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Unformatted text preview: Unit 1. Algebra as the Study of Structures Algebra started in ancient Egypt and Babylon where the preoccupation of early settlers was concentrated to solving linear equations (ax = b), quadratic equations (ax 2 + bx = c) and indeterminate equations (a x 2 +b y 2 = c z 2 ), whereby several unknowns are involved. Of course, the way they wrote their equations and the variables were very much different from how we them now. Interestingly, the ancient Babylonian’s solution of the quadratic equations resembled our present method of solving them today. The humble beginnings of Algebra continued a tradition of finding solutions of equations from very specific ones to the more general. This preoccupation spread across the Islamic world and later on to the continental Europe and the Americas over several centuries. Many renowned mathematicians, including Cardano, Ferrari, Abel and Galois, kept themselves busy with the quest for finding solutions of polynomial equations (ax n + bx n-1 + …+ c = 0) in their most general form. Even the philosopher Descartes contributed much to the development of the subject. However, it was during the time of Gauss when the modern phase of Algebra commenced. Rather than deal on the solution of equations, the focus was redirected to the study of the structure of mathematical systems composed of objects that behave in some common manners. From them on, Algebra has been considered as the study of mathematical structures. OUTLINE: 1. Sets, Set Operations and Number Sets: The Basic Objects of Algebra 2. The Real Number System as a Number Field a. Group properties b. Ring properties c. Field properties d. Ordered field properties e. Completely ordered field 3. The Complex Number System as a Number Field a. Group properties b. Ring properties c. Field properties 4. The Ring of Polynomials a. Addition and subtraction b. Multiplication 5. The Field of Algebraic Expressions a. Addition and subtraction b. Multiplication c. Division Unit 1. Algebra as the Study of Structures Section 1 page 2 _____________________________________________________________________________________ 1.1. The Basic Objects of Algebra: Sets, Set Operations and Number Sets The concept of “set” has pervaded almost all of mathematics so that it has become a fundamental concept. Due to this, it becomes impossible to define precisely in terms of more basic concepts. However, our real world experience has provided us with an intuitive knowledge of the notion of a set to rely on. Whenever a group of objects is formed, to our mind, a set is formed. Thus, it is east to accept that a set is simply a collection of objects, real or imagined . The only condition we impose on a collection to become technically a set is that it is possible to determine (in some manner) whether an object belongs to the given collection or not. For example, we may consider the following as sets: a) The set of freshman students of the present class b) The set of students in this class whose last name begins with the letter P...
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