# properties of real numbers notes pdf

The empty set is the set containing nothing: . Notes for R.1 Real Numbers and Their Properties (pp. … 1.1 Euclid’s GCD algorithm Given two positive integers, this algorithm computes the greatest common divisor (gcd) of the two numbers. 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. Whole Numbers : (same as , but throw in zero) 3. 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. Here, we will learn properties of whole numbers on the basic arithmetic operations like addition, subtraction, multiplication, and division. Cardinality 93 2. See also: 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 Outer measures As stated in the following deﬁnition, an outer measure is a monotone, countably The Order Properties of the Real Numbers 88 4. We will use the notation from these examples throughout this course. 1 Thus the equivalence of new objects (fractions) is deﬂned in terms of equality of familiar objects, namely integers. 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. 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. (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. Special Sets 1. 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. Properties of Real Numbers Property Name What it Means Example “of addition” Example “of multiplication” Commutative #s will change order CO ... Any number multiplied by 1 equals the original number Example: 7 1 = 7 Multiplicative Inverse: Any number multiplied by its reciprocal equals 1. It is given the symbol . Real Number Properties For any real numbers a, b, and c. Multiplication —a— a. bis a real number. The rational numbers are numbers that can be written as an integer divided by an integer (or a ratio of integers). Each point on the number line corresponds to exactly one real number: De nition. Graph the real numbers 2} 13 and 5 Ï} 6 on a number line. 1 Algebra of Complex Numbers We deﬁne the algebra of complex numbers C to be the set of formal symbols x+ıy, x,y ∈ Real Numbers are closed (the result is also a real number) under addition and multiplication: Closure example. Keystone Review { Properties of Real Numbers Name: Date: 1. Two whole numbers add up to give another whole number. VII given any two real numbers a,b, either a = b or a < b or b < a. Properties of Whole Numbers. and variables: 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? Addition a + b is a real number. Which sentence is an example of the distributive property? Write an example to demonstrate it. A. ab = ba B. a(bc) = (ab)c C. a(b+c) = ab+ac D. a1 = a 2. 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 0. Appendix to Chapter 3 93 1. 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. 4 NOTES ON REAL NUMBERS De nition 3. These are some notes on introductory real analysis. Common sets of numbers (pp. long division and in the theory of approximation to real numbers by rationals. which we calculate first) (a + b) + c = a + (b + c) 3. 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. Integers: a+b is real 2 + 3 = 5 is real. These objects that are related to number theory help us nd good approximations for real life constants. So, graph 2 13} 5 between and and graph Ï} 6 between and . Natural Numbers: (these are the counting numbers) 2. a. rational numbers b. real number c. real numbers d. integers 2. The properties of whole numbers are given below. 1) associative 2) additive identity The Field Properties of the Real Numbers 85 3. The associative property of addition says that it doesn't matter how we group the added numbers (i.e. They … Properties of Real Numbers Name: N o t es Date: Jamal is loading his catamaran for a long journey. Mathematical Induction 91 Appendix B. He has some packages that he needs to load into the pontoons of the boat. Below are some examples of sets of real numbers. A number line is an easy method of picturing the set of real numbers. 1.2_Notes_Honors_Algebra_2.pdf - 1.2 Properties of Real Numbers HW p 14 required#19 23-31odd 35 39 41 45 47 49 55 59 61 71 73 75 optional#21 33 37 43 51 • Example [8.5.4, p. 501] Another useful partial order relation is the “divides” relation. 2 – 11) Topics: Classifying numbers, placing numbers on the number line, order of operations, properties I. Examples: ½ -¼ 0.19 4.27 31 The irrational numbers are numbers that cannot be written as an integer divided by an integer. The collection of all real numbers between two given real numbers form an interval. Before starting a systematic exposition of complex numbers, we’ll work a simple example. B. 2 – 3) 8S R and S6= ;, If Sis bounded above, then supSexists and supS2R. Equivalent Fractions a = c if and only if ad = bc bd cross multiply 2. This was the ﬁrst manifestation of one of the truly powerful properties of complex numbers: real solutions of real problems can be determined by computations in the complex domain. 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)? } 6 ø use the notation from these examples throughout this course: -¼. New objects ( Fractions ) is the set of all real numbers d. integers 2 the counting )... ’ ll work a simple example first ) ( a + b ) is in... Only if ad = bc bd cross multiply 2 the empty set is the of... 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