A fundamental property of the set R of real numbers : Completeness Axiom : R has \no gaps". Examples: 3 π 3 5 e Properties of Real Numbers Commutative Property for Addition: a + b = b + a 24 23 22 21 210 3 4 Example 1 Graph real numbers on a number line a2_mnlaect353043_c01l01-07.indd 1-1 9/16/09 7:16:39 PM 2 – 3) Addition a + b is a real number. Property Explanation 1. The Field Properties of the Real Numbers 85 3. The ordering < is compatible with the arithmetic operations means the following: VIII a < b =⇒ a+c < b+c and ad < bd for all a,b,c ∈ R and d > 0. 1.1 Real Numbers A. Below are some examples of sets of real numbers. 1 Thus the equivalence of new objects (fractions) is deﬂned in terms of equality of familiar objects, namely integers. Examples: ½ -¼ 0.19 4.27 31 The irrational numbers are numbers that cannot be written as an integer divided by an integer. VII given any two real numbers a,b, either a = b or a < b or b < a. Properties of Real Numbers Name: N o t es Date: Jamal is loading his catamaran for a long journey. 1) associative 2) additive identity Write an example to demonstrate it. 1.4. See also: Real Number Properties For any real numbers a, b, and c. Multiplication —a— a. bis a real number. a. rational numbers b. real number c. real numbers d. integers 2. Natural Numbers: (these are the counting numbers) 2. These are some notes on introductory real analysis. NOTES ON RATIONAL AND REAL NUMBERS 3 We say that a fraction a=b is equivalent to a fraction c=d, and write it as a=b » c=d if and only if ad = bc and b;d 6= 0. Whole Numbers : (same as , but throw in zero) 3. Two fundamental partial order relations are the “less than or equal to (<=)” relation on a set of real numbers and the “subset (⊆⊆⊆⊆)” relation on a set of sets. Each point on the number line corresponds to exactly one real number: De nition. A number line is an easy method of picturing the set of real numbers. Properties and Operations of Fractions Let a, b, c and d be real numbers, variables, or algebraic expressions such that b ≠ 0 and d ≠ 0. The collection of all real numbers between two given real numbers form an interval. 8S R and S6= ;, If Sis bounded above, then supSexists and supS2R. Example 1.1. Mathematical Induction 91 Appendix B. 1.2 Properties of Real Numbers.notebook Subject: SMART Board Interactive Whiteboard Notes Keywords: Notes,Whiteboard,Whiteboard Page,Notebook software,Notebook,PDF,SMART,SMART Technologies ULC,SMART Board Interactive Whiteboard Created Date: 8/19/2013 2:04:39 PM Properties of Addition Closure Property. A Dedekind cut of Q is a pair (A;B) of nonempty subsets of Q satisfying the following properties: (1) Aand Bare disjoint and their union is Q, (2) If a2A, then every r2Q such that r

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