Lecture 16. Mathematics of Medieval India

The second period of Hindu mathematics Indian mathematics is also known asHindu 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 tomathematics in arithmetic, geometry and algebra. They were influenced by the civilizationat Alexandria, and other civilizations.

The most important mathematicians of the period are Aryabhata (475-550), Brah-magupta (598-668), Mahavira (9th cent.), and Bhaskara (1114-1185). Most of their workand that of Hindu mathematics generally was motivated by astronomy and astrology. Indianmathematicians made early contributions to the study of the concept of zero as a number,negative numbers, arithmetic, algebra, and trigonometry.

Figure 16.1 The Tai Mahal in India.

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

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contributions of Indian arithmetic. This counting system was accepted and improved byArabs in the 8th century, and is called the Hindu-Arabic numeral system and was evolvedinto the modern counting system that we are using today.

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

The first known use is by Brahmagupta about 628. Brahmagupta was born in 598, andlived until at least 665. His book the Brahma - sphuta - siddhanta describes him as theteacher from Bhillamala, which is a town now known as Bhinmal in the Indian state ofGujurat. Very little is known about his life except that he was prominent in astronomy aswell as mathematics. Brahmagupta introduced operations of 0 and negative numbers. Butfor solutions of a quadratic equation, Brahmagupta did not accept negative square roots.The Hindus did not unreservedly accept negative numbers, but negative numbers did gainacceptance slowly.

Figure 16.2 Gol Gumbaz at Bijapur

Irrational numbers Without rigorous proofs, Brahmagupta did some calculations ofirrational numbers. For example,

√3 +√

12 =

√(3 + 12) + 2

√3 · 12 =

√27 = 3

√3

and more generally√a +√b =

√(a + b) + 2

√ab.

The Hindus were less sophisticated than Greeks in that they failed to see the logicaldifficulties involved in the concept of irrational numbers. But their interests in calculation

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

Algebra and equations Like Diophantus, Indians used abbreviations. Moreover theyused more abbreviations than Diophantus did. For example, they used “ka”, from theword karana, to denote “square root.” When there are more than one unknowns, after thefirst unknown, they used color words “black, blue, yellow,” etc. to denote the remainingunknowns. 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

x =

√4ac + b2 − b

2a.1

This solution was not expressed in symbols but only applied to specific numbers, and thegeneral 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 thesame, 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 negative number was not allowed. Bhaskara (1114 - 1185) said thatthere 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 Sulbasutras. Sincesuch literary pieces were not designed to teach mathematics, there are no derivations, justassertions. Later commentators did give demonstrations. For example, in the BaudhayanaSulbasutra, which probably dated to around 600 B.C., Pythagoras Theorem was asserted:

The areas of the squares produced separately by the length and the breadth of arectangle together equal the area of the square produced by the diagonal. Thisis observed in rectangles having sides 3 and 4, 12 and 5, 15 and 8, 7 and 24, 12and 35, 15 and 36.

1In fact, as early as 2000 B.C., the Babylonians could, in today’s notation, solve pairs of equations

x+y = p and xy = q, i.e., x2+q = px. It was found that x, y = p2 ±√(

p2

)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.

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A proof of this theorem was given in the Yuktibhasa, written by Jyesthadeva (1530-1610)in the mid-sixteenth century. The idea is to draw the square on each of the two sides andon the hypotenuse (see above picture). If one cuts along each of the lines, then rotates thetriangles outside the large square, and two pieces together will fill up the square on thehypotenuse. As in the Chinese proof, not like as in Euclid’s Elements, there are no axiomsand rigorous proof. One just observes the diagram, rotates the pieces, and understands thatthe 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 acyclic quadrilateral. It states that if a quadrilateral has sides a, b, c and d, semi-perimeter sand all vertices on a circle, then its area is√

(s− a)(s− b)(s− c)(s− d).

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But there was no geometric proof offered by Brahmagupta. A proof for this formula firstappeared also in the Yuktibhasa. The proof is based on the formulas for the lengths of thediagonals AC and BD of the quadrilateral:

AC =

√(ac + bd)(ad + bc)

ab + cd, and BD =

√(ac + bd)(ab + cd)

ad + bc.

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

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