Mathematics in Ancient and Medieval India c6.s5: Value of √2

Mathematics in Ancient and Medieval India c6.s5

Language: English

Value of √2

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The value of √2 given by Baudhāyana² in his *Śulbasūtras* is : $√2 = 1 + ⅓ + 1/3.4 - 1/3.4.34 ( approx. ).$ This is also given by Āpastamba³ and Kātyāyana.⁴ In terms of decimal fractions, this becomes √2 = 1.4142156. According to modern calculation √2 = 1.414213…… .

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³ *Āpastambaśulba*, 1.6.

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⁴ *Kātyāyanaśulba*, 2.13.

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Thibaut, Rodet, Datta and others gave possible method of solution for arriving at the value as follows :–

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i ) *Thibaut’s proof* ⁵.

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⁵ Thibaut, G. ‘On the Śulbasūtras’, *Journal of the Asiatic Society of Bengal*, 44, Pt. I, No. 3, p. 239–41, 1875.

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We know 17² = 2.12² - 1. Thibaut argued, by how much side 17 must be diminished in order that the square on it may be 2.12² exactly. Since 2 × 17 × 1/34 = 1, he observed, two strips each of 1/34 (approximately) are to be cut off from a square with 17 as side to obtain the square 2.12² ( i.e. 12² + 12² ) $Hence ( 17 - 1/34 )² = 2.12²$ $or (17 - 1/34)/12 = √2$

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² *pramāṇaṃ tṛtīyena vardhayettacca caturthenātmaca-* *tustriṃśonena, saviśeṣa* ( *Baudhāyana-śulba*, 1.61–2 ).

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$Again, 17 - 1/34 = 12 + 4 + 1 - 1/34$ $or, 17 - 1/34 = 12 ( 1 + ⅓ + 1/3.4 - 1/3.4.34 )$ $or, 1/12 ( 17 - 1/34 ) = 1 + ⅓ + 1/3.4 - 1/3.4.34$ $or, √ 2 = 1 + ⅓ + 1/3.4 - 1/3.4.34$

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For Baudhāyana's selection of units of 12 *aṅgulas* (= one *pada*) and 34 *tilas* (= one *aṅgula*), Thibaut expressed no doubt for the selection of arbitrary relation 17² = 2.12² (approx.) contributing to the origin of the formula.

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ii ) *Rodet's approximation*¹.

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¹ Datta, B. *The Science of Śulba*, Calcutta University, 1932.

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According to Rodet, the approximation of √ 2 adopted by *Śulbakāras* may be obtained by successive approximation.

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$√(a² + r) = a + r/(2a + 1) + (r/(2a + 1) ( 1 - r/(2a + 1) ))/(2 ( a + r/(2a + 1) )) + e$

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where e is the fourth term approximation. Rodet might have obtained as follows :

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$√(a² + r) = a + r/(2a + 1) [ two terms approximation ]$

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where ( 2a + 1 ) is the difference of the squares of *a* and the next positive integer ( a + 1 ).

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For third term approximation, let

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$² + r = a + r/(2a + 1) + e₁$ $= (2a² + a + r)/(2a + 1) + e₁$

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Squaring and neglecting e₁², we get $2((2a²+a+r)/(2a+1)).₁=(a²+r)-((2a²+a+r)/(2a+1))²$ $=(r(2a+1-r))/((2a+1)²)$ $₁=(r(2a+1-r))/(2(2a+1)(2a²+a+r))=(r(2a+1-r))/(2(2a+1)²(a+r/(2a+1)))$ $=(r/(2a+1)(1-r/(2a+1)))/(2(a+r/(2a+1)))$

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Like wise the fourth term approximation is obtained. Obviously following above we write, $√2=√(1²+1)=1+⅓$ $let √2=1+⅓+e₁=4/3+e₁$

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Squaring both sides, and cancelling e₁² from both sides, we get $8/3e₁=2-16/9=2/9$ $₁=2/9×⅜=1/12=1/3.4$ $∴√2=1+⅓+1/3.4$ $Again let √2=1+⅓+1/3.4+e₂$ $=17/12+e₂$

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Squaring both sides and cancelling e₂² from both sides, $17/6e₂=2-(17/12)²=-1/144$

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$∴ e₂ = - 1/144 × 6/17 = - 1/12.34 = - 1/3.4.34$

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$∴ √2 = 1 + ⅓ + 1/3.4 - 1/3.4.34 ( approx. )$

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The method described later by Gurjar¹ and Gupta² is same and is no improvement over that of Rodet.

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¹ Gurjar, L. V. 'The Value of √2 given in the Sulbasūtras.' *Journal of the University of Bombay*, ns. 10, pt. 5, 6–10, 1942.

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² Gupta, R. C. 'Baudhāyana's value of √2, *The Mathematics Education*, *6*, No. 3, 77–79, 1972.

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iii) *Datta's Proof*:

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Datta has, however, explained that the value of √2 was possibly obtained from a geometrical method.³ This method consists in constructing a square with area equal to the sum of the areas of the two other squares having sides of one unit in length. For this, one of the two squares having side of unit length is divided into three equal parts by lines drawn parallel to one of its side. Each of these parts forms a rectangular piece of one unit in length and one third unit in width. Two of these rectangular parts are then joined lengthwise to the two adjacent sides of the other unit square. This leaves a square hole at one of the corners of the enlarged unit square. This square·hole will have a side of one-third unit in length. The remaining rectangular piece of the divided unit square is again subdivided into three equal parts each forming a square of side one-third unit length. One of these square is fitted into the square-hole formed at one of the corners of the enlarged squāre. Each of the remaining two squares is again subdivided into four equal rectangular pieces having length of ⅓ unit and width of 1/3.4 unit. Eight of these small rectangular slips are placed lengthwise side by side on the two adjacent sides of the enlarged square with four on each side. This again leaves a square-hole at the corner having a side of length 1/3.4 unit. Now two equal thin slips have to be deducted from the two adjacent sides of the enlarged square under construction, the width of each of the slips is therefore given by $((1/3.4)²)/(2 × (1 + ⅓ + 1/3.4)) = 1/3.4.34$

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³ Datta, B. *The Science of Śulba*, p. 192-94, Calcutta University 1932.

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Fig. 11

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Hence the side of the desired square is given approximately by 1 + ⅓ + 1/3.4 - 1/3.4.34 which gives the value of √2. This method is simpler and straight forward than that of Thibaut. Datta also discussed fully the knowledge of value of the irrationality of √2 of Hindus and concluded that the irrationality of √2 was known to the ancient Hindus, though no direct method is available in the Śulba texts. Sen has also discussed this in A Concise History of Science in India (p. 154–55).

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In Babylon, a small cunei-form tablet of the old Babylonian times (c. 1800–1600 B. C.) is now traced in the Yale Babylonian collection (No. 7289) which shows a square with its two diagonals.¹ Three numbers in sexagesimal system are inscribed on it. These three numbers as represented by Neugebauer² as the value of diagonal, a side and the value of √2 (since d = √2 a). Here √2 = 1,24,51,10. Now Hindu³ value √2 = 1 + ⅓ + 1/3.4 - 1/3.4.34, if changed into sexagesimal unit, gives √2 = 1,24,51,10,36…… . This has led to the speculation that the Hindu system of calculating the value of √2 was developed from that of the Babylonians. Two points are important in this connection so far as the Hindu origin is concerned : i) In the period, there was no use of sexagesimal system, ii) the rule was clearly stated thus : "the side of a square is to be increased by its third, the third again by its fourth part less its (fourth parts) thirty-fourth part". We have no evidence of any such method of which the Babylonians were aware. As regards Greek sources, many approximations to the values of √2 were current but such a methood and correct approximation of √2 did not occur.⁴

Linked texts (3)

Āpastambaśulbasūtra — Āpastamba, Sanskrit
Kātyāyanaśulbasūtra — Kātyāyana (Vājasaneyi sūtrakāra), Sanskrit
Baudhāyanaśulbasūtra — Baudhāyana, Sanskrit