Bhaskara Formula: The Quadratic Formula Called ‘Bhaskara’ in Brazil
The quadratic formula, known as the Bhaskara formula in Brazil, has a rich historical background that involves not just a single mathematician but also a method attributed to his predecessor, Sridhara. This article delves into the origins and teachings of these formulas and methods, as well as their significance in Brazilian mathematics education.
Origins and Historical Context
The quadratic formula, a fundamental concept in algebra, has been around for over a millennium. It is named after the Indian mathematician Bhāskara II, who lived in the 12th century. Bhāskara II's work, the 'Lilavati,' is celebrated for its collection of mathematical puzzles and problems. However, the method itself, as described in the Chakravala technique, was a part of a broader mathematical tradition that extended back even further.
The Discovery and Contribution of Sridhara
Indian mathematicians prior to Bhāskara II, such as Sridhara, are credited with significant contributions to the understanding and solving of quadratic equations. In the 9th century, Sridhara developed a method for solving quadratic equations, which is recorded in mathematical texts and is an essential precursor to the modern quadratic formula.
As mentioned in the research by Guedes Eduardo Gomes, the original text by Bhāskara attributes the method to Sridhara. This attribution highlights the collaborative and continuous nature of mathematical advancements where knowledge and methods are passed down through generations.
Quadratic Formula in Brazil
In the Brazilian mathematics curriculum, the quadratic formula is known as the 'Bhaskara formula,' a testament to the historical significance of this formula in global mathematics. The formula, as described in Brazilian textbooks, follows the original method detailed by Sridhara and later refined by Bhāskara II.
The credit given to Sridhara by both Indian and Brazilian mathematicians and researchers shows how the global mathematical community recognizes the importance of contributions from diverse cultures and historical periods.
Understanding the Method
The method for solving quadratic equations, as described by Sridhara, involves a literal description of the quadratic formula. This method can be summarized as follows:
Multiply each term by 4a to complete the the square by adding (b^2 - 4ac).Taking the square root of both sides and solving for x.This method, while seemingly complex, is essentially a visual and algebraic interpretation of the quadratic formula:
[ x frac{-b pm sqrt{b^2 - 4ac}}{2a} ]
Examples of Problems Solved Using the Quadratic Formula
To illustrate the application of the quadratic formula, consider two examples provided by the ancient Indian mathematicians:
Example 1: A group of bumblebees, half of which went to Malti flowers, and a ninth part of the remaining joined the lotus. After some time, one bee remained. Determine the total number of bumblebees.Example 2: Arjuna had a certain number of arrows. He used half of them to kill Karna, four times the root of the original number to stop the horses, and the rest to defeat other forces. Calculate the total number of arrows Arjuna had.The solution to the first example involves setting up and solving a quadratic equation using the method described. For the second example, a similar approach is applied, where the total number of arrows is the sum of various parts described in the problem.
The solutions to these problems demonstrate the practical and theoretical applications of the quadratic formula and its historical significance in both ancient and modern mathematics.
Conclusion
The Bhaskara formula, named after Bhāskara II but rooted in the work of his predecessor Sridhara, plays a significant role in Brazilian mathematics education. It underscores the importance of recognizing and honoring the contributions of mathematicians from different cultural and historical backgrounds. Understanding the historical context and the development of mathematical concepts such as the quadratic formula provides a richer appreciation of the subject and its applications in modern times.