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### P2-143

Course: MATH 331, Fall 2008
School: CSU Channel Islands
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Math Roybal 331 11/23/2007 Development of Algebra A true knowledge of algebra is an invaluable attribute in the world of mathematics. Algebra has been defined as the branch of mathematics in which letters are used to represent basic arithmetic relations (Encarta 1). The development of this branch of mathematics has been divided into three stages: rhetorical algebra, syncopated algebra, and symbolic algebra (Eves...

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Math Roybal 331 11/23/2007 Development of Algebra A true knowledge of algebra is an invaluable attribute in the world of mathematics. Algebra has been defined as the branch of mathematics in which letters are used to represent basic arithmetic relations (Encarta 1). The development of this branch of mathematics has been divided into three stages: rhetorical algebra, syncopated algebra, and symbolic algebra (Eves 179). Algebra is practical in its current form for many reasons: it uses letters to represent unknown or large numbers, provides a standardized method for solving equations, and provides a way to check answers. However, Algebra was not always such a practical mathematical tool; it was invented and developed through the labors of many mathematicians of previous civilizations. The algebraic styles of the Egyptians and Babylonians, two notable civilizations of antiquity, were rather similar in their was basic and practical nature. These early civilizations focused more on the necessity of survival than later civilizations, such as the philosophical Greeks, and their mathematics was used mainly to keep records and engineer. The Babylonian development of algebra was notably sophisticated for its time: By 2000 B.C. Babylonian arithmetic had evolved into a well-developed rhetorical, or prose, algebra (Eves 42). It is important to know that algebra was not always in the form that is now used; it originated as written words, rather than numbers and symbols. A mathematical exercise would have been a written question: An area A, consisting of the sum of two squares is 1000. The side of one square is less than 2/3 of the side of the other square. What are the sides of the squares? (Eves 58) This form of algebra could clearly make work more difficult than it is now. The algebra of Egyptians was slightly more developed than that of the Babylonians: There is some symbolism in Egyptian algebra. In the Rhind papyrus, we find symbols for plus and minus (Eves 55). These two symbols could shorten the length of written problems by replacing the phrases they represented. By approximately 1650 B.C., algebra was already progressing toward the symbolized form it now has. Over a millennium after the Egyptians came the Greek mathematicians who contributed greatly to the development of algebra. A crucial contributor to the Greek development of algebra was Diophantus: One of Diophantus outstanding contributions to mathematics was the syncopation of Greek algebra (Eves 179). Syncopated algebra is the style in which abbreviations are adopted for some of the more frequently recurring quantities and operations (Eves 179). This work of Diophantus did not affect the majority of the world until the fifteenth century (Eves 179). It is worth noting that Greek mathematical problems can be readily solved by geometrical algebra, but it is believed that they were actually solved arithmetically (Eves 179). Much of Greek mathematical work was in the field of geometry, so it follows that they would solve their problems with a style of algebra that favored their strength. After the Greeks had syncopated algebra, it was up to later civilizations to make the step to symbolic algebra. Algebra was eventually symbolized by Arabian mathematicians, as a result of extensive administration (Eves 232). This symbolism was used with different numeral systems proceeding from local numeral systems to Greek the system to the Hindu system. After the Hindu notation was introduced by those involved in commerce, later writers were influenced by Greek methods. Because of this, the Arabian mathematicians used a rhetorical style instead of the useful Hindu system (Eves 233). Symbolism, however, made a recovery, and modern mathematics is evidence to this. It is worth noting that it is from the Arabian mathematician al-Hkowrizms treatise on algebra, Hisb al-jabr wal-muq-balah, that we draw our word for the term: This title has been literally translated as science of the reunion and the opposition or, more freely, as science of transposition and cancellation. The textmade the word al-jabr, or algebra, synonymous with the science of equations (Eves 236). For the naming and symbolizing of algebra, the Arabian mathematicians are very important to the history and development of mathematics. The best way to appreciate what this development of algebra means for modern mathematicians, a single problem must be considered in all three algebraic styles: rhetorical, syncopated, and symbolic. Consider the following problem: One half of the monks in a mountaintop monastery decide to visit a nearby town to retrieve supplies for the next week. Given the number of monks in the monastery, the weight of supplies that each monk needs for the next week, the average weight of a monk, and the weight of the basket in which all will ride, how much weight will be pulled up the mountain when the monks are returning? This is the problem in its rhetorical form, surrounded with an interesting story to avoid monotony. The following is the problem in syncopated form: (one half) (number of monks) (average weight) + (weight of basket) + (number of monks) (weight of week of supplies) = (total weight). Now, the transition to the symbolic form requires the conversion of these terms into symbols: n=number of monks, WB=weight of basket, WM=average weight of a monk, WS=weight of one week worth of supplies, and WT=total weight. The following is the solut...

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CSU Channel Islands - MATH - 331
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