B. Use a calculator to approximate Ï} 6 to the nearest tenth: Ï} 6 ø . a+b is real 2 + 3 = 5 is real. Cardinality 93 2. A fundamental property of the set R of real numbers : Completeness Axiom : R has \no gaps". 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. 1 Thus the equivalence of new objects (fractions) is deﬂned in terms of equality of familiar objects, namely integers. The chart below is nice because it shows the addition and multiplication properties side by side do you can see the similarities and differences. 4 NOTES ON REAL NUMBERS De nition 3. Abstract. Mathematical Induction 91 Appendix B. Example 1.1. 2 – 11) Topics: Classifying numbers, placing numbers on the number line, order of operations, properties I. We will use the notation from these examples throughout this course. and variables: Notes for R.1 Real Numbers and Their Properties (pp. 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. 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. Properties of Whole Numbers. [a;b) is the set of all real numbers xwhich satisfy a x> < >>: x if x 0 x if x<0 Note. long division and in the theory of approximation to real numbers by rationals. • Example [8.5.4, p. 501] Another useful partial order relation is the “divides” relation. The rational numbers are numbers that can be written as an integer divided by an integer (or a ratio of integers). 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? 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: 1 Algebra of Complex Numbers We deﬁne the algebra of complex numbers C to be the set of formal symbols x+ıy, x,y ∈ 8S R and S6= ;, If Sis bounded above, then supSexists and supS2R. Sets A set is a list of numbers: We separate the entries with commas, and close off the left and right with and . He has some packages that he needs to load into the pontoons of the boat. Open and Closed Sets 96 … Graph the real numbers 2} 13 and 5 Ï} 6 on a number line. Two whole numbers add up to give another whole number. (that is, the set Shas a least upper bound which is a real number). 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. 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. Each point on the number line corresponds to exactly one real number: De nition. So, graph 2 13} 5 between and and graph Ï} 6 between and . Integers: 2 – 3) Equivalent Fractions a = c if and only if ad = bc bd cross multiply 2. Special Sets 1. These are some notes on introductory real analysis. 1) associative 2) additive identity 2. 2 and π are irrational numbers. Real Numbers are closed (the result is also a real number) under addition and multiplication: Closure example. The collection of all real numbers between two given real numbers form an interval. Property Explanation 1. a + 0 = a 6 + 0 = 6. a × 1 = a 6 × 1 = 6 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. These objects that are related to number theory help us nd good approximations for real life constants. Write an example to demonstrate it. 1.1 Euclid’s GCD algorithm Given two positive integers, this algorithm computes the greatest common divisor (gcd) of the two numbers. 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)? A. ab = ba B. a(bc) = (ab)c C. a(b+c) = ab+ac D. a1 = a 2. 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. 1.2_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 Examples: 3 π 3 5 e Properties of Real Numbers Commutative Property for Addition: a + b = b + a Keystone Review { Properties of Real Numbers Name: Date: 1. 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. Below are some examples of sets of real numbers. Outer measures As stated in the following deﬁnition, an outer measure is a monotone, countably 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. The properties of whole numbers are given below. 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 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. Before starting a systematic exposition of complex numbers, we’ll work a simple example. Solution Note that 2 13} 5 5 . Which sentence is an example of the distributive property? 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. 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 erties persist. Natural Numbers: (these are the counting numbers) 2. The Real Numbers are characterized by the properties of Complete Ordered Fields. The ordering < is compatible with the arithmetic operations means the following: VIII a < b =⇒ a+c < b+c and ad < bd for all a,b,c ∈ R and d > 0. 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

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