of the given number is erased (to get $12 \times 10$ ), and below this number, the double of the last digit ( $2 \times 5$ ) is placed.

\[\begin{array}{lll}
&2&5\\
1&2& \\
&1&0
\end{array}\]To the number at the top, is added the product of the second number $12 \times 10$ and the third number $10$ , the second number is shifted by one place (to represent $(2+10) \times 10^2$ ), and the third number (which has no further role) is then deleted.

\[\begin{array}{llll}
1&2&2&5\\
1&2& &
\end{array}\]As before, the square of the second digit of 12 is added to the top (i.e., $2^2 \times 10^2$ gets added), its double $2 \times 2=4$ is placed below, and the second digit itself is deleted.

\[\begin{array}{llll}
1&6&2&5\\
1& & & \\
&4& &
\end{array}\]To the number at the top, is added the product of the second number ( $1 \times 10^3$ ) and the third number (4); the second number is shifted one place (to represent $1 \times 10^4$ ) and the third number (which has no further role) is then deleted.

\[\begin{array}{lllll}
&5&6&2&5\\
1& & & &
\end{array}\]Adding the square of 1 at the top (i.e., adding $1^2 \times 10^4$ ), we get the required number $15625$ .

Likewise, there are algorithms (cf. [7], 162–169) for cubing a number using the formula

(a+b)^3=a^3+3a^2b+3ab^2+b^3.

An algorithm for square root

The standard Indian algorithms for extraction of square roots involve, apart from a mastery of the polynomial aspect of the decimal expansion (itself an algebraic principle), the place-value and the zero, the repeated and ingenious use of the formula

$(a+b)^2=a^2+2ab+b^2;$ the still more intricate algorithms for cube roots involve, apart from the above ideas of decimal notation, the formula

(a+b)^3=a^3+3a^2b+3ab^2+b^3.

We shall highlight the algorithm of Āryabhaṭa (499 CE) for extraction of the square root. For more details, see ([7], pp. 169–180).

To understand the steps, we consider a number $N$ whose square root is a 3-digit number with digits $x, y, z$ . Then

\[\begin{align*}
N&=(100x+10y+z)^2\\
& = (10x+y)^2 \times 10^2 +2(10x+y)z \times 10+z^2,\end{align*}\]

that is,

\[\begin{equation}\label{eq1}
N=x^2 \times 10^4+2xy \times 10^3+y^2 \times 10^2 +
2(10x+y)z \times 10+ z^2.
\end{equation}\]

Āryabhaṭa’s algorithm first seeks to identify the digit $x$ for which $10^4x^2$ is closest to $N$ . The number $N$ is of the form $N=10^4a+b$ where $b$ is the number formed by the last four digits of $N$ , and $a$ comprises the initial one or two digits of $N$ (according as $N$ comprises 5 or 6 digits). The algorithm therefore considers $x$ for which $x^2$ is closest to $a$ (but $x^2 <a$ ).

Then it considers the difference $N_1=N-10^4x^2$ , and finds $y$ for which $2xy \times 10^3$ is closest to $N_1$ , i.e., identifies the quotient $y$ when $N_1$ is divided by $2x \times 10^3$ . Again, $N_1$ is of the form $N_1=10^3a_1+b_1$ , where $b_1$ is the number formed by the last three digits of $N_1$ ; and it suffices to consider the division of $a_1$ by $2x$ .

Having obtained both $x$ and $y$ , the algorithm then considers first the difference $N_2= N_1-2xy \times 10^3$ and then the difference $N_3=N_2-y^2 \times 10^2 = 10a_3+b_3$ , where $b_3$ is the last digit of $N_3$ .

It remains to find $z$ for which $2(10x+y)z \times 10$ is closest to $N_3$ . As before, it suffices to consider the division of $a_3$ by $2(10x+y)$ and take $z$ to be the quotient. The rest fall in place.

For a numerical example, consider the number $N=54756$ . Its square root will be a 3-digit number $100x+10y+z$ . Here $N=5 \times 10^4+4756$ , and we look for $x$ for which $x^2$ is closest to $5$ . Clearly, $x=2$ .

Next, we consider the difference $N_1=N-10^4x^2=54756-40000=14756=14 \times 10^3+756$ . Now $y$ is the quotient obtained when $14$ is divided by $2x=4$ , i.e., $y=3$ .

Next, we obtain $N_2=N_1-2xy \times 10^3=14756-12000= 2756$
and then $N_3=N_2-y^2 \times 10^2=2756-900=1856=185 \times 10+6$ .

We have $2(10x+y)=46$ . Therefore, $z$ is the quotient obtained when $46$ divides $185$ , i.e., $z=4$ . The difference $N_4=N_3-2(10x+y)z \times 10 =1856-1840=16$ . Subtracting $z^2$ from $N_4$ , we obtain $16-4^2=0$ .

The algorithm is expressed very concisely by Āryabhaṭa in Āryabhaṭīya as follows (cf. [7], p. 170); the meanings are elaborated in the commentaries:

Always divide the even place by twice the square-root [up to the preceding place]; after having subtracted from the odd place the square [of the quotient], the quotient put down at the next place [in the line of the root] gives the root.

Marking the odd and even places by vertical and horizontal lines, the algorithm is arranged as follows. The roots are recorded in a separate column on the right-hand side.

\[\begin{array}{llllllllllllllllllll}
&&&&&&&|&-&|&-&|&&&&&&&& {\rm Root}\\
&&&&&&&5&4&7&5&6&&&&&&&& \\
\mathrm{ Subtract\ square} &&&&&&&4& & & & &&&&&&&& 2\\
\hline
\mathrm{Divide \ by \ twice \ the \ root}&&&&&&4)&1&4&(3&&&&&&&&&&23\\
&&&&&& &1&2& &&&&&&&&&& \\
\hline
&&&&&& &&2&7 &&&&&&&&&& \\
\mathrm{Subtract \ square \ of \ quotient}&&&&&& &&&9 &&&&&&&&&& \\
\hline
\mathrm{Divide \ by \ twice \ the \ root}&&&&&&&46)&1&8&5&(4&&&&&&&&234\\
&&&&&&& &1&8&4& &&&&&&&& \\
\hline
&&&&&&& &&&1&6 &&&&&&&& \\
\mathrm{Subtract \ square \ of \ quotient}&&&&&&& &&&1&6 &&&&&&&& \\
\hline
\end{array}\]The above method was used subsequently by Indian, Arab and European mathematicians for more than a thousand years. The method occurs in the work of P. Cataneo (1546 CE); see ([23], p. 147).

The modern method involves a slight contraction of the above method. To illustrate the modification, we return to the earlier example and notation of $N=(100x+10y+z)^2$ . The reader may have noticed that after computing $x$ , and defining $N_1=N-10^4x^2$ , then finding $y$ for which $2xy \times 10^3$ is closest to $N_1$ , one makes two subtractions in two successive steps: first, one performs the subtraction $N_1-2xy \times 10^3=N_2$ , and then another subtraction $N_2-y^2 \times 10^2=N_3$ , to obtain $N_3= N_1-2xy \times 10^3 -y^2 \times 10^2$ . In the modern version, one makes a single subtraction $N_3=N_1-(20x+y)y \times 10^2$ and the table is arranged accordingly. In the numerical example, having obtained $N_1=14756$ , one directly subtracts from it $43 \times 3 \times 10^2$ (rather than the earlier practice of first subtracting $4 \times 3 \times 10^3=12000$ and then subtracting $3^2 \times 10^2=900$ ). A similar contraction is made in subsequent steps too. Thus, the modern algorithm follows the following version of equation (\ref{eq1}):

\[\begin{equation}\label{eq2}
N=x^2 \times 10^4+(20x +y)y \times 10^2+(2(100x+10y)+z)z.
\end{equation}\]

The arrangement takes the following form:

\[\begin{array}{lllllllllllllllll}
&&&&\cdot&&\cdot&&\cdot&&&&&&&&\\
&&&&5&4&7&5&6&&&&&|& 2&3&4 \\
&&&&4& & & & &&&&&|&&& \\
\hline
&4&3&|&1&4&7& & &&&&&&&&\\
& &3&|&1&2&9& & &&&&&&&& \\
\hline
4&6&4&|& &1&8&5&6 &&&&&&&&\\
&&4&|& &1&8&5&6 &&&&&&&& \\
\hline
\end{array}\]The modern method occurs in Yuktibhāṣā (c. 1530) of Jyeṣṭhadeva;¹⁶ its precise antiquity in Indian mathematics is not known. In Europe, it occurs in the work of P. Cataldi (1613 CE); see ([23], p. 147).

The algorithms for finding square roots are usually learnt mechanically. A little reflection as to why they work reveals the richness and intricacy of algebraic thinking that went into the invention of the original algorithm.

Algebraic formulae in some special arithmetic methods

Indic mathematicians mention various algebraic identities which provide quick methods for multiplication, squaring or cubing in certain cases.

For instance, the iṣṭa-guṇana method of multiplication described by Brahmagupta ([7] p. 135) involves
a straightforward application of the algebraic formula

$ab=a(b-c)+ac {\rm or } ab=a(b+c)-ac$ for a judiciously chosen digit $c$ ; for instance
$135 \times 12= 135 \times (12-2) +135 \times 2 = 1350+270=1620;$

$135 \times 8=135 \times (8+2)-135 \times 2=1350-270=1080.$ Bhāskara II explains in Līlāvatī ([7], p. 149):

Multiply by the multiplicator diminished or increased by an assumed number, adding or subtracting (respectively) the product of the multiplicand and the assumed number.

Again, the identity
$n^2=(n-a)(n+a)+a^2$ can be used to quickly compute a square like

$15^2=10 \times 20+25=225.$ Brahmagupta (628 CE) writes (cf. [7], p. 160):

The product of the sum and the difference of the number (to be squared) and an assumed number plus the square of the assumed number give the square.

Śrīdhara (750 CE) mentions the identity

$n^2=1+3+5+\cdots+(2n-1)$ as a method for determining the square of a number (see [7], p. 160) and the identity

$n^3= {\displaystyle \sum_{1 \le r \le n}} \{3r(r-1)+1\}$ for cubing a number (cf. [7], p. 167); Mahāvīra (850 CE) mentions the formula

$n^3=n(n+a)(n-a)+a^2(n-a)+a^3$ for cubing (cf. [7], p. 167).

The Indian algorithm for long division is, in essence, the same as the modern method. But, for certain special cases, the quotient and remainder of a division can be readily obtained using an interesting algebraic observation of Brahmagupta. His rule can be symbolically put as the algebraic identity

$\frac{a}{b}=\frac{a}{b+h}+{\frac{a}{b+h}}{ \frac{h}{b}}$ which occurs in Chapter 12 (verse 57) of his treatise Brāhma-Sphuṭa-Siddhānta (cf. [19]). The chapter has been published with English translation in [15] (the rule for division occurs in p. 163).

The number $h$ (positive or negative) is to be chosen judiciously, by inspection, so that $b+h$ becomes a factor of $a$ . (Strictly speaking, the denominator on the right hand side has been described as $b \pm c$ where $c$ is a natural number.) For instance, using the above rule, one has:

\[\begin{align*}\frac{1920}{93} &=\frac{1920}{96}+ \frac{1920}{96} \times \frac{3}{96}\\ &=20 + 20 \times \frac{3}{93}=20+\frac{60}{93} \left(=20\frac{20}{31}\right)\\
\frac{9999}{97}&= \frac{9999}{99} + \frac{9999}{99} \times \frac{2}{97}\\
&= 101 + \frac{202}{97}\\
&=101+\frac{202}{101} + \frac{202}{101} \times \frac{4}{97}\\
&=103+\frac{8}{97}.\end{align*}\]

Thus, one can immediately see that when 1920 is divided by 93, the quotient is 20 and the remainder is 60; when 9999 is divided by 97, the quotient is 103 and the remainder is 8.

Brahmagupta has separate chapters for arithmetic and algebra (Chapters 12 and 18 respectively) in his treatise Brāhma-Sphuṭa-Siddhānta (cf. [19]). Being a rule for division, the above principle is naturally mentioned in the chapter on arithmetic. Consequently, the algebraic skill involved in the rule tends to get overlooked by historians.

Transmission of Indian Decimal Notation and Arithmetic

Early references to the Indian numerals

The Indian numerals have been found in tenth-century manuscripts of the geometry work of the Latin philosopher Boethius (c. 500 CE). That the fame of the Indian decimal notation had reached the banks of the Euphrates by the early seventh century can be seen in a passing reference by the Syrian astronomer-monk Severus Sebokht (662 CE):

I shall not now speak of the knowledge of the Hindus, … of their subtle discoveries in the science of astronomy—discoveries even more ingenious than those of the Greeks and Babylonians—of their rational system of mathematics, or of their methods of calculation which no words can praise strongly enough—I mean the system using nine symbols. (Quoted in [2], p. vi.)

A manuscript Codex Vigilanus of 976 CE¹⁷ by Vigila, a monk in a Spanish monastery, also records a similar appreciation:

The Indians have an extremely subtle intelligence, and when it comes to arithmetic, geometry and other such advanced disciplines, other ideas must make way for theirs. The best proof of this is the nine figures with which they represent each number no matter how large. (Quoted in [16], p. 362)

The role of Al-Khwārizmī

Several Arabic texts were written on decimal notation and Indian arithmetic. The Persian mathematician Al-Khwārizmī (780–850 CE), a scholar at the “House of Baghdad”, had developed a great liking for Indian mathematics, learnt Sanskrit, and wrote Arabic treatises on arithmetic and algebra. His arithmetic treatise, written around 820 CE, describes the features of the decimal notation—the numerals 1 to 9 and 0, and the place-value principle. It then explains the Indian procedures for carrying out the four fundamental mathematical operations—addition, subtraction, multiplication, and division—using the Indian decimal notation. The numerals were explicitly called hindisah (Indian) by Al-Nadīm (c. 987 CE).

The original text of Al-Khwārizmī is lost, but its Latin translation Liber Algorismi de numero Indorum (The Book of al-Khwarizmi on Indian numerals), composed in early 12th century, still survives. This Latin translation triggered off several adaptations and played a significant role in the spread of the decimal place-value notation and Indian arithmetic methods throughout medieval Europe. An account of the spread of Indian numerals is presented in [14].

The following passage, quoted in (cf. [1], p.168), conveys some flavour of Al-Khwārizmī’s exposition on the place-value principle with nine numerals and zero:

When I saw that Indians composed out of IX letters any number due to the position established by them, I desired to discover, God willing, what becomes of those letters to make it easier for the student. … Thus they created … IX letters, … The beginning of the order is to the right side of the writer, and this will be the first of them consisting of unities. If instead of unity they wrote X, it stood in the second digit, and their figure was that of unity, they needed a figure of tens similar to the figure of unity so that it becomes known that this was X. Thus they put before it one digit and wrote in it a small circle `o’, so that it would indicate that the place of unity is vacant.

Decimal notation in Europe

Lady Arithmetic is seen giving her judgement in favour of the arithmetician (to our left) working with Indian numerals and the zero, with the numerals also adorning her dress, in this wood-block engraving from Margarita philosophica (1503), by Gregor Reisch. — Lady Arithmetic is seen giving her judgement in favour of the arithmetician (to our left) working with Indian numerals and the zero, with the numerals also adorning her dress, in this wood-block engraving from *Margarita philosophica* (1503), by Gregor Reisch. Wikimedia Commons / Houghton Library, Harvard University

A vigorous effort to bring the Indian decimal system into common use in the West was made by Leonardo Fibonacci of Pisa (1180–1240). Acquainted with various systems of calculations, Fibonacci found all computational systems deficient compared to the Indian system: “quasi errorem computavi respectu modi indorum” (cf. [7], p. 95). His influential work Liber Abaci (Book of Computations) published in 1202 (and rewritten in 1228), based entirely on Indian numerals, prepared the ground for the spread and adoption of the Indian decimal system and computation methods. Several subsequent thinkers in Europe too advocated the system. A wood-block engraving Margarita Philosophica of Gregor Reisch (1503) contrasts the old and new systems by depicting a gloomy Pythagoras toiling with a counting board, a cheerful and serene Boethius comfortable with his computations (using the decimal system), and Lady Arithmetic giving her smiling approval of Boethius, with the Indian numerals adorning her dress. In another illustration by the Nuremberg artist Hans Sebald Beham (1550), a Winged Arithmetic is shown turning her back on the counting board, and pointing emphatically to a tablet with the new Indian numerals.

However, Roman numerals had become so deep-rooted in Europe, that it took around five centuries for the decimal notation to gain universal acceptance. Initially the new system “was too advanced for the merchants and too novel for the universities” (cf. [7], p. 95) and there was a determined resistance against it from the abacists.

With the passage of time, more people began to realise its importance and decimal notation eventually came into common use in the seventeenth century.

Impact on modern civilization

The Indian decimal notation gradually replaced the Roman system prevalent in Europe. By the seventeenth century of the Common Era, it became the standard system of writing natural numbers.

To see an example of the economy achieved by the Indian system, note that while a number like eight hundred and eighty-eight (888) needs only three digits in decimal notation, it requires twelve symbols (DCCCLXXXVIII) in the Roman system. And it is practically impossible to represent the number “one million” in the Roman system of notation—one has to write M a thousand times!

In Part 2 of the article, we had already discussed some examples of the impact made by the decimal notation (more generally, the decimal expansion), especially in mathematics and astronomy. It made an enormous simplification of arithmetic, facilitated computations with large numbers in mathematics and the sciences, and influenced the algebraic thinking of mathematicians.

The adoption of the decimal notation and Indian arithmetic methods were among the major factors for the commercial, mathematical and scientific renaissance in Europe. The pivotal role played by the decimal notation in shaping modern civilization has been widely acknowledged by scholars and thinkers. Due to their simplicity, children across the world now learn the fundamental arithmetic methods at an early stage. The facility with basic mathematics has enabled a vast majority of people to have access to a considerable body of scientific and technical knowledge which would otherwise have remained restricted to a gifted few.

Etymology of algorithm and algebra

We have seen that algebraic insights are involved in the invention of the ancient Indian “algorithms” for arithmetic operations. Ironically, both the words “algorithm” and “algebra” are associated with Al-Khwārizmī.

As the Latin translation was made from the Arabic treatise of Al-Khwārizmī, the ten numerals (digits) came to be referred to as “Arabic” numerals, although the signs for the numerals had been adapted from the Indian number-symbols for digits. Even today, many scholars even in India retain the impression that our present decimal numerals are of Arabic origin!

Further, Al-Khwārizmī’s name got so closely associated in Europe with the new arithmetic methods based on the Indian decimal system, that the word algorismus (later algorithmus), derived from Algoritmi, the Latin form of Al-Khwārizmī, was given to any Indian computational method. The word algorismus got contracted to algorism which later became algorithm. Thus, etymologically, the word “algorithm” denotes any Indian method for the fundamental arithmetic operations. Later, the term “algorithm” would denote any well-defined step-by-step logical procedure (a computer-implementable instruction in modern terminology) to perform a computation or to solve a class of specific problems.

The word “algebra” comes from the Arabic al-jabr which occurs in the title of an elementary treatise on the subject, again written by Al-Khwārizmī. Thus, although an astonishingly sophisticated chapter on algebra had been composed by the Indian mathematician Brahmagupta (628 CE) two centuries earlier (cf. [10]), the name of Al-Khwārizmī is still associated with the foundation of the subject, thanks to its Arabic name.

So strong has been this association, that too often one comes across highly educated Indians who would severely frown at any suggestion that Indics had already made brilliant developments in the subject centuries before the Persians and Arabs!

Appendix: Āryabhaṭa’s Alphabetical Coding of Large Numbers

Texts in ancient Indian science were often composed in verses and, as emphasised in the Introduction, brevity was considered a desirable attribute for a scientific treatise. The richness and flexibility of the Sanskrit language were used effectively to devise terms in mathematics, astronomy and other sciences which were not only suggestive of the underlying concepts but also enabled important scientific statements to be encapsulated briefly in verse form.

Authors of texts on astronomy had to devise efficient coding of large numbers while preserving the metres of the verses. Āryabhaṭa blended his linguistic and mathematical skills to make ingenious use of the phonetic Sanskrit alphabet for representing large numbers with only a few letters. His scheme was, in essence, akin to a place-value system with base hundred: the consonants of the Sanskrit alphabet denoting numerals and the vowels attached to the consonants indicating their respective place-values (in the sense of “powers of 100” and not literal positions).

We now explain his scheme. We first discuss his representation of numbers up to $10^{18}$ . Let the vowels of the Sanskrit alphabet (in usual order) denote the powers $100^n(=10^{2n})$ ( $0 \leq n \leq 8$ )—the same value is to be attached to a short vowel and its corresponding long vowel. This step must have been taken in order to avoid confusion during oral transmission. Thus the assignment:

a = ā = $1$ ; i = ī = $10^2$ ; u = ū = $10^4$ ; ṛ = $10^6$ ; ḷ = $10^8$ ; e = $10^{10}$ ; o = $10^{12}$ ; ai = $10^{14}$ ; au = $10^{16}$ .

Assign to the twenty-five “varga” (classed) consonants from k to m the numbers (rather numerals) from 1 to 25:

\[ \begin{align*}
&k = 1; & kh = 2;&
&g = 3; && gh = 4; &&\dot{\mbox{n}} = 5; \\
&c = 6; &ch = 7;
&&j = 8; &&jh = 9; &&ñ = 10; \\
&ṭ = 11; &ṭh = 12; &&ḍ = 13;
&&ḍh = 14; &&ṇ = 15; \\
&t = 16; &th = 17; &&d = 18; &&dh = 19; &&n = 20; \\
&p = 21; &ph = 22; &&b = 23; &&bh = 24; &&m = 25.
\end{align*} \]The above 25 letters are called vargākṣara as they are classified into 5 vargas (classes)—k-varga, c-varga, etc, each with 5 letters.

The syllable formed by joining a vowel with value $10^{2x}$ to a varga letter with value $z$ is assigned the number $z \times 10^{2x}$ . Thus, khu = $2 \times 10^4$ ; ghṛ = $4 \times 10^6$ ; ṅi} = $5 \times 10^2$ ; ṇḷ = $15 \times 10^8$ ; bu = $23 \times 10^4$ ; etc. This already enables one to represent any number of the form $z \times 10^{2x}$ for $1 \leq z \leq 25$ and $0 \leq x \leq 8$ with just a single alphabetical symbol. In particular, any number of the form $z \times 10^{2x}$ ( $1 \leq z \leq 9$ ) or $q \times 10^{2x+1}$ ( $1 \leq q \leq 2$ ), where $0 \leq x \leq 8$ , can be represented by a single syllable.

Now use the eight “avarga” (unclassed) consonants to represent a digit $q$ , $3 \leq q \leq 10$ , occurring in a number of the form $q \times 10^{2x+1}$ ( $0 \leq x \leq 8$ ), by assigning values:

y = $30$ ; r = $40$ ; l = $50$ ; v = $60$ ; ś = $70$ ; ṣ = $80$ ; s = $90$ ; h = $100$ .

Thus yu = $30 \times 10^4$ ( $=3 \times 10^5$ ); śi = $70 \times 10^2$ ( $=7 \times 10^3$ ); ṣṛ = $80 \times 10^6$ ( $= 8 \times 10^7$ ).

The largest number covered so far is hau $=10^{18}$ . For numbers larger than $10^{18}$ , use the nine distinct vowels suitably, as above, in blocks of 18. (For instance, vowels with anusvāra (ṁ ) could be used to denote the places between $10^{18}$ and $10^{36}$ .)

While each of the numbers from 1 to 25 and the first ten multiples of 10 can be represented by a single letter, all other numbers within 100 can be represented by two letters. Thus an $n$ -digit number gets represented verbally using at most $n$ alphabetical characters; usually much fewer symbols are needed. For instance, the two-syllabic “ khyughṛ ” denotes the 7-digit number $(2+30) \times 10^4 + 4 \times 10^6 = 432 \times 10^4$ ; while the five-syllabic “ ṅiśibuṇḷṣkhṛ ” denotes the 10-digit number $5 \times 10^2 + 70 \times 10^2 + 23 \times 10^4 + 15 \times 10^8 + (80+2) \times 10^6 = 1582237500$ .

Since the role of “place-values” in this scheme is played by vowels and not by actual “places”, a permutation of positions of the syllables does not alter the numbers (unlike the decimal place-value notation)—for instance, both mani and nima represent the same number 2025. Thus, while Āryabhaṭa’s scheme extracts that aspect of the place-value notation (the idea of expressing numbers through powers of $x$ , by using $x$ numerals) which results in conciseness, it bypasses the rigidity of position inherent in a place-value scheme. Also note that numbers can have various representations—the number 30 could be denoted by the first avarga letter ya as well as by a combination of the two varga letters ṅa} and ma [=5+25]; 43 by rāga [40+3] or by naba [20+23]. For the purpose of versification, such flexibility and scope for variations were desirable.

In Gītikā 2, the first technical verse in Āryabhaṭīya (Gītikā 1 being an Invocation), Āryabhaṭa gives a strenuously terse description of the system in the form of a rule giving a correspondence between his centesimal alphabetical notation and the decimal place value notation:

vargākṣarāṇivarge’varge’vargākṣarāṇikāt nṁau yaḥ
khadvinavake svarā nava varge’varge navāntyavarge vā

[varga : class, classed, block, square; akṣara : letter; kha : zero, void, hollow, sky; dvi : two, double; nava : nine; svara : vowel; antya : last, following; vā : or, and, as, like.]

The varga letters, beginning with k, [are to be used] in the varga [places], the avarga letters in the avarga [places]; [in such a way that] ṅa} plus ma equals ya. The nine vowels [are to mark the eighteen] zeros formed by the nine pairs of varga and avarga [places]. A like [procedure is to be repeated using] nine [vowel-symbols] for the subsequent blocks [of eighteen varga, avarga places].

The readers would have noticed a double entendre in the verse. In Sanskrit, the two terms varga and avarga denote, respectively, classed and unclassed (consonants) as well as perfect squares and non-squares. Here the decimal places are also being called varga or avarga depending on whether the underlying power of ten is a perfect square $10^{2x}$ or is a non-square $10^{2x+1}$ . In each block of 18 decimal places, each of the nine distinct vowels is associated with two consecutive decimal places—one varga ( $10^{2x}$ ) and one avarga ( $10^{2x+1}$ ): a is associated with the unit’s place and the ten’s place, i with the hundred’s place and the thousand’s place, and so on. The varga consonants are meant for varga places; the avarga for avarga places. For a number like 2500, denoted mi, one has to conceive the varga numeral m for 25 as being attached to the varga place $10^2$ while for a number like 9000, denoted si, one sees the avarga digit s for 9 as being put in the avarga place $10^3$ .

Note the explicit use of a term for zero (kha) to denote a notational place—commentators like Bhāskara I and Sūryadeva have clarified that kha denotes śūnya and that khadvinavake refers to the eighteen [places] marked by zeros. Thus, unless a specific consonant-numeral occupies a certain place ( $10^x$ ) in a number, by default, the place gets marked by a zero. We see here the occurrence of the concept of the mathematical zero as a place-marker in a place-value system.

Āryabhaṭa does not mention (or use) the alphabetical notation in the Gaṇita (mathematics) section. The role of this innovation was consciously restricted to the concise representation of large astronomical numbers in the Gītikā section. The notation was not suitable (and was not meant to be used) for performing arithmetic operations. Efficient systems, based on the decimal place value and zero, were already in vogue in India for that purpose.

It appears that Āryabhaṭa’s system is not used by subsequent astronomers (except during commentaries on his relevant verses). Some of the words become too complicated for pronunciation (as the readers would have noticed). Besides, the system does not provide sufficient variety for sustained versification. Later Indian astronomers either continue with the standard verbal decimal terminology (now prevalent) and the Bhūtasaṅkhyā or adopt the alphabetical decimal notation kaṭapayādi. The originator of kaṭapayādi is not known; but it is presumed that Āryabhaṭa knew the kaṭapayādi system—his commentator Śūryadeva remarks that the letters kāt has been used by Āryabhaṭa (in Gītikā 2) to distinguish his method from the kaṭapayādi (cf. [7], p. 71.)

The master stroke of Āryabhaṭa’s consonant-numeral system lies in the judicious application of the place-value idea—the successful imparting of high place-values to the vowels (which, in the Sanskrit script, tend to merge with the consonants)—to achieve an extreme compression in the verbal depictions of numbers (making it much shorter than even the decimal place-value notation). Considering the central importance attached to Āryabhaṭīya, and the use of the scheme in crucial verses at the very outset of this influential treatise, the understanding of Āryabhaṭa’s coding in Gītikā 2 must have been indispensable for all serious astronomy scholars. Subsequent astronomer-mathematicians would have imbibed its subtleties and Āryabhaṭa’s verse would thus have played a significant role not only in making the ideas embedded in the place-value principle take deeper roots in the Indian mind, but also in pushing it towards an organised development of symbolic algebra. By the time of his brilliant successor Brahmagupta (628 CE), symbolism had become firmly established in Indian mathematics and algebra began to flourish as a distinct discipline. It is significant that Brahmagupta mentions varṇa—which means “letters of the alphabet” as well as “colour”—as symbols for the unknowns. In later times, unknown variables are explicitly named by distinct colours like kālaka (black), nīlaka (blue), pītaka (yellow), lohita (red) and so on, and symbolically represent the first letter of the respective colour-names—kā, nī, pī, lo, etc. The systematic use of letters of the alphabet to denote unknown variables is a great step for the rapid progress of mathematics.

Thus, even though Āryabhaṭa’s notation itself was not adopted, it is likely to have had a profound impact on the mathematical thought in Classical India. $\blacksquare$

References

A.K. Bag and S.R. Sarma (eds.), The Concept of Śūnya, Indira Gandhi National Centre for the Arts, Indian National Science Academy and Aryan Books International, New Delhi, (2003).
A.L. Basham, The Wonder that was India, Sidgwick and Jackson, London, (1954).
N. Bourbaki, Elements of the History of Mathematics, Masson, Paris; reprinted by Springer-Verlag (1994).
B. Datta, Early literary evidence of the use of the zero in India. American Mathematical Monthly, 33, 449–454, 1926; $\&$ 38, 566–572, (1931).
B. Datta, The Bakhshali Mathematics. Bulletin of the Calcutta Mathematical Society, 21, 1–60, (1929(i)).
B. Datta, The Jaina School of Mathematics. Bulletin of the Calcutta Mathematical Society, 21, 115–145, (1929(ii)).
B. Datta and A.N. Singh, History of Hindu Mathematics Part I, Motilal Banarasidass, (1935); reprinted by Asia Publishing House (1962); $\&$ Bharatiya Kala Prakashan (2004).
P.P. Divakaran, The Bakhshali Manuscript and the Indian Zero. Bhāvanā, 2(4), 13–21, (2018).
A.K. Dutta, Decimal System in India, Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures (Ed. H. Selin), Springer, (2016).
A.K. Dutta, The bhāvanā in Mathematics, Bhāvanā, 1(1), 13–19, (2017).
A.K. Dutta, Mathematics in Ancient India Part 1: Geometry in Vedic and Sūtra Literature, Bhāvanā, 6(1), 23–33, (2022).
A.K. Dutta, Mathematics in Ancient India Part 2: Computational Mathematics in Vedic and Sūtra Literature, Bhāvanā, 6(2), 43–61, (2022).
S. Ganguli, The Indian Origin of the Modern Place-Value Arithmetical Notation. American Mathematical Monthly, 39(5), 251–256, $\&$ 39(7), 389–393, (1932); 40(1), 25–31, $\&$ 40(3),154–157, 1933).
R.C. Gupta, Spread and Triumph of Indian Numerals. Indian Journal of History of Science 18(1), 23–38, (1983).
V.D. Heroor (ed.), Brahmaguptagaṇitam, Chinmaya International Foundation Shodha Sansthan, Ernakulam, (2014).
G. Ifrah, The Universal History of Numbers, John Wiley and Sons, (2000).
P. Mukhopadhyay, Much ado about nothing, Ramakrishna Mission Residential College Narendrapur, West Bengal, (2021).
K. Plofker, Mathematics in India, Princeton University Press, (2009); reprinted by Hindustan Book Agency, (2012).
R.S. Sharma (ed.), Brāhma Sphuṭa Siddhānta of Brahmagupta, Indian Institute of Astronomical Research, (1966).
K.S. Shukla and K.V. Sarma (ed.), Āryabhaṭīya of Āryabhaṭa, Indian National Science Academy, (1976).
K.S. Shukla (ed.), Āryabhaṭīya of Āryabhaṭa with the commentary of Bhāskara I and Someśvara. Indian National Science Academy, (1976).
A.K. Singh, Zero in Early Indian Inscriptions, Purātattva, 38, 121–126, (2007).
D.E. Smith, History of Mathematics, Vol II, originally 1925; Dover (1958).

Footnotes

In fact, 60 is a “superior highly composite number”. As the definition of this concept is technical, we omit it. ↩
The original statement of P.S. Laplace occurs in his Exposition du Systeme du Monde, Volume II, Paris, 1799, p. 294. The first English translation of the book appears to have been made by H.H. Harte: The System of the World by M. Le Marquis de Laplace, Volume II, Dublin, 1830; the quoted statement appears there on pp. 222–223. We have quoted a lucid English version of the essence of Laplace’s statement which occurs in Number: The Language of Science by T. Dantzig, Macmillan NY, 1930, p. 19. ↩
G.B. Halsted, On the foundation and technique of Arithmetic, Chicago (1912), p.20. ↩
L. Hogben, Mathematics for the Million, W.W. Norton $\&$ Co., New York, Fourth Edition (1968), p. 39. ↩
van der Waerden, Science Awakening, P. Noordhoff Ltd., Groningen, Holland (1954), p. 56. ↩
M.D. Pandit, Zero in Pāṇini, University of Poona (1990). ↩
http://www.ams.org/publicoutreach/feature-column/fc-2018-06. ↩
For further discussion see A.V. Raman: The Katapayadi Formula and the Modern Hashing Technique, IEEE Annals of the History of Computing, 19 (1997) 49–52. ↩
A group of eminent mathematicians who authored a series of meticulously rigorous and systematic texts under this common name. ↩
For further discussion, see A. Coomaraswamy, “Kha and Other Words denoting `Zero’ in connection with the metaphysics of space’’, Bulletin of the School of Oriental Studies, Univ. London, 7(3), (1934), pp. 487–497. Reprinted in `Perceptions of the Veda’ IGNCA Publications, New Delhi (2000). ↩
The verse from Sārīraka-Bhāṣya, with translation, is quoted in [13], p. 255. ↩
Dated between 385 and 465 CE in M. Singh, Subandhu, Sahitya Akademi (1993), p. 11. ↩
Translated in Swami Nikhilananda, The Gospel of Sri Ramakrishna, Sri Ramakrishna Math, Madras, 4th edition (1964), p. 643. A slightly edited translation of Sri Ramakrishna Kathamrita (in Bengali) by M. Gupta. ↩
The original Bengali statement is quoted in Nolini Kanto Sarkar, Asa Jaoar Majkhane Part 2 (in Bengali), Mitra and Ghosh (1985), p.6. ↩
T.L. Heath (ed), Euclid, The Thirteen Books of The Elements Vol. 2, 2nd edition, Dover, pp. 296–299; originally Cambridge University Press (1925). ↩
Gaṇita-Yukti-Bhāṣā of Jyeṣṭhadeva, edited with English translation by K.V. Sarma, with explanatory notes by K. Ramasubramanian, M.D. Srinivas and M.S. Sriram, Vol. 1, Hindustan Book Agency (2008), reprinted by Springer (2009); p.19. ↩
Now kept in the Library of El Escorial, a monastery, about 50 Km. away from Madrid. ↩

Amartya Kumar Dutta is at the Stat-Math Unit of the Indian Statistical Institute, Kolkata. He is a corresponding editor of Bhāvanā.