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Bhaskara


Bhaskara Akaria lived from approximately 1114 to 1185 in India. Born into a traditional family of Indian astrologers, he followed the family's professional tradition, but with a scientific orientation, focusing more on mathematical and astronomical aspects (such as calculating the date and time of eclipses or the positions and conjunctions of planets) that supports Astrology. Its merits were soon recognized and very soon it reached the post of director of the Ujjain Observatory, India's largest center for mathematical and astronomical research at the time.

He wrote two mathematically important books and because of this he became the most famous mathematician of his time.

His most famous book is the Lilavati, a very elementary book devoted to simple problems of Arithmetic, Flat Geometry (measures and elementary trigonometry) and Combinatorics. The word Lilavati it is a woman's proper name (the translation is Graciosa), and the reason she gave this title to her book is because she probably would have wanted to make a pun comparing the elegance of a woman of the nobility to the elegance of the methods of arithmetic.

In a Turkish translation of this book, 400 years later, the story was invented that the book would be a tribute to the daughter who cannot marry. It is precisely this invention that has made it famous among people with little knowledge of mathematics and the history of mathematics. It also seems that teachers are very willing to accept romantic stories in such an abstract and difficult area as mathematics; it seems to humanize her more.

Bhaskara's other work was:

Undetermined equations or diophantines
We call the equations (polynomials and integer coefficients) with infinite integer solutions, such as:

  • y-x = 1 that accepts all x = a and y = a + 1 as solutions, whatever the value of The
  • the famous Pell x equation2 = Ny2 + 1
    Bhaskara was the first to succeed in solving this equation by introducing the chakravala (or spray) method.

But what about Bhaskara's formula?

  • EXAMPLE:
    to solve the quadratic equations of the form ax2 + bx = c, the Indians used the following rule:
    "multiply both members of the equation by the number that is worth four times the square coefficient and add to them a number equal to the square of the original unknown coefficient. The desired solution is the square root of it."

It is also very important to note that the lack of an algebraic notation, as well as the use of geometric methods to derive rules, made Rule Age mathematicians have to use various rules to solve quadratic equations. For example, they needed different rules to solve x2= px + q and x2+ px = q. It was not until the Formula Age that attempts to give a single procedure to solve all equations of a given degree began.

Bhaskara knew the rule above, but the rule was not discovered by him. The rule was already known to at least the mathematician Sridara, who lived more than 100 years before Bhaskara.

Summarizing Bhaskara's Involvement with Quadratic Equations:

  • For DETERMINED equations of the second degree:
    In Lilavati, Bhaskara does not deal with certain quadratic equations, and what he does about it in Bijaganita is a mere copy of what other mathematicians had already written.
  • Regarding undetermined quadratic equations:
    Then he really made great contributions and these are on display in the Bijaganita. It can be said that these contributions, especially the invention of the iterative method of chakravala and its modification of the classic method kuttaka they correspond to the apex of classical Indian mathematics, and it may be added that it is only with Euler and Lagrange that we will again find technical resourcefulness and fertility of comparable ideas.

Bibliography: Information from the UFRGS website.