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History of Hindu Mathematics: A Source Book v1.c8.s15: 15. The Mathematics of Zero

History of Hindu Mathematics: A Source Book v1.c8.s15

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15. The Mathematics of Zero

v1.c8.s15.1

It has been shown that the zero was invented in India about the beginning of the Christian era to help the writing of numbers in the decimal scale. The Hindu mind did not rest satisfied till it evolved the complete arithmetic of zero. The Hindus included zero among the numbers (*saṅkhyā*), and it was used in their arithmetic at the time when the original of the Bakhshâlî Manuscript was written, about the third century A.D. The operation of addition and subtraction of zero are incidentally mentioned in the *Pañcasiddhântikâ* of Varâhamihira (505). The complete decimal arithmetic is found in the commentary of Bhâskara I (*c.* 525) on the *Âryabhaṭîya*. The results of operations by zero are found stated in the work of Brahmagupta (628) and in all later mathematical treatises. The treatment of zero in the arithmetic of the Hindus is different from that found in their algebra. In order, therefore, to bring out this difference clearly, we give separately the results found in *pâṭîgaṇita* (arithmetic) and in *bîjagaṇita* (algebra).

v1.c8.s15.2

**Zero in Arithmetic.** The Hindus in their arithmetic define zero as the result of the operation $a - a = 0$ This definition is found in Brahmagupta's work¹ and is repeated in all later works. It is directly used in the operation of subtraction. In carrying out arithmetical operations, the results of the operations of addition, subtraction and multiplication of zero and by zero are required. The Hindus did not recognise the operation of division by zero as valid in arithmetic; but the division of zero by a number was recognised as valid. Nârâyaṇa in his *pâṭîgaṇita* (arithmetic) has clearly stated this distinction:

v1.c8.s15.3

¹ *BrSpSi*, p. 309. *Cf.* B. Datta, *BCMS*, XVIII, pp. 165-176 for some other details regarding operations with zero.

v1.c8.s15.4

“Here in *pâṭîgaṇita*, division by zero is not recognised, and therefore, it is not mentioned here. As it is of use in *bîjagaṇita* (algebra), so I have mentioned division by zero in my *Bîjagaṇita*.”²

v1.c8.s15.5

² *GK*, remark subjoined to i. 30.

v1.c8.s15.6

The following are the results of the operations in relation to zero mentioned in Hindu works on *pâtî-gaṇita*:

v1.c8.s15.7

Śrîdhara says:

v1.c8.s15.8

"In addition cipher makes the sum equal to the additive; when cipher is subtracted (from a number), there is no change in the number. In multiplication and other¹ operations on zero the result is zero. Multiplication of a number by cipher also gives zero."²

v1.c8.s15.9

¹ By other operations are meant: (1) division of zero by a number, (2) squaring and square-root of zero, (3) cubing and cube-root of zero—in the sequence in which these operations are mentioned in Hindu arithmetic.

v1.c8.s15.10

² *Triś*, p. 4. It should be noted that distinction is made between a × 0 and 0 × a, although the result in each case is zero.

v1.c8.s15.11

Āryabhaṭa II, in the chapter on *pâṭîgaṇita* in his *Mahâ-siddhânta*, writes:

v1.c8.s15.12

"If zero is added to a number, the number is unchanged; the same is true when zero is subtracted. In the operation of multiplication or division of zero (by another number) or in evolution and involution (of zero), the result is zero."³

v1.c8.s15.13

³ *MSi*, p. 146.

v1.c8.s15.14

Nârâyaṇa in his *pâṭîgaṇita* states:

v1.c8.s15.15

"When zero is added to or subtracted from a number, it (the number) remains unchanged; that (number) multiplied by zero becomes zero. In the multiplication and other operations on zero, the result is zero. When a quantity is added to zero the result is that which is added."⁴

v1.c8.s15.16

⁴ *GK*, i. 30.

v1.c8.s15.17

Mahâvîra in his *Gaṇita-sâra-saṁgraha* writes:

v1.c8.s15.18

"A number multiplied by zero is zero, and that number remains unchanged when it is divided by, combined with, or diminished by zero. Multiplication and other operations on zero give rise to zero and in the operation of addition, zero becomes the same as what is added to it.”¹

v1.c8.s15.19

¹ *GSS*, p. 6. (49).

v1.c8.s15.20

Mahâvîra’s statement that a number remains unchanged when divided by zero is obviously incorrect. The correct result of the operation was known to Brahmagupta who preceded him by more than two centuries. It is, therefore, strange to find Mahâvîra making such a statement. He probably thought that, so far as arithmetic was concerned, division by zero was no division at all.

Zero in Algebra

v1.c8.s15.t1.1

The earliest treatment of zero in algebra is found in the *Brâhma-sphuṭa-siddhânta*² (628):

v1.c8.s15.t1.2

² *BrSpSi*, pp. 309-310, rules 31-35. These rules contain also, the rules of the operations with the positive and the negative in algebra. The results relating to zero only are mentioned here.

v1.c8.s15.t1.3

“Negative less cipher is negative; positive (less cipher is) positive; cipher (less cipher is) nought . . . .

v1.c8.s15.t1.4

“The product of cipher and negative, or of cipher and affirmative, is cipher; of two ciphers is cipher.

v1.c8.s15.t1.5

“Cipher, divided by cipher is nought . . . . Positive or negative divided by cipher is *taccheda* (a fraction with that for denominator), or cipher divided by negative or affirmative (is either zero or is expressed by a fraction with zero as numerator and the finite quantity as denominator).”³

v1.c8.s15.t1.6

³ The result of the operation given in the bracket is according to the interpretation of S. Dvivedî.

v1.c8.s15.t1.7

Bhâskara II in the *Lîlâvatî* as well as in his *Bîjagaṇita* gives the result of the operations with zero. In the *Lîlâvatî* he writes:

v1.c8.s15.t1.8

“In addition cipher makes the sum equal to the additive. In involution and evolution the result is cipher. A number divided by zero is *kha-hara* (that number with zero as denominator). The product of (a number and) zero is zero, but it must be retained as a multiple of zero (*kha-guṇa*), if any further operations impend. Zero having become a multiplier (of a number), should zero afterwards become a divisor, the number must be understood to be unchanged. So likewise any number, to which zero is added, or from which it is subtracted (is unaltered).”¹

v1.c8.s15.t1.9

In the *Bîjagaṇita*, the same results are given with the addition that if a quantity is subtracted from zero, its sign is reversed, while in the case of addition the sign remains the same.

v1.c8.s15.t1.10

**Zero as an Infinitesimal.** It will be observed that Brahmagupta directs that the results of the operations x ÷ 0 and 0 ÷ x should be written as x/0 and 0/x respectively. It is not possible to tell exactly what he actually meant by these forms. It seems that he did not specify the actual value of these forms, because the value of the variable x is not known. Moreover, the zero seems to have been considered by him as an infinitesimal quantity which ultimately reduces to nought. If this surmise be correct, Brahmagupta is quite justified in stating the results as he has done.

v1.c8.s15.t1.11

The idea of zero as an infinitesimal is more in evidence in the works of Bhâskara II. He says: “The product of (a number and) zero is used the infinitesimal increment f'(x)δ x of the function f(x), due to a change δ x in x.

v1.c8.s15.t1.12

The commentator Kṛṣṇa proves the result 0 × a = 0 = a × 0 as follows:

v1.c8.s15.t1.13

“The more the multiplicand is diminished, the smaller is the product; and, if it be reduced in the utmost degree, the product is so likewise: now the utmost diminution of a quantity is the same with the reduction of it to nothing; therefore, if the multiplicand be nought, the product is cipher. In like manner, as the multiplier decreases, so does the product; and, if the multiplier be nought, the product is so too.”

v1.c8.s15.t1.14

In the above zero is conceived of as the limit of a diminishing quantity.

v1.c8.s15.t1.15

**Infinity.** The quotient of division by zero of a finite quantity has been called by Bhâskara II as *kha-hara*, which is synonymous with *kha-cheda* (the quantity with zero as denominator) of Brahmagupta. Regarding the value of the *kha-hara*, Bhâskara II remarks:

v1.c8.s15.t1.16

“In this quantity consisting of that which has cipher for its divisor, there is no alteration, though many may be inserted or extracted; as no change takes place in the infinite and immutable God, at the period of the destruction or creation of worlds, though numerous orders of beings are absorbed or put forth.”\1

v1.c8.s15.t1.17

\1 *BBi*, pp. 5-6. G. Thibaut (*Astronomie, Astrology und Mathematik*, Strasbourg, 1899, p. 72) thought that this passage was an interpolation. There appears no justification for considering this as an interpolation, as the passage occurs in the oldest known commentary and in all copies of the work so far found. *Cf.* Datta, *l.c.*, p. 174.

v1.c8.s15.t1.18

From the above it is evident that Bhâskara II knew that a/0 = ∞ and ∞ + k = ∞.

v1.c8.s15.t1.19

Gaṇeśa remarks that a/0 is "an indefinite and unlimited or infinite quantity: since it cannot be determined how great it is. It is unaltered by the addition or subtraction of finite quantities: since in the preliminary operation of reducing both fractional expressions to a common denominator, preparatory to taking their sum or difference, both numerator and denominator of the finite quantity vanish."

v1.c8.s15.t1.20

Kṛṣṇa remarks:

v1.c8.s15.t1.21

"As much as the divisor is diminished, so much is the quotient increased. If the divisor is reduced to the utmost, the quotient is to the utmost increased. But, if it can be specified, that the amount of the quotient is so much, it has not been raised to the utmost: for a quantity greater than that can be assigned. The quotient, therefore, is indefinitely great, and is rightly termed infinite."

v1.c8.s15.t1.22

Regarding the proof of a/0 ± k = a/0 Kṛṣṇa makes the same remarks as Gaṇeśa. He, however, goes a step further when he says that $a/0 = b/0.$

v1.c8.s15.t1.23

This is illustrated by him through the instance of the shadow of a gnomon, which at sun-rise and sun-set is infinite; and is equally so whatever height be given to the gnomon, and whatever number be taken for the radius. "... Thus, if the radius be 120; and the gnomon be 1, 2, 3 or 4; the expression deduced from the proportion, as sine of sun's altitude is to sine of zenith distance, so is gnomon to shadow, becomes 120/0, 240/0, 360/0 or 480/0. Or, if the gnomon be, as it is usually framed, 12 fingers, and radius be taken as 3438, 120, 100 or 90, the expression will be 41256/0, 1440/0, 1200/0 or 1080/0, which are all alike infinite."¹

v1.c8.s15.t1.24

¹ All the above passages are taken from the respective commentaries. They have been noted by Colebrooke, *l.c.*

v1.c8.s15.t1.25

**Indeterminate Forms.** Brahmagupta has made the incorrect statement that $(°)/(°) = °$

v1.c8.s15.t1.26

Bhâskara II has sought to correct this mistake of Brahmagupta. According to him $Lim (ε→°) (a.ε)/(ε) = a.$

v1.c8.s15.t1.27

His language, however, in stating this result is defective, for he calls the infinitesimal ε zero, not being in possession of a suitable technical term. That, in the above case, he actually meant by zero a small quantity tending to the limiting value zero, is abundantly clear from the use he makes of the result in his Astronomy. Taylor² and Bapu Deva Sâstrî³ are also of this opinion.

v1.c8.s15.t1.28

³ His *Bîja-gaṇita* (in Hindi), Pt. I, Benares, 1875, p. 179 *et sq.*

v1.c8.s15.t1.29

Bhâskara has given three illustrative examples. They are:

v1.c8.s15.t1.30

(i) Evaluate ((x × ° + (x × °)/2))/(°) = 63.

v1.c8.s15.t1.31

From this he derives the result x = 14, which is correct if we consider ° = ε, a small quantity tending to zero. His other examples are:

v1.c8.s15.t1.32

(ii) (x/(°) + x - 9)² + (x/(°) + x - 9) ° = 90

v1.c8.s15.t1.33

² *Lîlâwatî*, Bombay, 1816, p. 29.

v1.c8.s15.t1.34

giving x = 9; and $(iii) (x+x/2)× 0²+2(x+x/2)× 0÷ 0 = 15,$ giving x=2.¹

v1.c8.s15.t1.35

¹ The answers of this and the previous example are incorrect because 0² has been taken to be equal to 0.

v1.c8.s15.t1.36

Bhâskara II's result $a/0× 0 = a$ is, however, not quite correct, as the form is truly indeterminate and may not always have the value a. His attempt, however, at such an early date to assign a meaning to the form 0/0, and his partial solution of the problem are very creditable, seeing that in Europe mathematicians made similar mistakes upto the middle of the nineteenth century A. D.²

v1.c8.s15.t1.37

² Martin Ohm (1828) says: "If a is not zero, but b is zero, then the quotient a/b has no meaning" for the quotient "multiplied by zero gives only zero and not a, as long as a is not zero." *Lehrbuch der niedern Analysis*, Vol. I, Berlin, 1828, pp. 110, 112.

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