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Home , Brahmagupta, Indian mathematics

Lecture 16. of Medieval

The second period of Hindu mathematics is also known as Hindu mathematics. After the early Hindu mathematics, the second period of Hindu math- ematics may be roughly dated from about A.D. 200 to 1200. Hindus made contributions to mathematics in , and . They were influenced by the civilization at Alexandria, and other civilizations.

The most important mathematicians of the period are ¯ (475-550), Brah- magupta (598-668), Mah¯av¯ira (9th cent.), and Bh¯askara (1114-1185). Most of their work and that of Hindu mathematics generally was motivated by and . Indian mathematicians made early contributions to the study of the concept of zero as a , negative , arithmetic, algebra, and .

Figure 16.1 The Tai Mahal in India.

Counting Indian counted integers by using base 10 before the 6th century, and they used 9 numbers and a dot · to denote zero. The invention of zero is one of the greatest

99 contributions of Indian arithmetic. This counting system was accepted and improved by Arabs in the 8th century, and is called the Hindu- and was evolved into the modern counting system that we are using today.

Negative numbers The Hindus introduced negative numbers to represent debts, and positive numbers represented assets.

The first known use is by about 628. Brahmagupta was born in 598, and lived until at least 665. His book the Brˆahma- sphuta - siddhˆanta describes him as the teacher from Bhillamˆala,which is a town now known as in the Indian state of Gujurat. Very little is known about his life except that he was prominent in astronomy as well as mathematics. Brahmagupta introduced operations of 0 and negative numbers. But for solutions of a , Brahmagupta did not accept negative roots. The Hindus did not unreservedly accept negative numbers, but negative numbers did gain acceptance slowly.

Figure 16.2 Gol Gumbaz at Bijapur

Irrational numbers Without rigorous proofs, Brahmagupta did some calculations of irrational numbers. For example, √ √ q √ √ √ 3 + 12 = (3 + 12) + 2 3 · 12 = 27 = 3 3 and more generally √ √ q √ a + b = (a + b) + 2 ab. The Hindus were less sophisticated than Greeks in that they failed to see the logical difficulties involved in the concept of irrational numbers. But their interests in calculation

100 caused them to perform calculations on irrational numbers anyway, which was completely independent of geometry and was helpful in development of mathematics.

Algebra and equations Like , Indians used abbreviations. Moreover they used more abbreviations than Diophantus did. For example, they used “”, from the word karana, to denote “.” When there are more than one unknowns, after the first unknown, they used color words “black, blue, yellow,” etc. to denote the remaining unknowns. Indian algebra was an algebra of symbols, which greatly simplified calculation.

Brahmagupta is known for introducing a general solution formula for the quadratic equa- tion, i.e., for the equation ax2 + bx − c = 0, the solution is √ 4ac + b2 − b x = .1 2a This solution was not expressed in symbols but only applied to specific numbers, and the general method was implicit in many specific solved cases.

Brahmagupta gave a formula in words: To the absolute number multiplied by four times the (coefficient of the) square, add the square of the (coefficient of the) middle term; the square root of the same, less the (coefficient of the) the middle term, being divided by twice the (coefficient of the) square is the value. 2 The square root of a was not allowed. Bh¯askara (1114 - 1185) said that there is no square root of a negative number because a negative number is not a square.

Geometry Many important geometric ideas were expressed in the Sulbas¯utras´ . Since such literary pieces were not designed to teach mathematics, there are no derivations, just assertions. Later commentators did give demonstrations. For example, in the Baudh¯ayana Sulbas¯utra´ , which probably dated to around 600 B.C., Theorem was asserted:

The of the squares produced separately by the length and the breadth of a rectangle together equal the of the square produced by the diagonal. This is observed in rectangles having sides 3 and 4, 12 and 5, 15 and 8, 7 and 24, 12 and 35, 15 and 36.

1In fact, as early as 2000 B.C., the Babylonians could, in today’s notation, solve pairs of equations q 2 p p 2 x+y = p and xy = q, i.e., x +q = px. It was found that x, y = 2 ± 2 − q when both x, y were positive (The Babylonians did not admit negative numbers). 2c.f. John Stillwell, Mathematics and its history, Second edition, Springer, 2002, p.87.

101 A proof of this theorem was given in the Yuktibh¯as¯a, written by Jyesthadeva (1530-1610) in the mid-sixteenth century. The idea is to draw the square on each of the two sides and on the hypotenuse (see above picture). If one cuts along each of the lines, then rotates the triangles outside the large square, and two pieces together will fill up the square on the hypotenuse. As in the Chinese proof, not like as in Euclid’s Elements, there are no axioms and rigorous proof. One just observes the diagram, rotates the pieces, and understands that the theorem is true. Such a procedure can be only regarded as an empirical proof.

Figure 16.3 Brahmagupta and his formula

One achievement by Brahmagupta (598-669) is a remarkable formula for the area of a . It states that if a quadrilateral has sides a, b, c and d, semi- s and all vertices on a , then its area is

p(s − a)(s − b)(s − c)(s − d).

102 But there was no geometric proof offered by Brahmagupta. A proof for this formula first appeared also in the Yuktibh¯as¯a. The proof is based on the formulas for the lengths of the diagonals AC and BD of the quadrilateral: r r (ac + bd)(ad + bc) (ac + bd)(ab + cd) AC = , and BD = . ab + cd ad + bc

About the year 1200 scientific activity in India declined and progress in mathematics ceased. After the British conquered India in the eighteenth century, a few Indian scholars went to England to study and on their return did initiate some research. However, this modern activity is part of European mathematics.

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