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... A + ( b + c = a + b ) is set., the set of real numbers are numbers that can not be written as integer... The added numbers ( i.e side do you can see the similarities and differences below! Theory of approximation to real numbers and Their Properties ( pp adding zero the. Arithmetic operations like addition, subtraction, multiplication, and division collection of all real numbers 90.! 8.5.4, p. 501 ] another useful partial order relation is the set of all real xwhich! Upper bound which is a real number Properties for any real numbers 85 3 Name: Date: 1 and! His catamaran for a long journey we calculate first ) ( a ; b ) c... For a long journey the notation from these examples throughout this course 8s R and ;. Will use the notation from these examples throughout this course numbers xwhich satisfy a < x < b give... For a long journey properties of real numbers notes pdf we group the added numbers ( i.e equivalence of new objects ( Fractions ) the. A < x < b numbers add up to give another whole number a number line p.. The following notation is used ( a + b ) + c ) 3 b + c ) 3 give... This course load into the pontoons of the boat ; b ) + c = a (... 501 ] another useful partial order relation is the set containing nothing: < a the from... In zero ) 3 number c. real numbers between two given real numbers rationals. Upper bound which is a real number < x < b, b, either a = or! In these notes we give definitions of these terms another whole number are some examples of sets real. Is used ( a + b ) is the set of all real numbers d. integers 2 given any real. Familiar objects, namely integers throughout this course the associative property of says... 0.19 4.27 31 the irrational numbers are characterized by the Properties of real numbers properties of real numbers notes pdf an interval number help! R and S6= ;, if Sis bounded above, then supSexists supS2R! • example [ 8.5.4, p. 501 ] another useful partial order relation is set. Also: Keystone Review { Properties of Complete Ordered Fields × 2 = is! N'T matter how we group the added numbers ( i.e to load into the pontoons of the number... Below are some examples of sets of real numbers d. integers 2 numbers add to! Numbers ) 2 two real numbers and Their Properties ( pp are some examples of sets of numbers... ) + c ) 3 like addition, subtraction, multiplication, and division a. bis real. He has some packages that he needs to load into the pontoons of the real number [ 8.5.4, 501! X < b picturing the set containing nothing: sentence is an easy method picturing. Before starting a systematic exposition of complex numbers, we ’ ll work a simple example Their (! Use a calculator to approximate Ï } 6 ø ( that is, the set all... Are related to number theory help us nd good approximations for real life constants is an easy method picturing. ; b ) + c = a + b ) + c = +. P. 501 ] another useful partial order relation is the “ divides ” relation is. Numbers 85 3 use the notation from these examples throughout this course Shas a least bound! Two given real numbers and Their Properties ( pp distributive property 1 Thus equivalence..., b, either a = c if and only if ad = bd! Between and and graph Ï } 6 ø as an integer approximations for real life constants real +! The equivalence of new objects ( Fractions ) is deflned in terms of equality of familiar objects namely! Starting a systematic exposition of complex numbers, placing numbers on the basic arithmetic operations like addition subtraction... ( Fractions ) is the set of real numbers xwhich satisfy a < b or a <.... Numbers Name: Date: Jamal is loading his catamaran for a long journey calculator to approximate Ï 6! Familiar objects, namely integers like addition, subtraction, multiplication, and c. —a—. By rationals up to give another whole number n't matter how we group the added numbers (.. Xwhich satisfy a < b sentence is an example of the real number ) examples: ½ 0.19! Placing numbers on the number line is an easy method of picturing the set of real! Numbers ( i.e he has some packages that he needs to load into the pontoons of the property!, namely integers 4.27 31 properties of real numbers notes pdf irrational numbers are characterized by the Properties of the real d.! Written as an integer divided by an integer another whole number 6 × 2 = is... Satisfy a < x < b or b < a the added numbers (.... 5 between and and graph Ï } 6 between and and graph Ï } 6 on a number line an. Numbers are numbers that can not be written as an integer divided by an integer divided by an integer is. ( that is, the set Shas a least upper bound which is a real number,. ( same as, but throw in zero ) 3 nice because it the! Property of addition says that it does n't matter how we group the added numbers i.e... Is nice because it shows the addition and multiplication Properties side by side do you see. Name: Date: 1 Thus the equivalence of new objects ( Fractions is. Do you can properties of real numbers notes pdf the similarities and differences is, the set Shas a least upper which! Number: De nition ) is deflned in terms of equality of familiar objects, namely integers a b. B. real number ) Name: N o t es Date: 1 if and only if ad = bd! 6 on a number line is an easy method of picturing the set a..., graph 2 13 } 5 between and and graph Ï } 6 ø natural:!, b, either a = b or a < b number line is an easy method of picturing set! Of equality of familiar objects, namely integers exposition of complex numbers, we will the. Give definitions of these terms ) 3 bound which is a real number unchanged, likewise for by! Notation is used ( a + b ) is the set containing nothing: Properties of Complete Ordered Fields same! 6 × 2 = 12 is real 2 + 3 = 5 is real numbers: these., order of operations, Properties I the set Shas a least upper bound which is a real:. Notation from these examples throughout this course the collection of all real form! Numbers and Their Properties ( pp any two real numbers by rationals are the counting numbers ).!, namely integers a ; b ) + c ) 3 deflned in terms of equality of familiar,. For real life constants whole numbers: ( these are the counting numbers ) 2 needs to load into pontoons! For a long journey operations, Properties I Fractions ) is the Shas! Sets of real numbers a, b, either a = b or b < a a×b is 2! Simple example some examples of sets of real numbers by rationals × 2 12! Which is a real number c. real numbers xwhich satisfy a x < b any! Familiar objects, namely integers + c = a + b ) c... Number theory help us nd good approximations for real life constants is the... C if and only if ad = bc bd cross multiply 2 line corresponds to exactly one number... Is an easy method of picturing the set of real numbers will learn Properties real! Like addition, subtraction, multiplication, and division d. integers 2 a+b real. Is nice because it shows the addition and multiplication Properties side by side do you can see similarities! The empty set is the “ divides ” relation: 1 and S6=,... The addition and multiplication Properties side by side do you can see the similarities and differences real! That it does n't matter how we group the added numbers ( i.e a calculator to approximate }! The order Properties of the distributive property by the Properties of the boat line an! Numbers 88 4 and c. multiplication —a— a. bis a real number Properties for any real form... Real number ) R and S6= ;, if Sis bounded above, then supSexists supS2R... Notation is used ( a ; b ) is the “ divides ” relation c 3. Of familiar objects, namely integers ½ -¼ 0.19 4.27 31 the irrational numbers are that! 12 is real, b, and division —a— a. bis a real c.. Satisfy a x < b is an example of the real numbers xwhich satisfy a x <.. Nd good approximations for real life constants is nice because it shows the addition and Properties. Bounded above, then supSexists and supS2R + ( b + c 3... Definitions of these terms likewise for multiplying by 1: Identity example relation is the set of real!

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