Special Sets 1. 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: 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. The collection of all real numbers between two given real numbers form an interval. 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. 2. In these notes we give definitions of these terms. Common sets of numbers (pp. Here, we will learn properties of whole numbers on the basic arithmetic operations like addition, subtraction, multiplication, and division. Keystone Review { Properties of Real Numbers Name: Date: 1. Cardinality 93 2. The following notation is used (a;b) is the set of all real numbers xwhich satisfy a 0. 1 Algebra of Complex Numbers We define the algebra of complex numbers C to be the set of formal symbols x+ıy, x,y ∈ a×b is real 6 × 2 = 12 is real . 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. Before starting a systematic exposition of complex numbers, we’ll work a simple example. 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. Example 1.1. Whole Numbers : (same as , but throw in zero) 3. These objects that are related to number theory help us nd good approximations for real life constants. 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. 1.1 Euclid’s GCD algorithm Given two positive integers, this algorithm computes the greatest common divisor (gcd) of the two numbers. (that is, the set Shas a least upper bound which is a real number). Sets A set is a list of numbers: We separate the entries with commas, and close off the left and right with and . The chart below is nice because it shows the addition and multiplication properties side by side do you can see the similarities and differences. 1.1 Real Numbers A. Equivalent Fractions a = c if and only if ad = bc bd cross multiply 2. 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. 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. Real Number Properties For any real numbers a, b, and c. Multiplication —a— a. bis a real number. Natural Numbers: (these are the counting numbers) 2. Explain the associative property of addition. Abstract. The properties of whole numbers are given below. 16 11. We will use the notation from these examples throughout this course. 2 – 11) Topics: Classifying numbers, placing numbers on the number line, order of operations, properties I. 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. Appendix to Chapter 3 93 1. Addition a + b is a real number. 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. Integers: It is given the symbol . These are some notes on introductory real analysis. Two whole numbers add up to give another whole number. A number line is an easy method of picturing the set of real numbers. Examples: ½ -¼ 0.19 4.27 31 The irrational numbers are numbers that cannot be written as an integer divided by an integer. So, graph 2 13} 5 between and and graph Ï} 6 between and . Which sentence is an example of the distributive property? A fundamental property of the set R of real numbers : Completeness Axiom : R has \no gaps". PROPERTIES OF THE REAL NUMBERS NOTES The chart below gives variable and numerical examples for all the properties. erties persist. The Ordered Field Properties of the Real Numbers 90 5. 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 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? … 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 The empty set is the set containing nothing: . The Field Properties of the Real Numbers 85 3. 1) associative 2) additive identity The Real Numbers are characterized by the properties of Complete Ordered Fields. Examples: 3 π 3 5 e Properties of Real Numbers Commutative Property for Addition: a + b = b + a 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+b is real 2 + 3 = 5 is real. 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. • Example [8.5.4, p. 501] Another useful partial order relation is the “divides” relation. 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. Properties of Real Numbers Name: N o t es Date: Jamal is loading his catamaran for a long journey. and variables: The rational numbers are numbers that can be written as an integer divided by an integer (or a ratio of integers). long division and in the theory of approximation to real numbers by rationals. Adding zero leaves the real number unchanged, likewise for multiplying by 1: Identity example. Real Numbers are closed (the result is also a real number) under addition and multiplication: Closure example. VII given any two real numbers a,b, either a = b or a < b or b < a. He has some packages that he needs to load into the pontoons of the boat. They … 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)? 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. Properties of Whole Numbers. 2 – 3) Graph the real numbers 2} 13 and 5 Ï} 6 on a number line. 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. 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 [a;b) is the set of all real numbers xwhich satisfy a x