History of Hindu Mathematics: A Source Book v1.c8.s10: 10. Checks on Operations

History of Hindu Mathematics: A Source Book v1.c8.s10

Language: English

10. Checks on Operations

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The earliest available description of a method of checking the results of arithmetical operations, the direct as well as the inverse, is found in the *Mahâsiddhânta*¹ (*c.* 950). It says:

"Add together the own digits of the numbers forming the multiplicand, multiplier, and product upto one place;² such should be done with the dividend, divisor, quotient and remainder, etc. Then if the number (of one digit) obtained from the product of those numbers (that have been already obtained) from the multiplicand and the multiplier be equal to that obtained from the product, the multiplication is true. If the number, which results from the product of those obtained from the quotient and the divisor, added to that from the remainder, be equal to that obtained from the dividend, the division is true. Add together the digits of a number, its (nearest) square-root (in integers) and of the remainder. If the number, obtained from the square of that (number) which is obtained from the square-root plus the number obtained from the remainder, be equal to that which results from the given number, the root-extraction is true. If the number, resulting from the cube of the number obtained by adding the digits of the cube-root plus the number obtained from the remainder, be equal to the number resulting from the given number, then the operation is correct. Such are the easy tests for correctness of multiplication etc."

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² That is, the digits of the number should be added together; the digits of the sum thus obtained should be again added and the process should be continued until there remains a number of one digit only.

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The *rationale* of the above rules will be clear from the following: Let $n = d_m d_m-1 d₂ d₁$ be a number of *m* digits written in the decimal place-value notation. Let S₁ denote the sum of its digits, S₂ the sum of the digits of S₁, and so on.

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Then $n = d₁ + 10d₂ + + 10m-1d_m,$ $S₁ = d₁ + d₂ + d₃ + + d_m$ so that $n - S₁ = 9(d₂ + 11d₃ + ).$

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Therefore, $n S₁ (mod. 9),$ Similarly $S₁ S₂ (mod. 9),$ $S₂ S₃ (mod. 9),$ $$ $$ $S_k-1 S_k (mod. 9),$ where S_k is a number of one digit only, say n', which is certainly less than or equal to 9.

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Adding the congruences, we obtain $n n' (mod. 9).$

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Thus the number obtained by adding the digits of a number repeatedly is equal to the remainder obtained by dividing that number by nine.

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Now, if there be a number N which is equal to the continued product of p other numbers n₁, n₂, n₃, , n_p plus or minus another number R, then we write $N = n₁ . n₂ . n₃ n_p ± R$

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Now, let $n₁ n'₁ (mod. 9)$ $n₂ n'₂ (mod. 9)$ $$ $$ $n_p n'_p (mod. 9)$

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Multiplying the congruences, we obtain $n.n₂ n_p n′₁.n′₂ n′_p 9.$ Further let $R r′ 9$ Therefore $n₁.n₂.n₃ n_p± R n′₁.n′₂ n′_p ± r′ 9.$ Hence $N n′₁.n′₂ n′_p ± r′ 9.$ In particular, if $n₁ = n₂ = = n_p = n, say$ then will $n′₁ = n′₂ = = n′_p = n′.$ Therefore, $N = np ± R$ and $N n′ p ± r′ 9.$

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From the above follow easily the rules of the *Mahâsiddhânta*.

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The following rule for testing multiplication is given by Nârâyaṇa¹ (1356):

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¹ Quoted by S. Dvivedi, *History of Mathematics* (in Hindi), Benares, 1910, p. 79.

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"The remainders obtained on division of each of the multiplicand and the multiplier by an optional number are multiplied together and then divided by the optional number. If the remainder so obtained be equal to the remainder obtained on dividing the product (of the multiplicand and the multiplier) by the optional number, then, it is correct."

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It must be noted here that a complete set of rules for checking by nines is first found in India. Methods for testing multiplication and division were probably known to the Hindus much earlier. But as these tests were not considered to be among the fundamental operations, they were not mentioned in the works on *pâṭî-gaṇita*.¹ Nârâyaṇa seems to be the first Hindu mathematician to give rules for testing operations by the casting out of any desired number.

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¹ Besides the above works, the check of nines is also quoted by Sumatiharṣa (1618) from an anterior writer Bîjadatta, in his commentary on the *Karaṇa-kutûhala* of Bhâskara II, ed. by Mâdhava Śâstrî, Bombay, 1901, i. 7.

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In the works of early Arab writers the methods of testing multiplication, and division without remainder, by the check of nines are given, while a complete set of rules for testing all operations is found first in the works of Avicenna² (*c.* 1020) who calls his method the "Hindu" method. Maximus Planudes³ also ascribes an Indian origin to the check of nines.

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² F. Woepcke, *JA*(6), I, 1863, pp. 500 *et sq.*

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³ Vide Delambre, *Histoire de l'Astronomie Ancienne*, t. I, Paris, 1817, pp. 518 ff.

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There is thus no doubt as to the Hindu origin of the check of nines. Before Âryabhaṭa II, it was probably used to test multiplication and division only. It was perhaps in this imperfect form when it was communicated to the Arabs. Thereafter, the method was probably perfected independently both in Arabia and India. This would account for the difference in the formulation of the rules by the Arabs and by Âryabhaṭa II, the author of the *Mahâsiddhânta*.⁴ It is, however, certain that the Hindus did not borrow the method from the Arabs, because Âryabhaṭa II wrote before Avicenna. Behâ Eddîn⁵ (*c.* 1600) gives the check of nines in exactly the same form as Âryabhaṭa II.

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⁴ Noted by B. Datta, *JASB*, XXIII, 1927, p. 265.

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⁵ *Kholâsat al-hisâb*, French translation by A. Marre, *Nouvelles Annales d. Math.*, t. v, 1846, p. 263.

Linked texts (2)

Mahāsiddhānta (Āryasiddhānta) — Āryabhaṭa II, Sanskrit
Karaṇakutūhala — Bhāskara II, Sanskrit