Historical Significance of Algebra
Algebra is a branch of mathematics that emerged in the 16th century in Europe. Developed by Francois Viete, algebra involves computation with non-numerical mathematical objects almost the same to those of arithmetic. Later in the 19th century, algebra evolved and was adopted in the theories of equations such as fundamental theory of algebra (Kleiner & Israel, 45). Algebra is an Arabic word Al-Jabar from Muhammad ibn Musa al-Khwarizmi’s the Compendious Book on Calculation by Completion and Balancing, a treatise written in 820 by the medieval Persian mathematician. (Smory Aski & Craig, 125). The treatise had systematic solution for linear and quadratic equations. Computation was done by cancellation of similar terms in the opposite side of the equation to balance and reduce the equation.
Before 19th century, algebra used symbols in different stages. First was the rhetoric algebra in which equations were written in full sentences. For instance x + 2 = 3, was rhetorically written as ‘the thing plus two equals three’. This rhetoric algebra was developed by ancient Babylonians and was used until 16th century. The second stage was the syncopated algebra which represented symbolic algebra although not in all the equation. Symbols were used at one side of the equation and first appeared in Diophantus Arithmetic (Hettle & Cyrus, 72). The last stage was the development of symbolic algebra. In this period, full symbolism was used in the equation. Most of the materials credits to the Islamic mathematician such as al-Qalasadi and Ibn al-Banna although it is Francois Viete who fully developed symbolic algebra.
Quadratic equations also played an important role in the development of early algebra until all the quadratic equations were classified and belonged to one of three categories. Between the rhetoric stage and the syncopated stages of symbolic algebra, there was geometry constructive algebra. It was developed by the classical Greek and Vedic Indian mathematicians where geometry was used to solve algebraic equations. The conceptual stage included geometric stage, static equation-solving stage, dynamic function stage and abstract stage (Stedall and Jacqueline, 135).
Babylonian algebra is one of the oldest forms of algebra and was mainly used for approximation. It dates back to 1900-1600 BCE. Algebraic mathematics was encrypted in tablets and unlike the Egyptian algebra which was mainly linear equations, Babylonian algebra was more advanced and included both quadratic and cubic equations. Another place where algebra is said to have been studied is in Greek, China and Islamic Arab Empire. European algebra is considered to be more recent and modern as well as more detailed. Abstract algebra was developed more in the 19th century where Cambridge university students used physical variables such as time, space and weight to do exercises (Frejd &Peter, 92).
There were lacks of information to credit either Hellenistic mathematician, Diophantus or Al-khwarizmi as the ‘father’ of algebra. This was because there were various types of algebra that were developed differently. First, algebra found in Al-jabr is elementary and found in arithmetic that are both associated
with Diophantus. On the other hand, Al-khwarizmi is said to have provided algebraic solution of quadratic equations for positive root and taught algebra in an elementary form (Hettle & Cyrus, 312). Secondly, Diophantus dealt with theory of numbers while Al-khwarizmi introduced reduction and balancing as means to solve algebraic equation where cancellation was done on the opposite side of the equation.
Contribution of Carl Friedrich Gauss in Area of Mathematics
The 19th century was also an important age in the development of mathematics. There was unprecedented increase in complexity of mathematical concept and several mathematicians from France and Germany were credited to the revolution. Among them was Carl Friedrich Gauss, famously referred to as the ‘prince of mathematics’. He had a wide field of study covering mathematics and science. He ranked among the most influential mathematicians in the 19th century. He was able to distinguish his ability in mathematics by correcting his father’s pays lip error at the age of 3 years. At the age of 7 years, he summed up the integers from 1 to 100 by pairing them (Andrea, & Fiocca, 52). At the age of 12, he was able to challenge Euclid’s geometry.
Gauss attended Collegium Carolinum at the age of 15 years through a Duke of Brunswick and then joined university of Gottingen where he discovered several theorems. He was the first mathematician to discover and find the pattern of occurrence of prime numbers by graphing the incidence of prime numbers as they increased. At 19 years, Gauss constructed hitherto using a compass and a ruler which had seventeen sides. This was a major step as only Greeks had such knowledge at the time. He later had many discoveries and contributions in mathematics especially in complex numbers (Herivel & Gauss, 42). He highlighted the clear exposition of complex numbers. He showed how real numbers connected with imaginary numbers which he did by expressing them graphically. It is after this exposure that the theory of complex numbers received huge attention. At the age of 22, he made a breakthrough in algebra by proving fundamental theorem of algebra. This was an important step in understanding the field of complex numbers in closed algebraic.
In his book Disquisitiones arithmeticae published in 1801, he laid the foundation of modern number theory. It showed Gauss’s method for solving modular arithmetic and proved the law of quadratic reciprocity. Gauss was also interested in theoretical astronomy and was able to predict the position of Planetoid Ceres while working as the director of the astronomical observatory at Gottingen.
One of the major contributions of Gauss is in the field of statistics where he introduced Gaussian distribution, error curve and function. In addition, one of the most notable invention in mathematics he made and used in number theory, computer science, abstract algebra, visual and music art and cryptography, is the modular arithmetic (Gauss & Dunnington, 69). He also had some contribution in physics where he invented the first electric telegraph and his name was used to state the internal unit of magnetic induction due to his contribution in electromagnetism. He also collaborated with Wilhelm Weber in measuring the earth magnetic field.
References
Del Centina, Andrea, and Alessandra Fiocca. ‘The Correspondence between Sophie Germain and Carl Friedrich Gauss’. Arch. Hist. Exact Sci. 66.6 (2012): 585-700. Web. 20 June 2015.
Frejd, Peter. ‘Old Algebra Textbooks: A Resource for Modern Teaching’. BSHM Bulletin: Journal of the British Society for the History of Mathematics 28.1 (2013): 25-36. Web. 24 June 2015.
Gauss, Carl Friedrich, and G. Waldo Dunnington. ‘The Historical Significance Of Carl Friedrich Gauss In Mathematics And Some Aspects Of His Work’. Mathematics News Letter 8.8 (1934): 175. Web.
Herivel, J. ‘Carl Friedrich Gauss: A Biography by Tord Hall. Pp 176. The M.I.T. Press, London. 1970. E3.75’. Endeavour 30.111 (1971): 155-156. Web.
Hettle, Cyrus. ‘The Symbolic and Mathematical Influence of Diophantus’s Arithmetica’. Journal of Humanistic Mathematics 5.1 (2015): 139-166. Web. 24 June 2015.
Kleiner, Israel. A History of Abstract Algebra [electronic resource] Springer e-books; Boston, MA : Birkh Auser Boston, 2007. Empire State College E-Book Catalog
Smory Aski, Craig. History of Mathematics [electronic resource] Springer eBooks; New York, NY: 2008. Empire State College E-Book Catalog
Stedall, Jacqueline A. A Discourse Concerning Algebra. New York: Oxford University Press, 2002. Print.
