# History of Hindu Mathematics: Book 1, Chapter I, Section 12: The Zero Symbol

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Page 75

### 12. The Zero Symbol

Earliest Use. The zero symbol was used in metrics by Piṅgala (before 200 B.C.) in his Chandaḥ-sûtra. He gives the solution of the problem of finding the total number of arrangements of two things in n places, repetitions being allowed. The two things considered are Page 76 the two kinds of syllables "long" and "short", denoted by l and g respectively. To find the number of arrangements of long and short syllables in a metre containing n syllables, Piṅgala gives the rule in short aphorisms:

"(Place) two when halved;"1 "when unity is subtracted then (place) zero;"2 "multiply by two when zero;"3 "square when halved."4

The meaning of the above aphorisms will be clear from the calculations given below for the Gâyatrȋ metre which contains 6 syllables.5
 A : B Place the number 6 2 Halve it, result 3 Separately place 2 3 cannot be halved, therefore, subtract 1, result 2 "   " 0 Halve it, result 1 "   " 2 1 cannot be halved, therefore, subtract 1, result 0 "   " 0

The process ends.

The calculation begins from the last number in column B. Taking unity double it at 0, this gives 2; at 2 square this (2), the result is 2²; then at zero double (2²), the result is 2³; ultimately at 2 square this (2³), the result is 26, which gives the total number of ways Page 77 in which two things can be arranged in 6 places.1

It will be calculated that two symbols are required in the above calculation to distinguish between two kinds of operations, viz. (1) that of halving and (2) that of the 'absence' of halving and subtraction of unity. These might have been denoted by any two marks arbitrarily chosen.2 The question arises: why did Piṅgala select the symbols "two" and "zero"? The use of the symbol two can be easily explained as having been suggested by the process of halving—division by the number two. The zero symbol was used probably because of its being associated, at the time, with the notion of 'absence' or 'subtraction.' The use of zero in either sense is found to have been common in Hindu mathematics from early times. The above reference to Piṅgala, howewv, shoes that the Hindus possessed a symbol for zero (śûnya), whatever it might have been, before 200 B.C.

The Bakhshâlȋ Manuscript (c. 200) contains the use of zero in calculation. For instance, on folio 56 verso, we have:
 " 880 964 multiplied becomes 848320 84 168 14112

The square of forty different places is 1600. On subtracting this from the number above (numerator), the remainder is
 846720 14112
. On removal of the common factor, it becomes 60."

Page 78 There are a large number of passages of this kind in the work. It will be noticed that in such passages the sentences would be incomplete without the figures, so the figures must have been put there at the time of the original composition of the text, and cannot be suspected of being later interpolations. For an explicit reference to zero and an operation with it, we take the following instance from the work:

"
 0 2 3 4 visible 200 1 1 1 1 1
1 Adding2 unity to zero 1 2 3 ......"3

In the Pañca-siddhântikâ (505) zero is mentioned at several places. The following is an instance:

"In Aries the minutes are seven, in the last sign six; in Taurus six (repeated) thrice; five (repeated) twice; four; four; in Gemini they are three, two, one, zero (śûnya) (each repeated) twice."4

Zero is here conceived as a number of the same type as three, two or one. It cannot be correctly interpreted otherwise. Addition and subtraction of zero are also used in expressing numbers in this work for the sake of metrical convenience. For instance:

"Thirty-six increased by two, three, nine, twelve, nine, three, zero (śûnya) are the signs."5

Instances of the above type all occur in those Page 79 sections of the Pañca-siddhântikâ which deal with the teachings of Puliśa. It seems, therefore, that such expressions are quotations from the Puliśa-siddhânta. As it is known that the word numerals were employed by Puliśa (c. 400) it can be safely concluded that he was conversant with the concept of the zero as a numeral.

The writings of Jinabhadra Gaṇi (529–589), a contemporary of Varâhamihira, offer conclusive evidence of the use of zero as a distinct numerical symbol. While mentioning large numbers containing several zeros, he often enumerates, obviously for the sake of abridgement, the number of zeros contained. For instance: 224,400,000,000 is mentioned as "twenty-two forty-four, eight zeros;"1 3,200,400,000,000 as "thirty-two two zeros four eight zeros."2 At another place in his work

241960407150483920 = 2419604071548392
is described thus:

"Two hundred thousand forty-one thousand nine hundred and sixty; removing (apavartana) the zeros, the numerator is four-zero-seven-one-five, and the denominator four-eight-three-nine-two."3

It should be noted that the term apavartana means what is known in modern arithmetic as the reduction of a fraction to its lowest terms by removing the common factors from the numerator and the denominator. Hence the zero of Jinabhadra Gaṇi is certainly not a mere concept of nothingness but is a specific numerical symbol used in arithmetical calculation.

Page 80 Another contemporary mathematician, Bhâskara I (c. 525), refers to the subtraction of zero in his Mahâbhâskarȋya. In his commentary on the Âryabhaṭȋya he uses the place-value numerals with zero. As has been pointed out before (p. 66) zero is also used by him to denote the notational places.

Siddhasena Gaṇi who lived in the sixth century, has, in his commentary on the Tattvârthâdhigama-sûtra of Umâsvâti, used zero in calculation, as is evidenced by the following two typical instances taken from his work:1

"....the remainder is this, 3,534,400,000,000. The square-root of this is extracted; half of the eight zeros are four zeros; The root of the ramaining portion is one-eight-eight; hence the resulting root is this, 1,880,000."

"On removing the four zeros, the quotient obtained after that is 100,000."

All known Hindu treatises on arithmetic and algebra contain a section dealing with the fundamental operations with zero, including involution and evolution. Details regarding these operations will be given later on; but it must be pointed out here, that these arithmetical operations with zero, certainly presuppose its existence as a numeral denoted by some specific symbol.2

Page 81 Form of the Symbol. The above quotations from ancient works prove conclusively that the zero has been considered a number in India from the earliest centuries of the Christian era, and that there existed some symbol to denote this number. What exactly was the form of this symbol is doubtful. The Bakhshâlȋ Manuscript employs a dot for zero, but as the present copy of the work dates from probably the eighth or the ninth century, it cannot be said whether the form of the symbol was the same when the Bakhshâlȋ work was written, i.e. in the third century A.D. or earlier. Evidence as to the use of the dot for zero is also furnished by the writings of Subandhu, a poet litterateur who flourished about the close of the sixth century. In the Vâsavadattâ of Subandhu we meet with the following metaphor:

"And at the time of the rising of the moon with its blackness of night, bowing low, as it were, with folded hands under the guise of closing blue lotuses, immediately the stars shone forth,.......... like zero dots (śûnya-bindu), because of the nullity of metempsychosis, scattered in the sky as if on the ink-blue skin rug of the Creator who reckoneth the sum total with a bit of the moon for chalk."1

The term bindu ("dot") has been used for zero in word numerals as well as in later literature,2 when a small circle was in use to denote zero, the dot having Page 82 been given up long before. The quotation from Subandhu cannot, therefore, be taken as a definite proof of the use of the dot as a symbol for zero in his time. All that we can infer is that at some period before subandhu, the dot was in use. We may go further and state that very probably, the earliest symbol for zero was a dot and not a small circle.

The earliest epigraphical record of the use of zero is found in the Ragholi plates1 of Jaivardhana II of the eighth century. The Gwalior inscriptions of the reign of Bhojadeva2 also contain zero. The form of the symbol in these inscriptions is the small circle. This is the form that has been in common use from quite early times, probably from before the eighth century.

Other Uses of the Symbol. In the present elementary schools in India, the student is taught the names of the several notational places and is made to denote them by zeros arranged in the line. These, zeros are written as

........ 0 0 0 0 0 0 0 0 0
The teacher points out the first zero on the right and says 'units', then he proceeds to the next zero saying 'tens' and so on. The student repeats the names after the teacher. This practice of denoting the notational places by zeros can be traced back to the time of Bhâskara I, who, as already pointed out on page 66, in his commentary on the Âryabhaṭȋya, Gaṇita-pâda,2, says:

'Writing down the places, we have

0 0 0 0 0 0 0 0 0 0."

In all works on arithmetic (pâṭȋgaṇita) zero has Page 83 been used to denote the unknown. This use of zero can be traced back to the third century A.D. It is used for the unknown in the Bakhshâlȋ arithmetic. In algebra, however, letters or syllables have been always used for the unknown. It seems that zero for the unknown was employed in arithmetic, really to denote the absence of a quantity, and was not a symbol in the same sense as the algebraic x (), for it does not appear in subsequent steps as the algebraic symbols do. This use of zero is mostly found in problems on proportion—the Rule of Five, Rule of Seven, etc. The Arabs also under Hindu influence used zero for the unknown in similar problems. Similar use of zero for the unknown quantity is found in Europe in a Latin manuscript of some lectures by Gottfried Wolack in the University of Erfürt in 1467–68.1 The dot placed over a number has been used in Hindu Gaṇita to denote the negative. In this case it denotes the 'absence' of the positive sign. Similar use of the dot is found in Arabia and Europe obviously under Hindu influence.2

Page 76 1 Piṅgala Chandraḥ-sûtra, ed. by Sri Sitanath, Calcutta, 1840, viii. 28.

2 Ibid, viii. 29.

3 Ibid, viii. 30.

4 Ibid, viii. 31.

5 For 7 syllables the steps are:
 Subtract 1 place 0 Double 2·26 = 27 Halve " 2 Square 26 Subtract 1 " 0 Double 2·22 = 23 Halve " 2 Square 22 Subtract 1 " 0 Double 1 = 2

Page 77 1 This method of calculation is not peculiar to the Piṅgala Chandraḥ-sûtra. It is found in various other works on metrics as well as mathematics. The zero symbol has been similarly employed in this connection in later works also. Vide infra.

2 E.g., Pṛthudakasvâmȋ uses va (from varga, "square") and gu (from guṇa, "multiply"), while Mahâvȋa uses the numerals 1 and 0. Vide infra.

Page 78 1 The zeros given here are represented in the manuscript by dots. The statement in modern symbols is equivalent to the equation,

x + 2x + 3x + 4x = 200.

2 The Sanskrit word is yutaṁ meaning literally "adding", but what is meant is "putting" unity for the unknown (zero).

3 BMs, folio 22, verso.

4 PSi, vi. 12.

5 PSi, xviii. 35; other instances of this nature are in iii. 17; iv. 7; iv. 8; iv. 11; xviii. 44; xviii, 45; xviii. 48; xviii. 51.

Page 79 1 Bṛhat-kṣetra-samâsa, ed. with the commentary of Malayagiri, Bombay, i. 69.

2 Ibid, i. 71. Other such instances are in i. 90, 97, 102, 108, 113, 119, etc.

3 Ibid, i. 83.

Page 80 1 Tattvârthâdhigama-sûtra of Umâsvâti, with his own gloss, elucidated by Siddhasena Gaṇi, ed. by H. R. Kapadia, Bombay, 1926, iii. 11 (com.).

2 Smith and Karpinski (Hindu-Arabic Numerals , p. 53) state, "... the Gaṇita-Såra-Saṅgraha of Mahâavirâchârya (c. 830 A.D.), while it does not use the numerals with place value, has a similar discussion with zero." The first part of the statement is incorrect, because Mahâvȋra has always used numerals with place-value. In fact, no trace of numerals without place-value is to be found in the Gaṇita-Sâra-Saṅgraha. J. Tropfke's statement (Geschichte d. Elementar-Mathematik, Bd. II, 1926, p. 56) that zero was no regarded as a number before the seventeenth century A.D., is incorrect. Cf. B. Datta, "Early literary evidence of the use of the zero in India," American Math. Monthly, XXXVIII, 1931, p. 569.

Page 81 1 Vâsavadattâ of Subandhu, edited by F. Hall (Calcutta, 1859, p. 182) and translated into English by Louis H. Gray (New York, 1913, pp. 99f).

2 E.g. the Hindi poet Bihârȋ in one of his couplets remarks: "The dot on her forehead is increasing her beauty ten-fold, just as a dot increases a number ten-fold."

Page 82 1 No. 4 in the list of inscriptions given before.

2 Nos. 19 and 20 in the list.

Page 83 1 Smith and Karpinski, l.c., pp. 53–54.

2 The occasional use by Al-Battani (929) of the Arabic negative , to indicate the absence of minutes (or seconds), noted by Nallino (Verhandlungen des 5 congresses der Orientalisten, Berlin, 1882, Vol. II, p. 271), is similar to the use of the zero dot to denote the negative.