properties of real numbers notes pdf

December 30, 2020

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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 The chart below is nice because it shows the addition and multiplication properties side by side do you can see the similarities and differences. which we calculate first) (a + b) + c = a + (b + c) 3. Here, we will learn properties of whole numbers on the basic arithmetic operations like addition, subtraction, multiplication, and division. Number Systems Notes Mathematics Secondary Course MODULE - 1 Algebra 3 1 ... illustrate the extension of system of numbers from natural numbers to real (rationals and irrational) numbers. Equivalent Fractions a = c if and only if ad = bc bd cross multiply 2. So, graph 2 13} 5 between and and graph Ï} 6 between and . Property Commutative Associative Identity Inverse Closure Distributive a (b + c) = ab + acand (b + = ba+ ca Rational numbers can be expressed as a ratio g where a and b are integers and b is not zero. A.N.1: Identifying Properties: Identify and apply the properties of real numbers (closure, commutative, associative, distributive, identity, inverse) 1 Which property is illustrated by the equation ax+ay =a(x+y)? 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. 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. In these notes we give definitions of these terms. Each point on the number line corresponds to exactly one real number: De nition. Properties of Addition Closure Property. 1) associative 2) additive identity Which sentence is an example of the distributive property? B. Abstract. [a;b) is the set of all real numbers xwhich satisfy a x 0. A fundamental property of the set R of real numbers : Completeness Axiom : R has \no gaps". 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 Use a calculator to approximate Ï} 6 to the nearest tenth: Ï} 6 ø . These objects that are related to number theory help us nd good approximations for real life constants. Basic Properties of Real Numbers Commutative Laws: a+ b= b+ a, ab= ba Associative Laws: (a+ b) + c= a+ (b+ c), (ab)c= a(bc) Distributive Law: a(b c) = ab ac Cancellation Law: If c6= 0 then ac bc = a b An important consequence of the Cancellation Law is that the only way a product of two numbers can equal 0 is if at least one of the factors is 0. He has some packages that he needs to load into the pontoons of the boat. Natural Numbers: (these are the counting numbers) 2. The Field Properties of the Real Numbers 85 3. 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? The associative property of addition says that it doesn't matter how we group the added numbers (i.e. 4 NOTES ON REAL NUMBERS De nition 3. 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. Integers: • Example [8.5.4, p. 501] Another useful partial order relation is the “divides” relation. 1 Thus the equivalence of new objects (fractions) is deflned in terms of equality of familiar objects, namely integers. The empty set is the set containing nothing: . Cardinality 93 2. Adding zero leaves the real number unchanged, likewise for multiplying by 1: Identity example. 16 11. Example 1.1. Appendix to Chapter 3 93 1. 2 – 11) Topics: Classifying numbers, placing numbers on the number line, order of operations, properties I. Below are some examples of sets of real numbers. 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. ab = ba B. a(bc) = (ab)c C. a(b+c) = ab+ac D. a1 = a 2. The absolute value of a real number x, denoted by jxj, refers to the distance from that number to the origin of the number line, the point corresponding to 0. jxj= 8 >> < >>: x if x 0 x if x<0 Note. 1.1 Euclid’s GCD algorithm Given two positive integers, this algorithm computes the greatest common divisor (gcd) of the two numbers. Sets A set is a list of numbers: We separate the entries with commas, and close off the left and right with and . We will use the notation from these examples throughout this course. Properties of Real Numbers Name: N o t es Date: Jamal is loading his catamaran for a long journey. 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 Write an example to demonstrate it. Before starting a systematic exposition of complex numbers, we’ll work a simple example. Notes for R.1 Real Numbers and Their Properties (pp. They … Properties of Whole Numbers. (that is, the set Shas a least upper bound which is a real number). 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. The rational numbers are numbers that can be written as an integer divided by an integer (or a ratio of integers). a. rational numbers b. real number c. real numbers d. integers 2. 8S R and S6= ;, If Sis bounded above, then supSexists and supS2R. 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. 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. A number line is an easy method of picturing the set of real numbers. 1.1 Real Numbers A. Outer measures As stated in the following definition, an outer measure is a monotone, countably THE REAL NUMBER SYSTEM 5 1.THE FIELD PROPERTIES. 1 Algebra of Complex Numbers We define the algebra of complex numbers C to be the set of formal symbols x+ıy, x,y ∈ Keystone Review { Properties of Real Numbers Name: Date: 1. The Real Numbers are characterized by the properties of Complete Ordered Fields. 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. 2 – 3) long division and in the theory of approximation to real numbers by rationals. erties persist. The Order Properties of the Real Numbers 88 4. (Note that π ̸= 22 7 because π = 3.14159... whereas 22 7 = 3.14285...) We will use the following notation: R = the set of all real numbers, R+ = the set of positive real numbers, and R− = the set of negative real numbers. Graph the real numbers 2} 13 and 5 Ï} 6 on a number line. Common sets of numbers (pp. Mathematical Induction 91 Appendix B. These are some notes on introductory real analysis. Property Explanation 1. Addition a + b is a real number. See also: Real Numbers are closed (the result is also a real number) under addition and multiplication: Closure example. Examples: ½ -¼ 0.19 4.27 31 The irrational numbers are numbers that cannot be written as an integer divided by an integer. Whole Numbers : (same as , but throw in zero) 3. VII given any two real numbers a,b, either a = b or a < b or b < a. a+b is real 2 + 3 = 5 is real.

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