A public monument to the mean vaule theorem in Bejing. Can you imagine such a thing in America? We have a hard enough time putting our great mathematicians on stamps!
Bhaskara the Teacher
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Whilst making love a necklace broke. A row of pearls mislaid. One sixth fell to the floor. One fifth upon the bed. The young woman saved one third of them. One tenth were caught by her lover. If six pearls remained upon the string How many pearls were there altogether? --Bhaskara, Lilavati |
Moving on a bit, Bhaskara was one of the more interesting mathematicians of the middle ages. Hailing from twelfth century India, he was a pioneer in his time, whose mathematical achievements would take centuries to be rivaled. He developed the proof of the Pythagorean Theorem discussed in the previous episode, but his most famous work is in arithmetic and algebra.
One of his first works, named after his daughter Lilavati, is about arithmetic. The book was writing as though he were writing it for his daughter, with him posing exercises for the reader as though he were asking Lilavati a question. The place value system was well understood in Bhaskara’s time, and we easily recognize some of the methods he teaches in the beginning of the book. “Oh Lilavati, intelligent girl,” Bhaskara writes, “if you understand addition and subtraction, tell me the sum of the amounts 2, 5, 32, 193, 18, 10, and 100, as well as [the remainder of] those when subtracted from 10000.” Easy enough, right?
Well, it gets harder. “Tell quickly, intelligent calculator, what are the quantities whose difference is eight, and the difference of whose squares is four hundred?” This one is more of a head scratcher. He wants two numbers x and y, such that
Not all of Bhaskara’s flair was fancy arithmetic tricks. He was also the first to generalize the solution to the quadratic equation, which is usually remembered today to the tune of “Pop Goes the Weasel”: “x equals negative b / plus or minus the square root / of b squared minus four a c / all over two a.” Bhaskara wrote of division by zero (thought it was infinity, much as we think of it today) and wondered about the square root of negative one (thought there was no solution; we’ve expanded the number system since then). (MacTutor) He also developed some of the basic methods that would later make up the foundations of differential calculus, and stated Rolle’s Theorem, a special case of the mean value theorem, one of the most important results in all of differential calculus (in fact, it’s used to prove the fundamental theorem of calculus).
This is a pretty impressive resume for one of the forgotten mathematicians of the East. Few things that Bhaskara wrote we would think of as formal proofs today, at least not in the Euclidian tautological sense. Bhaskara was a teacher, and preferred to show his readers how something worked without spelling out all of the little bits for them. Still, if he had proven Rolle’s Theorem rather than just saying that it’s true, perhaps we’d call it Bhaskara’s Theorem today.
A public monument to the mean vaule theorem in Bejing. Can you imagine such a thing in America? We have a hard enough time putting our great mathematicians on stamps! |
Bhaskara II, Lilavati. Translated by Brown University History of Mathematics Department. c. 1150. http://www.brown.edu/Departments/History_Mathematics/lilavati.html. Accessed 9/22/08.
O’Connor, John & Robertson, Edmund. "Bhaskara". MacTutor History of Mathematics Archive. http://www-history.mcs.st-andrews.ac.uk/Biographies/Bhaskara_II.html. Accessed on 11/12/08.
Photograph by Vmenkov of Wikimedia Commons, licensed CC-BY-SA.
