See also: 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. 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. In these notes we give definitions of these terms. 1.1 Real Numbers A. Properties of Real Numbers Name: N o t es Date: Jamal is loading his catamaran for a long journey. A. ab = ba B. a(bc) = (ab)c C. a(b+c) = ab+ac D. a1 = a 2. 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: Solution Note that 2 13} 5 5 . The Order Properties of the Real Numbers 88 4. 2 – 3) Here, we will learn properties of whole numbers on the basic arithmetic operations like addition, subtraction, multiplication, and division. Outer measures As stated in the following definition, an outer measure is a monotone, countably Use a calculator to approximate Ï} 6 to the nearest tenth: Ï} 6 ø . The properties of whole numbers are given below. 1.1 Euclid’s GCD algorithm Given two positive integers, this algorithm computes the greatest common divisor (gcd) of the two numbers. Properties of Addition Closure Property. THE REAL NUMBER SYSTEM 5 1.THE FIELD PROPERTIES. … [a;b) is the set of all real numbers xwhich satisfy a x> < >>: x if x 0 x if x<0 Note. Keystone Review { Properties of Real Numbers Name: Date: 1. 4 NOTES ON REAL NUMBERS De nition 3. Integers: erties persist. Notes for R.1 Real Numbers and Their Properties (pp. • Example [8.5.4, p. 501] Another useful partial order relation is the “divides” relation. 8S R and S6= ;, If Sis bounded above, then supSexists and supS2R. PROPERTIES OF THE REAL NUMBERS NOTES The chart below gives variable and numerical examples for all the properties. The rational numbers are numbers that can be written as an integer divided by an integer (or a ratio of integers). 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 a. rational numbers b. real number c. real numbers d. integers 2. The collection of all real numbers between two given real numbers form an interval. (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. 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. A fundamental property of the set R of real numbers : Completeness Axiom : R has \no gaps". Cardinality 93 2. So, graph 2 13} 5 between and and graph Ï} 6 between and . The Real Numbers are characterized by the properties of Complete Ordered Fields. The Field Properties of the Real Numbers 85 3. A number line is an easy method of picturing the set of real numbers. Sets A set is a list of numbers: We separate the entries with commas, and close off the left and right with and . 2. 16 11. We will use the notation from these examples throughout this course. 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. Real Number Properties For any real numbers a, b, and c. Multiplication —a— a. bis a real number. Appendix to Chapter 3 93 1. Equivalent Fractions a = c if and only if ad = bc bd cross multiply 2. 1) associative 2) additive identity Common sets of numbers (pp. Natural Numbers: (these are the counting numbers) 2. 2 and π are irrational numbers. Examples: 3 π 3 5 e Properties of Real Numbers Commutative Property for Addition: a + b = b + a Which sentence is an example of the distributive property? 1 Thus the equivalence of new objects (fractions) is deflned in terms of equality of familiar objects, namely integers. long division and in the theory of approximation to real numbers by rationals. 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 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. 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)? 1 Algebra of Complex Numbers We define the algebra of complex numbers C to be the set of formal symbols x+ıy, x,y ∈ Below are some examples of sets of real numbers. Properties of Whole 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. a + 0 = a 6 + 0 = 6. a × 1 = a 6 × 1 = 6 and variables: The associative property of addition says that it doesn't matter how we group the added numbers (i.e. These are some notes on introductory real analysis. a×b is real 6 × 2 = 12 is real . (that is, the set Shas a least upper bound which is a real number). a+b is real 2 + 3 = 5 is real. 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. It is given the symbol . The following notation is used (a;b) is the set of all real numbers xwhich satisfy a 0. Each point on the number line corresponds to exactly one real number: De nition. 1.4. Mathematical Induction 91 Appendix B. 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. Special Sets 1. The Ordered Field Properties of the Real Numbers 90 5. Write an example to demonstrate it. 2 – 11) Topics: Classifying numbers, placing numbers on the number line, order of operations, properties I. 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. B. These objects that are related to number theory help us nd good approximations for real life constants. 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. Before starting a systematic exposition of complex numbers, we’ll work a simple example. Example 1.1. They … Property Explanation 1. The chart below is nice because it shows the addition and multiplication properties side by side do you can see the similarities and differences. He has some packages that he needs to load into the pontoons of the boat. 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 Addition a + b is a real number. 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. Abstract. Explain the associative property of addition. Adding zero leaves the real number unchanged, likewise for multiplying by 1: Identity example. VII given any two real numbers a,b, either a = b or a < b or b < a. 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 Is deflned in terms of equality of familiar objects, namely integers an method... 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