Pramāṇa · Search · About · API

History of Hindu Mathematics: A Source Book v1.c8.s15.t1: Zero in Algebra

History of Hindu Mathematics: A Source Book v1.c8.s15.t1

Language
English

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.

Cites (3)