Bhaskara II.ppt

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1、Bhaskara II,Casey Gregory,Background Information,One of most famous Indian mathematicians Born 1114 AD in Bijjada Bida Father was a Brahman (Mahesvara) and astrologer Nicknamed Bhaskaracharya “Bhaskara the Teacher” Studied Varahamihira and Brahmagupta at Uijain,What he knew,Understood zero and negat

2、ive numbers Except how to divide by it Knew x2 had 2 solutions * Had studied Pells equation and other Diophantine problems,His Accomplishments,First to declare a/0 = * First to declare + a = Wrote 6 works including Lilavati (mathematics) Bijaganita (algebra) Siddhantasiromani Vasanabhasya (commentar

3、y on Siddhantasiromani) Karanakutuhala (astronomy) Vivarana,Lilavati,O girl! out of a group of swans, 7/2 times the square root of the number are playing on the shore of a tank. The two remaining ones are playing with amorous fight, in the water. What is the total number of swans?,Lilavati,13 Chapte

4、rs definitions; arithmetical terms; interest; arithmetical and geometrical progressions; plane geometry; solid geometry; the shadow of the gnomon*; the kuttaka; combinations. 2 Methods for multiplication* 4 methods for squaring Rules of three, five, seven and nine Kuttaka Method Example: “Say quickl

5、y, mathematician, what is that multiplier, by which two hundred and twenty-one being multiplied, and sixty-five added to the product, the sum divided by a hundred and ninety-five becomes exhausted.” Bhaskaracharya is finding integer solution to 195x = 221y + 65. He obtains the solutions (x,y) = (6,5

6、) or (23,20) or (40, 35) and so on.,Bijaganita,12 Chapters Including: positive and negative numbers; zero; the unknown; surds*; the kuttaka*; indeterminate quadratic equations; simple equations; quadratic equations; equations with more than one unknown; quadratic equations with more than one unknown

7、; operations with products of several unknowns; and the author and his work Quadratic equation - 700 A.D. Brahmagupta who also recognized 2 roots in the solution. 1100A.D. ANY positive number has 2 square roots,Tried to prove a/ 0 = , however if that were true, *0 = a, therefore proving all numbers

8、equal Shows that the kuttaka method to solve indeterminate equations such as ax + by + cz = d has more than one solution. His conclusion shows his poetic and passionate nature: “A morsel of tuition conveys knowledge to a comprehensive mind; and having reached it, expands of its own impulse, as oil p

9、oured upon water, as a secret entrusted to the vile, as alms bestowed upon the worthy, however little, so does knowledge infused into a wise mind spread by intrinsic force.”,Siddhanta Siromani,Picture of Goladhyaya.,Siddhanta Siromani,Wrote Siddhanta Siromani (1150 AD) Leelavati (arithmetic) Bijagan

10、ita (algebra) Goladhayaya (spheres, celestial globes) Grahaganita (mathematics of the planets),Topics Covered in Siddhanta Siromani,Astronomy Related Latitudes and problems of astronomical calculations.,Further Information in Siddhanta,First time trigonometry was studied as its own entity, rather th

11、an how it related to other calculations.sin(a + b) = sin a cos b + cos a sin b sin(a - b) = sin a cos b - cos a sin b.,His 7th work?,There exists a 7th work, but it is thought to be a forgery.,After Bhaskara II,Bhaskara II dies in 1185 A HUGE scientific lull after invasion by muslims 1727, next impo

12、rtant Hindu mathematician Sawai Jai Singh II Several of Bhaskaras findings were not explored heavily after his death, and ended up being “discovered” later by European mathematicians.,Bhaskara II Rediscovered,chakrawal, or the cyclic method, to solve algebraic equations. * 6 centuries later, Galois,

13、 Euler and Lagrange rediscovered this and called it “inverse cyclic“. differential calculus Rediscovered as “differential coefficient“ “Rolles theorem“ Newton and Leibniz receive credit Bhaskara is renowned for his concept of Tatkalikagati (instantaneous motion).,Works Cited,http:/ http:/www.bbc.co.uk/dna/h2g2/A2982567 http:/www-groups.dcs.st-and.ac.uk/%7Ehistory/Mathematicians/Bhaskara_II.html http:/www.math.sfu.ca/histmath/India/12thCenturyAD/Bhaskara.html,