GENERAL ARTICLE Brahmagupta, Mathematician Par Excellence C R Pranesachar Brahmagupta holds a unique position in the his- tory of Ancient Indian Mathematics. He con- tributed such elegant results to Geometry and Number Theory that today's mathematicians still marvel at their originality. His theorems leading to the calculation of the circumradius of a trian- gle and the lengths of the diagonals of a cyclic quadrilateral, construction of a rational cyclic C R Pranesachar is involved in training Indian quadrilateral and integer solutions to a single sec- teams for the International ond degree equation are certainly the hallmarks Mathematical Olympiads. of a genius. He also takes interest in solving problems for the After the Greeks' ascendancy to supremacy in mathe- American Mathematical matics (especially geometry) during the period 7th cen- Monthly and Crux tury BC to 2nd century AD, there was a sudden lull in Mathematicorum. mathematical and scienti¯c activity for the next millen- nium until the Renaissance in Europe. But mathematics and astronomy °ourished in the Asian continent partic- ularly in India and the Arab world. There was a contin- uous exchange of information between the two regions and later between Europe and the Arab world. The dec- imal representation of positive integers along with zero, a unique contribution of the Indian mind, travelled even- tually to the West, although there was some resistance and reluctance to accept it at the beginning. Brahmagupta, a most accomplished mathematician, liv- ed during this medieval period and was responsible for creating good mathematics in the form of geometrical theorems and number-theoretic results. This is besides Keywords his contribution to astronomy. Brahmasphutasiddhanta, zero as a digit, Pythagorean triples, He was born in a village called Bhillamala in North West rational triangles, rational cyclic Rajastan in the year 598 AD and wrote his ¯rst book quadrilaterals, second degree Brahmasphutasiddhanta (the Opening of the Universe) integer equations. RESONANCE March 2012 247 GENERAL ARTICLE “As the sun eclipses in the year 628 AD. He wrote a second book Khan- the stars by his dakhadyaka later. The ¯rst book contains 1008 slokas brilliance, so (verses) in 25 chapters and deals with arithmetic, alge- does the man of bra, geometry and number theory. (See Box 1 for some knowledge eclipse slokas.) He was the ¯rst to introduce zero as a digit. the fame of others in This was translated into Arabic with the title Sindhind. assemblies The second book has 194 slokas and deals with astro- of people if he nomical calculations in 9 chapters. proposes Algebraic He was certainly a mathematician of preeminence for his problems, and still times, but he also had the habit of criticising his pre- more if he decessors sharply for some of their faults and omissions. solves them.” There is a sequel to his second book which deals with Brahmagupta some corrections of his earlier work. He was the head of Ujjain observatory. He passed away in the year 668 AD. Brahmagupta's Works 1. Brahmagupta gave a general formula for the so- called Pythagorean triples, namely, (2mn; m2 n2; m2 + n2). This was known to others also. ¡ 2. (a) Given a side a of a right-angled triangle other than the hypotenuse, a formula was given for the sides of the triangle: 1 a2 1 a2 a; m ; + m : 2 m ¡ 2 m µ ¶ µ ¶ (b) Given the hypotenuse c, of a right-angled tri- 2mnc (m2 n2)c ¡ angle, the sides are given by c; m2+n2 ; m2+n2 . These result in rational right triangles. 3. Given a rational altitude x of a triangle, if the sides are given by a = 1 x2 + x2 p q ; b = 2 p q ¡ ¡ 1 x2 1 x2 ³ ´ 2 p + p ; c = 2 q + q , then we have a sca- len³e trian´gle with ra³tional s´ides, rational area and rational altitudes (and rational circumradius). 248 RESONANCE March 2012 GENERAL ARTICLE Box 1. Slokas from Brahmasphutasiddhanta (BrSpSi XII.38) This sloka describes the method of obtaining a rational cyclic quadrilateral using two nonsimilar rational right triangles. (This seems to be slightly di®erent from the method described below.) (BrSpSi XVIII.64) This sloka describes the equation Nx2 c2 = y2. § (BrSpSi XVIII.67) This sloka describes the method of transforming the equa- 2 2 2 ®¯ 2 ¯2 ®2 tion N® + 4 = ¯ into N 2 + 1 = 4 + N 4 . ¡ ¢ ³ ´ ¡ ¢ ¡ ¢ ³ ´ ®¯ 2 (BrSpSi XVIII.68) This sloka gives the transformation of N 2 + 1 = 2 ¯2 ®2 4 + N 4 . ¡ ¢ ³ ´ ³ ´ (BrSpSi XVIII.3-5) These slokas describe the method of obtaining a (general) solution of the ¯rst degree indeterminate equation ax + by = c. RESONANCE March 2012 249 GENERAL ARTICLE Brahmagupta gave 4. The product of any two sides of a triangle is equal a simple method to to the product of its circumdiameter and altitude construct cyclic drawn on the third side. This result which is easily quadrilaterals with proved by similarity of triangles leads to a formula integer sides, for the circumradius of a triangle. integer diagonals 5. The area of a (cyclic) quadrilateral with sides a; b; and integer area. c; d and semiperimeter s is (s a)(s b)(s c)(s d) . ¡ ¡ ¡ ¡ p(Brahmagupta did not mention the word cyclic). 6. The diagonals of a cyclic quadrilateral with sides (ab+cd)(ac+bd) (ad+bc)(ac+bd) a; b; c; d are ad+bc and ab+cd . q q 7. Integer cyclic quadrilaterals: Brahmagupta gave a simple method to construct cyclic quadrilater- als with integer sides, integer diagonals and inte- ger area. Take two di®erent (nonsimilar) right- angled triangles with sides (a; b; c) and (x; y; z), where c and z are the hypotenuses. Magnify the ¯rst by factors of x and y to get two triangles O1A1D1 and O2C1B1 and the second by factors of a and b to get two more triangles O3C2D2 and O4A2B2. Assemble these four right triangles so that the O's, A's, B's, C's and D's coincide. Then we have a cyclic quadrilateral ABCD with inte- ger sides bz; cy; az; cx and integer diagonals ay + 1 bx; ax+by and integer area 2 (ax+by)(ay+bx).(See Figure 1.) Figure 1. 250 RESONANCE March 2012 GENERAL ARTICLE This single marvellous result needs a little elabo- ration. The integer quadrilateral constructed by Brahmagupta has perpendicular diagonals. Do there exist integer cyclic quadrilaterals with non- orthogonal diagonals? In the 19th century AD, Kummer, a German mathematician found all ra- tional cyclic quadrilaterals. For this one needs to start with two nonsimilar triangles, the sines and cosines of whose angles are all rational (with one angle of one triangle supplementary to one angle of the other). Further the circumradius of such quadrilaterals also turns out to be rational! 8. Brahmagupta gave a beautiful method to generate in¯nitely many integer solutions of the single equa- tion Nx2+1 = y2, where N is a non-square integer, starting with one trial solution. In fact, if (x; y) = (x1; y1) and (x; y) = (x0; y0) are two solutions one can easily see that (x; y) = (x1y0 + x0y1; Nx1x10 + y1y0) is another solution. Hence if we have found one solution of Nx2 + 1 = y2, say (x; y) = (a; b), then taking (x1; y1) = (a; b); (x0; y0) = (a; b), one 2 2 gets a second solution (x2; y2) = (2ab; Na + b ). Again by taking (x1; y1) = (a; b); (x0; y0) = (x2; y2), we get a third solution and so on. Thus the nth solution (xn; yn) is generated by taking (x1; y1) = (a; b) and (x0; y0) = (xn 1; yn 1). It is believed that Brahmagupta did not ¡menti¡on how the ¯rst trial solution can be found. For small values of N this is easily guessed. For example, if N = 2, then we may take (a; b) = (2; 3) as the ¯rst solution of 2x2 + 1 = y2 and generate the solutions succes- sively: (12; 17); (77; 90); (408; 577); ::: . If N = 3, then we may take (a; b) = (1; 2) as a ¯rst solu- tion of 3x2 + 1 = y2 and get the other solutions: (4; 7); (15; 26); (56; 97); :::. Finding the ¯rst solution is not easy in all cases; for example, if N = 61, then x and y are really RESONANCE March 2012 251 GENERAL ARTICLE very large. This feat was achieved by Bhaskara II of 12th century AD. In fact he found that (x; y) = (223 153 980; 1766 319 049) was the smallest solu- tion of 61x2 + 1 = y2. The method used to obtain a solution is called `chakravala'. These equations were later explored by European mathematicians thoroughly and now there is a rich and interesting theory created by number-theorists. Continued fractions play a big role in this. Brahmagupta's contribution to second-order inter- polation for ¯nding sine ratios accurately also de- serves a mention. The readers may note that the seeds of trigonometry were sown in India. Brah- magupta gave formulae for the sum of squares and cubes of ¯rst n natural numbers. He also solved the general quadratic equation. Bhaskara II aptly gave the title `Ganakachakra Chudamani' to Brahmagupta. While mathemat- ics was described as the jewel of all sciences, Brah- magupta accordingly deserves to be described as Address for Correspondence `a brightest star in the galaxy of mathematicians'. C R Pranesachar Mathematical Olympiad Cell HBCSE, TIFR Suggested Reading at Department of Mathematics [1] S Balachandra Rao, Indian Mathematics and Astonomy, 3rd Edition, Indian Institute of Science 2004, Lakshmimudranalaya, Bangalore. Bangalore 560012, India. [2] www.wikipedia.org Email: [3] http://www.gap-system.org/~history/Projects/Pearce/Chapters/ [email protected] Ch8_3.html 252 RESONANCE March 2012.