Mathematics and Economics Research Paper: The History of the Quadratic Equation

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The History of the Quadratic Equation
(2519 words)
1.0 General Introduction
As taught in most high schools, the quadratic formula x1,2=(-b/2a) ± (1/2a)(b2-4ac)1/2 provides a solution to a generic equation in the form of ax2 + bx + c = 0. The development of mathematical concepts and equations take the nature of deducibility, logicality, and rectilinearity as much as possible, and the historical development of these concepts has been continuous, logical, and rectilinear. This implies that one mathematician has been picking up ideas from where another one left it. However, this paper uses a quadratic formula to show the historical development of mathematics has not followed the rectilinearity of other mathematical concepts, but there have been parallel developments, confluences, and interconnections in its development and the paper will show the interrelated social, political, cultural, and religious issues connected with the development of the quadratic equation. To explain the history of the quadratic equation, this paper reviews a brief history of the quadratic equation focusing on the Egyptian contributions, the Babylonian and Chinese input, the concepts of Pythagoras and the Euclid, the Brahmagupta’s concepts, and the Al-Khwarizmi’s ideas, and finally the ideologies of Rene Descartes. To supplement this discussion, this paper provides a discussion on the practical application of the quadratic equation in determining the sales that earn a profit to a business.
2.0 A brief history of quadratic equations
The engineers in Egypt, China, and Babylon were smart and intelligent humans who understood that the area of a square scales with its side lengths. The engineers knew that it was possible to store nine times more bales of hay by tripling the square of a loft. Besides, they understood how to compute areas of more complex designs such as T-shapes, rectangles, and many other shapes. The only problem they encountered was how to calculate the sides of the shapes, such as the length of the sides beginning from a known area. Unfortunately, this is what most of their clients required. Since this was the original problem of understanding how to scale a certain shape with the total area and knowing what is needed in terms of sides and length and the walls to make a working floor plan, it troubled the engineers to work out a method to resolve the issue (Mathnasium). This problem could be solved using a quadratic equation as understood historically and by today’s mathematicians.
Fig 1 The modern quadratic equation (Delbert)
1 Egyptian contribution
One of the most original concepts that resulted in a quadratic equation was the realization that it was related to the very pragmatic issues that as a result required a dirty and quick solution. However, Egyptian engineers did not understand numbers and equations the way they are understood by modern mathematicians. They followed rhetorical, descriptive, and hard to understand concepts. However, most Egyptian Wisemen, including priests, scribes, and engineers knew the shortcomings of these approaches and came up with methods to circumvent this issue. Instead of learning a formula or an operation that could find the sides of an area, they computed the area of shapes of rectangles, and squares with their possible sides and constructed a look-up table. The method they made works so much with what is currently available for learning multiplication tables in modern schools by heart instead of working out proper operations. This means that if an engineer wanted to know a loft with a certain capacity and shape to store different bales of papyrus, they would refer back to the table and find the most fitting design (Lawrence and Storm). However, the Egyptian engineers did not have the time to find out all shapes and sides to construct their tables. Instead, they used a reproduction of a master look-up table and the copyists poorly understood if the table they were copying made a meaning or not as they never understood anything about mathematics. Therefore, they at times made serious errors and the copies of other copies were not devoid of errors and were not trustworthy. As these tables still exist today, it is possible to point out where the errors crept in as the copyists made copies of the documents.
2 Babylonian and the Chinese contribution
While the Egyptian method could work fine, mathematicians wanted a more general solution that did not need tables as a desirable method. At this point, the Babylonian mathematicians came into play and had a significant advantage over the Egyptian mathematicians. In particular, the Babylonians applied a number system that nearly looked like the one modern mathematician use today, although they used a hexagesimal basis or base-60. It was easy to carry out additions and multiplications with this system and the engineers during 1000 BC could double-check the values in their tables (Lawrence and Storm). During 400 BC, mathematicians found a more general method known as the “completing the square” in solving generic problems that involved the areas. However, there is no evidence that the Chinese and the Babylonians made use of any specific mathematical method in finding out the solutions, and this points to some educated guessing was largely involved in the process. Around the same period, or at least a bit later, the method also appears in the Chinese documents (Lawrence and Storm). The Chinese people, like the Egyptians, did not have the numeric system but a method that tends to appear in the Chinese documents. Like the Egyptians, the Chinese never used the numeric method but rather a double-checking of simple operations in mathematics that was made extremely easy by the widespread application of the abacus.
1 Pythagoras and Euclid contribution
Perhaps the first attempts to discover the general formula used in solving a quadratic equation can be traced back to the geometry to-bananas Pythagoras who lived in Croton, Italy in 500 BC and Euclid in 300 BC in Alexandria, Egypt who employed a stricter geometric approach to finding a general procedure of solving a quadratic equation. During this time, Pythagoras discovered a general procedure that could solve a quadratic equation. Using the method, Pythagoras found out that the ratios between the areas of a square and the length of the side of the square root, were not always an integer, but then refused to allow for the promotion of other methods than the rational. Pythagoras always hated the idea of rational numbers and 268 years later, Euclid had to prove him wrong (Lawrence and Storm). However, according to Pythagoras, the ratios between the area of a square and the respective side length of the side of the square roots were not always an integer, but instead reduced to allow for proportions different from the rational ones.
Fig 2 Pythagoras (Burton)
Euclid, through his book, “The Elements” Euclid expanded upon the idea of Pythagoras to discover that the proportion was not always rational and provided the existence of irrational numbers. By the use of a strictly geometric approach, Euclid was able to advance the mathematical principle of a quadratic equation. He postulated that if a straight line will be cut into both equal and unequal segments, the rectangle that would be contained by the unequal segments of the whole together with the square on the straight line between the points of a section would be equal to the square on the half. For instance, if line AB would be cut into equal segments at C and unto unequal segments at D and if the rectangle contained at AD, DB together with the equal squire on CD would be equal to the square on CB. The proposition would be true since the rectangle AL and would have unequal content. In the 17th century, mathematicians applied proportions of II-5 and II-6 to justify the geometric of the standard algebraic solutions of quadratic equations (Lawrence and Storm).
2 Brahmagupta’s(Indian/Hindu) contribution
Since 600 A.D., Hindu mathematicians have used the decimal system that is presently used and one of the most important contributions of the system on Indian mathematics was its applications in commerce. Average Hindu merchants were fast in simple math as they understood t...