Unit 1 Section 2 - 1.2. The Real Number System as a Number...

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1.2. The Real Number System as a Number Field In this section, we will attempt to present the set of real numbers together with the defined operations on the set as a mathematical system . By this, we mean, we are going to construct the real numbers by use a minimum number of axioms (generally accepted propositions or statements) and proceeding to deduce more statements (generally called theorems) that will characterize the system. Since most statements will be about algebraic properties, it is natural to call this system an algebraic system (or algebraic structure ). Numbers have several uses; naming, ordering, counting, and measuring, to name a few. In this course, we shall be using numbers for counting and measuring. We have seen numbers used to count the elements of a set, hence the term “counting numbers”. In some instances, we will be using numbers to measure the length of a line segment or the area of a region; thus, the counting numbers will not suffice for these purposes. These numbers will be collective called real numbers and we shall denote by R the set of all real numbers. First let us state the axioms that deal with equality of real numbers. These properties are so basic that we sometimes take them for granted. Axiom E1 . Reflexive Property of Equality For any a R, a = a. Axiom E2 . Symmetric Property of Equality For a, b R, if a = b then b = a. Axiom E3 . Transitive Property of Equality For a, b, c R , if a = b and b = c, then a = c. In elementary mathematics, particularly in arithmetic, we learned two basic operations on real numbers, namely addition and multiplication . We will assume we all know how to perform these operations . Addition of real numbers is denoted by “+” while multiplication is denoted by ”. Thus we write, 2 + 3 = 5 and 2 3 = 6. The result of addition is called the sum , while performing multiplication results to a product . In the axioms that follow, there will be one form for addition and one for multiplication and we will indicate this by placing an “A” or an “M” after the number of the axiom. The next axioms are so familiar that reference to them is often neglected but they are important in the study of algebraic systems. Axiom 1A . Closure Property for Addition For any a, b R , a + b R . (The sum of two real numbers is a real number.) Axiom 1M . Closure Property for Multiplication For any a, b R , a b R . (The product of two real numbers is a real number.)
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The closure property guarantees that an operation between two real numbers results to a real number. In short, we say, R is closed under addition and multiplication . The next set of axioms deals with the way we group real numbers whenever we add or multiply more than two real numbers. Axiom 2A
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This note was uploaded on 05/13/2011 for the course MATH 17 taught by Professor Dikopaalam during the Spring '11 term at University of the Philippines Los Baños.

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Unit 1 Section 2 - 1.2. The Real Number System as a Number...

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