They cover the properties of the real numbers, sequences and series of real numbers, limits of functions, continuity, di erentiability, sequences and series of functions, and Riemann integration. The Order Properties of the Real Numbers 88 4. 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. 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. … 16 11. a + 0 = a 6 + 0 = 6. a × 1 = a 6 × 1 = 6 1 Algebra of Complex Numbers We deﬁne the algebra of complex numbers C to be the set of formal symbols x+ıy, x,y ∈ 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 properties of whole numbers are given below. The Field Properties of the Real Numbers 85 3. These are some notes on introductory real analysis. So, graph 2 13} 5 between and and graph Ï} 6 between and . Solution Note that 2 13} 5 5 . Natural Numbers: (these are the counting numbers) 2. Mathematical Induction 91 Appendix B. a×b is real 6 × 2 = 12 is real . 1 Thus the equivalence of new objects (fractions) is deﬂned in terms of equality of familiar objects, namely integers. Properties of Real Numbers identity property of addition_Adding 0 to a number leaves it unchanged identity property of multiplication_Multiplying a number by 1 leaves it unchanged multiplication property of 0_Multiplying a number by 0 gives 0 additive Inverse & definition of opposites_Adding a number to its opposite gives 0 o Every number has an opposite and variables: [a;b) is the set of all real numbers xwhich satisfy a x> < >>: x if x 0 x if x<0 Note. SWBAT: identify and apply the commutative, associative, and distributive properties to simplify expressions 4 Algebra Regents Questions 1) The statement is an example of the use of which property of real numbers? Open and Closed Sets 96 … Adding zero leaves the real number unchanged, likewise for multiplying by 1: Identity example. Write an example to demonstrate it. perfectly valid numbers that don’t happen to lie on the real number line.1 We’re going to look at the algebra, geometry and, most important for us, the exponentiation of complex numbers. A. ab = ba B. a(bc) = (ab)c C. a(b+c) = ab+ac D. a1 = a 2. The rational numbers are numbers that can be written as an integer divided by an integer (or a ratio of integers). 8S R and S6= ;, If Sis bounded above, then supSexists and supS2R. We define the real number system to be a set R together with an ordered pair of functions from R X R into R that satisfy the seven properties listed in this and the succeeding two sections of this chapter. Explain the associative property of addition. Algebra II Accelerated Name _____ 1.1 Properties of Real Numbers – Notes Sheet Date _____ Digits – Natural (Counting) Numbers – Whole Numbers – Integers – Rational Numbers – Irrational Numbers – Example 1: Write each rational number as a fraction and list what sets of numbers each belong to: a) b) c) Create a Number Line showing all of the numbers from Example 1: 2 – 3) B. Cardinality 93 2. Use a calculator to approximate Ï} 6 to the nearest tenth: Ï} 6 ø . THE REAL NUMBER SYSTEM 5 1.THE FIELD PROPERTIES. The Ordered Field Properties of the Real Numbers 90 5. Before starting a systematic exposition of complex numbers, we’ll work a simple example. Appendix to Chapter 3 93 1. Keystone Review { Properties of Real Numbers Name: Date: 1. a+b is real 2 + 3 = 5 is real. 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 See also: Notes for R.1 Real Numbers and Their Properties (pp. He has some packages that he needs to load into the pontoons of the boat. Our use of extended real numbers is closely tied to the order and monotonicity properties of R. In dealing with complex numbers or elements of a vector space, we will always require that they are strictly ﬁnite. Deﬁnition 0.1 A sequence of real numbers is an assignment of the set of counting numbers of a set fang;an 2 Rof real numbers, n 7!an. 4x3 y5 = Power Property: Multiply exponents when they are inside and outside parenthesis EX w/ numbers: (53)4 = EX w/ variables: (y3)11 = EX w/ num. 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