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Mathematics in Ancient and Medieval India c5.s14: Surd Number

Mathematics in Ancient and Medieval India c5.s14

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Surd Number

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In the *Śulbasūtra*¹, a surd was technically called *karaṇī*. Thus *dvikaraṇī* means √2, *trikaraṇī* = √3, *catuṣkaraṇī* = √4, *saptamakaraṇī* = √7 etc. By *karaṇī*, root of both square and non-square numbers were meant. The idea of terms ‘audible’ for rational number and ‘inaudible’ for irrational number used by the Arabian scholar al-Khwārizmī (c. 825 A.D.) perhaps developed from Hindu term *karaṇī*¹. The Śulba authors knew the operations of addition and multiplication on surd numbers. Only in one instance, Śulba authors viz., Baudhāyana, Āpastamba and Kātyāyana calculated the value of √2 in terms of fraction by following a method of approximation. It has been discussed in the chapter on geometry. This is :

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¹ Datta, B. *The Science of śulba*, University of Calcutta, p. 188, 1932.

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¹ The term *karaṇī* was however changed to *karṇa* meaning ‘ear’ which may be classified as ‘bad ear’ and ‘good ear’. ‘Bad ear’ means ‘inaudible’ i.e., where root cannot be exactly determined and ‘good ear’ means ‘audible’ whose root can be determined.

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$√(a²+r)=a +r/(2a+1)+e₁ where e₁ is the third term$ approximation.

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$The formula √A=√(a²+r)=a+r/2awhere a=$ greatest root, r=original minus the square of the greatest root=A-a², was given by a Jaina canon writer (1st century A.D.)² The *Bakhshālī Manuscript*³ (4th century A.D.) gave a more accurate formula for finding an approximate value of the square root of a non-square number as follows :

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² Datta, B. ‘Jaina School of Mathematics’ *Bulletin of the Calcutta Mathematical Society*, 21, p. 132, 1929.

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³ Datta, B. ‘The Bakhshālī Mathematics’ *Bulletin of the Calcutta Mathematical Society*, 21, p. 11-12, 1929.

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$√A = √(a² + r) = a + r/2a - ((r/2a)²)/(2(a + r/2a)) (approx.)$

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By following this, √41 is found as √41 = 6 + 5/12 - ((5/12)²)/(2(6 + 5/12)). This formula is also used to calculate the values of √41, √105, √487, √889, √339009. The rule is attributed to Greek Heron (200 A.D.) and is restated by al-Ḥaṣṣarār (c. 1175 A.D.?) and medieval Arabian algebraists.¹

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¹ Heath, T. L. *History of Greek Mathematics*, 2, p. 324; Smith, D. E. *History of Mathematics*, Dover Publication, 2, p. 254 (f. n.).

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Bhāskara I has made use of surds in his *Āryabhaṭīya-bhāṣya*.

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He knew, √x . √y = √xy x . √y = √(x²y)

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√xy + √yz = √y (√x + √z) etc. He has admitted that the rules on surds has been quoted from earlier works. For approximate value of non-square number, Śrīdhara² and Munīśvara (*Pāṭīsāra* v. 117) used the following formula,

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$√A = √((A . a²)/(a²)) (where a is a large number chosen)$

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² *rāśermūladasyā hatasya vargena kenacit mahatā* । *mūlaṃ Śeṣeṇa vinā vibhajet guṇavarga-mūlena* । (Sūtra 118) Similar rules are also available in the *Mahāsiddhānta* (xv. 55), *Siddhāntaśekhara* (xiii. 36), *Gaṇita Kaumudi*, BK. II, p. 33, lines 4-7; and *Paṭīsāra*, v. 117.

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$or √A=(√(p²+r))/a=p/a$

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For example, √10=√((10×49)/49)=√((22²+6)/7)=22/7(approx.)

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Bhāskara II (1150 A.D.) used the following formula : ¹

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¹ Chakravarti, G. ‘Surd in Hindu Mathematics’, *Journal of the Department of Letters*, Calcutta University, 24, p. 29–58, 1934.

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$√A=√(a+r)=(a+A/a)/2$

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The same rule was used by some Arabian scholars² in the 14th century A. D. to approximate the value of √10. Bhāskara II³ (1150 A. D.) gave also methods of extracting the square root of surd quantities like a+√b, a+√b+√c+√d where a, b, c, d are rational. He considered also the problem of the type √(10+√24+√40+√60)=√2+√5+√3; and many others. Bhāskara II, Nārāyaṇa and others perhaps, also knew to calculate the approximate values of the surd quantities like √N by the method of continued fraction (vide chapter on Algebra).

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² Smith, D. E. *History of Mathematics*, Dover Publication, 2, p. 259. The relation √A=a+(A-a²)/2a was used by the Arab scholar. This is equivalent to √A=(a+A/a)/2

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³ Vide his *Bījagaṇita*—Section on *Karaṇī*, See also Srinivas-iengar, C. N. *The History of Ancient Indian Mathematics*, p. 90, World Press, 1967.

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Nārāyaṇa¹ ( 1357 A. D. ), however, has given the following method :

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¹ Since Aa² + 1 = b², we have √A = √((b² - 1)/(a²)) = b/a √(( 1 - 1/(b²) )). If b is large, b/a is a close approximation to √A. The error is less than 1/(2x² √N) and greater than 1/2xy. This can be verified at once by writing √A = y/x √(1 - 1/(y²)) and expanding the right hand side by the binomial theorem.

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*mūlaṃ grāhyaṃ yasya ca rūpakṣepaje pade tatra* |
*jyeṣṭhaṃ hrasvapadenoddharenamūlamāsannam* |
( *Bījagaṇita* of Nārāyaṇa, Bk. 1, Rule 88 )

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“Obtain the roots ( of a square nature ) having unity as the additive and the number whose square root is to be determined ( as the multiplier ). Then the greater root divided by the lesser root will be the approximate value of the square root.”

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If x = a, y = b be a solution of Ax² + 1 = y², then

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√A = y/x = b/a approximately.

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Nārāyaṇa calculated the values of √10 and √(⅕) from 10 × ² + 1 = y². and ⅕ × ² + 1 = y² to give illustrations of this method.

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In the first example, according to Nārāyaṇa, “the roots for the additive unity are ( 6, 19 ), ( 228, 721 ), ( 8658, 27379 ). Then dividing the greater root by the smaller root the approximate values are :

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. √10 = 19/6, 721/228, (27379")/8658 likewise, √(⅕) = 9/20, 161/360, 2889/6460.

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These are arbitrary solutions, for how x = a and y = b were selected for solution of Ax² + 1 = y² is not clearly pointed out. However, a similar method of closer approximations to the value of surd was given by Euler¹ in 1732 A. D. Nārāyaṇa's method was not followed by Jñānarāja² ( 1503 A. D. ) who for approximating the value of √N, found the nearest perfect square a² to A and formed closer approximation as follows :

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¹ Datta, B. “Nārāyaṇa's Method for values of Surds”. *Bulletin of the Calcutta Mathematical Society*, 23, p. 187–194, 1931.

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² *āsannamūlena hatātsvavargāllabdhena mūlaṃ sahitaṃ vibhaktaṃ bhavettadāsannapadaṃ tato'pi muhurmuhuḥ syātsphuṭamūlameva* ( *Siddhāntasundara* of Jñānarāja, the chapter on Algebra).

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a₁ = ½ ( a + A/a ). A next approximation a₂ = ½ ( a₁ + N/a₁ ) is formed and so on. Jñānarāja illustrated his rule by evaluating √8, √20, etc.

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“Divide its square by the root of the nearest square number, the quotient together with that approximate root being halved will be a root more approximate than that. Values more and more accurate can certainly be found by proceeding in the same way repeatedly”

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The rule of Jñānarāja has been reproduced by his son Sūryadāsa in his commentary on the algebra of Bhāskara II. It reappears in the works of Kamalākara¹ (1658 A. D.). From the standpoint of actual computation, the approximate values of surds as obtained by Jñānarāja were stated by Greek Heron² (200 A. D.), who however gave no method of his calculation.

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¹ *svāsannamūlena hatādvargā labdhisṭaḍāsanna padaikya khaṇḍam* । *yat tat sphuṭāsannapadena sūkṣam tadvat sphuṭasannapadam* *muhuḥ syāt* ॥ (*Siddhāntatattvaviveka*, Spaṣṭādhikāra, v. 14).

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² Heath, T. L. *History of Greek Mathematics*, 2, p. 324. Heath mentions, some of the mathematicians of the middle ages and of modern times have applied the same processes.

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