The Development of Algebra

The word algebra is usually derived from multiple types of mathematical techniques and concepts which tend to correlate to manipulation of formal abstract symbols directly and to find equations solution. Algebra is a sub-discipline of mathematics which has been derived from the historical evolution of concepts. The notion of Algebra is a discipline of mathematics is derived from the natural and straightforward idea of equations which involves a significant amount of own mathematical assumptions and mutual interactions in which each of them has been obtained from an intricate and lengthy historical procedure (Button, 2010). The consolidated concept of an equation is derived from the works of Viète in the historical period of the sixteenth century which offers a mathematical idea comprising of two parts on which the mathematical operations can be performed concurrently (Button, 2010). By conducting the mathematical procedure, the equation tends to remain constant but the when undertaking the mathematical method, we resolve the numerical figure of the unknown quantities. In discovering the numerical value, an individual is required to conduct three simple steps that will be used to provide the solution of the equation to the algebraic equation. 

In 1650 BC, Egyptians developed their mathematical concepts. The mathematical texts of the Egyptians were able to offer solutions to linear equations. In the period of 300 BC, shreds of evidence reveal that similar solutions to a mathematical equation that involved two unknown quantities (Button, 2010). However, the equation which included the two numbers did not require the amount in solving the mathematical equation. In this historical period, the mathematical solution equation was uttered and at the same time solved in a verbal manner. 

The Babylonians developed their mathematical concepts which date from 1800 BC (Button, 2010). The mathematical concepts were derived from cuneiform texts that were stored on clay tablets. The arithmetic’s of the Babylonians were based on the positional sexagesimal criteria and well elaboration of the mathematical concepts. However, the Babylonian arithmetic’s had an inconsistency in the use of the numerical figure zero. The mathematical concepts of the Babylonians included tables, exponentials squares, reciprocal and multiplication tables, cube roots, and square. However, besides that the Babylonian mathematics using tables they also provide solutions of unknown numbers. Furthermore, the Babylonian text tends to offer the steps that were being followed to solve the mathematical equation. The first step would involve creating relationships between absolute numbers and unknown values. The solved number could be a square numbers root, the stone’s weight, and a triangle’s length. The derived solutions were didactical figures but not practical exercises. 

Greeks made a significant contribution to the mathematical world of arithmetic. The Greek mathematics was a milestone discovery which was created by Pythagoreans in a historical period of 430 BC. The Pythagoreans developed specific ratios which included pairs of magnitudes which had an indirect correlation towards simple rations in whole numbers (Button, 2010). The astonishing fact is that fundamental metaphysical Pythagoreans beliefs became vivid that the links between the geometrical magnitudes. The invention of such quantities led to the design of the logic of proportion in the world of Algebra. The concept of proportion in a that was developed led it to be a revolutionary tool in the world of Mathematics. However, there is a comparison between the Greek proportion and the modern identity as no principle od-f equation can be derived from the two concepts. For example, the proportion tends to provide an equation of ratios between components of the line, say B, A, is similar to the ratio between the two areas S, R. The Greek would suggest that the equation was a form verbal fashion. Furthermore, shreds of evidence of shorthand expression were not discovered in the Greeks texts. The concept of proportions was a milestone discovery as it offered solutions yet it could not derive the original equation compared to the new equation which is being calculated by scholars. The main feature of Greek mathematics is that it could not compare simultaneous solutions which could only be made among similar magnitudes. The fundamental demand for uniformity would be preserved in all mathematical derivations which were being retrieved from Greek mathematical sources till the contribution of Descartes. 

Geometrical construction was made by Greek mathematicians which tend to appear in Euclid’s elements. However, when the components are translated were translated by modern mathematicians it vividly indicated algebraic identities and solving of quadratic equations. However, the mathematical concepts of the Greeks were being derived for the idea of proportions as not only symbols could be retrieved from the mathematical texts of the Greeks but also the whole algebraic theory (Hannah, 2015). The unknown quantities that were present in the Algebraic solution indicated that the answers could be calculated through the method of manipulation. However, the discipline of geometry was also used in offering solutions through symbolic manipulation (Dieudonne, 2017). The arithmetic operations played a critical role in Greeks logics of the algebra of preferring domain rather than algebraic one (Dieudonne, 2017). In the classical Greek concept of arithmetic, which is derived from the Books VII-XI of Euclid’s components, a numerical figure is a form units collection which is known in the contemporary world as a natural number (Hannah, 2015). In the concept, negative numbers are included, and zero is not added to the picture. Furthermore, the status of one in the picture is considered as a matter of ambiguity in specific texts as it not included in Euclid’s collections. Eliud’s contribution in the disciple of algebra led to the development of a more flexible and elaborate idea of a number which is a primary factor in the development of algebra. For example: 

x2 = (a – x). a 

Diophantus of Alexandria who was a Greek mathematician made a significant impact in the world of algebra. The contributions that Diophantus of Alexandria made in algebra was solving equation problems that were involved in second-degree equations or equation which had several variables. Diophantus of Alexandria relied on Greek mathematical concepts by including rational and positive solutions. The mathematician did not involve negative numbers in creating mathematical solutions, and to him negative numbers were absurd. Furthermore, his contributions did not offer general techniques that would be considered appropriate in solving standards equations. Diophantus of Alexandria solutions tend to offer more than one answer to the equation and would sometime go an extra mile to offer an infinite answer. However, Diophantus of Alexandria would suddenly stop after identifying the first solution. In equations involving quadratic equations, he never suggested that the equations could offer two solutions as h never tried to identify more solutions to the quadratic expression. Diophantus of Alexandria was the first Greek who introduced a form of systematic symbolism that could easily be manipulated. The idea was a type of short-hand writing rather than actual symbols that would easily be manipulated. The use of actual symbols would arise in instances that were within the stricture of limited possibilities. The original works of Diophantus always begun with an unknown quantity which would be used to solve a specific problem using specific values that would facilitate the easy solution to the algebraic problem. For example 

s=3 

a=30 

r=2 

b=50 

2x4 – x3 – 3x2 + 4x + 2 

Indian and Chinese also made a significant contribution to the world of algebra. India had mathematicians such as Bhaskara and Brahmagupta who established non-symbolic procedures that were used in solving. However, the primary contribution of the Hindu was the illustration of decimal and positioning of numerical figures which led to the establishment of symbolic algebra in traditional Europe. Moreover, in Hindu arithmetic, the establishment of a correct and consistent set of regulations for operation in both negative and positive numbers. The numerical figure if zero was treated as a number even in problems which involved division. However, it took long decades for the European mathematicians to integrate the ideas of Hindu concepts of algebra in establishing the concept of algebra. The Chinese also made their contributions in the discipline of algebra as they created their unique techniques in solving of quadratic equations using radicals and classifying such solutions. The Chinese were unsuccessful in solving higher degree equations using the same criteria. Yang Hui of 12 AD developed approximation techniques of high accuracy. 

The Islam’s made a significant impact in the world of mathematics particularly in the period of A.D 825 when a mathematician known as Muhammad ibn Musa al-Khwarizmi. However, by the 19th century, his works had been translated in the form of Arabic language through the works of mathematicians such as Archimedes, Euclid, Ptolemy, Apollonius, and Diophantus Apollonius through the mathematical corpus. Furthermore, Hindu mathematics and Babylonian mathematics were also available to scholars during the traditional period. The Islamic contributions that wee integrated involved the use of a numeric system of the Hindu which lacked sections of decimal fractions in their algebraic equations. 

Al-Khwarizmi’s included a practical value that offered steps in solving six kinds of equations which included squares equal numbers, squares similar roots, squares, numbers same roots and squares, roots equal numbers and the number corresponding to a square. Negative values and zero were considered as legitimate solutions to the algebraic equations (Clotfelter, Ladd, and Vidgor, 2018). The techniques were described verbally. The works of Al-Khwarizmi’s also relied on the concept of proportions in solving the algebraic equations. For example: 

bx = c 

ax2 + bx = c 

ax2 + c = bx 

bx + c = ax2 

Leonardo Pisano made a significant contribution in the discipline of algebra in his Liber Abaci which was written in 1202. However, his contributions were no innovations as it only adhered to specific steps of Islamic theory in solving and formulating algebraic problems in what was considered as a rhetorical fashion by many. The works reflected a useful communication tool to the Latin world. For example R. V: cu.R. 108 p: 10 m: R. V: cu. R. 108m: 10. 

Meaning x= -

Gerolamo Cardano was a famous physician of Italian descent who was a prolific writer and had personal interests towards mathematics (Clotfelter, Ladd, and Vidgor, 2018). Gerolamo made a significant contribution towards solving of third-degree equations and fourth-degree equations. Furthermore, he significantly used symbolic figures in solving of the algebraic equations Viète was a prolific lawyer who has an enormous interest in mathematics. Established high cryptographic skills which were put into use in the reign of King Henri III (Clotfelter, Ladd, and Vidgor, 2018). Viète used well-known symbols with unknown symbols and the use of known consonants. The technique facilitated generality and flexibility in solving of algebraic equations. Viète also ensured that his works showed the links that were available between the coefficient values and the original algebraic equations. Viète also made some contributions towards the classical algebra. For example: 

C plano + A cubus in A aequatus D solido 

contemporary terms: d= x3 + cx 

Descartes’ contribution was a commencing point for the definite change of polynomials into an autonomous element of mathematical interest. Algebra has developed to an extent and the polynomials theory. The polynomial approach of algebraic equation advocated for a systematic and coherent reformulation which would be integrated to offer algebraic equations (Arano, de Laat, and Wahl, 2018). However, the algebraic techniques could be used in solving algebraic equations that were more than a degree of four. Furthermore, polynomial equations would generate more than one solution from a specific algebraic equation. A breakthrough of solving an algebraic equation of which was more than four degrees was made in 1770 by mathematicians such as Lagrange, Gauss, Abel, and Ruffini who use basic methods to acquire the solutions to the algebraic equation. 

References 

Arano, Y., de Laat, T., & Wahl, J. (2018). The Fourier Algebra of a Rigid C*-Tensor Category.  Publications of the Research Institute for Mathematical Sciences ,  54 (2), 393-411. 

Burton, D. (2010).  History of Mathematics . London: McGraw-Hill Publishing. 

Clotfelter, C. T., Ladd, H. F., & Vigdor, J. L. (2015). The aftermath of accelerating algebra evidence from district policy initiatives.  Journal of Human Resources ,  50 (1), 159-188. 

Dieudonne, S. (2017).  History Algebraic Geometry . Routledge. 

Hannah, J. (2015). Taming the Unknown&58; A History of Algebra from Antiquity to the Early Twentieth Century by Victor J. Katz and Karen Hunger Parshall edd.  Aestimatio: Critical Reviews in the History of Science ,  12 (unknown), 13-20.