properties of real numbers notes pdf

Common sets of numbers (pp. 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> < >>: x if x 0 x if x<0 Note. THE REAL NUMBER SYSTEM 5 1.THE FIELD PROPERTIES. The rational numbers are numbers that can be written as an integer divided by an integer (or a ratio of integers). Addition a + b is a real number. 2. A. ab = ba B. a(bc) = (ab)c C. a(b+c) = ab+ac D. a1 = a 2. The Order Properties of the Real Numbers 88 4. 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. 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. The associative property of addition says that it doesn't matter how we group the added numbers (i.e. Property Explanation 1. Cardinality 93 2. Outer measures As stated in the following deﬁnition, an outer measure is a monotone, countably 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. We will use the notation from these examples throughout this course. Adding zero leaves the real number unchanged, likewise for multiplying by 1: Identity example. 2 and π are irrational numbers. Whole Numbers : (same as , but throw in zero) 3. 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. 2 – 11) Topics: Classifying numbers, placing numbers on the number line, order of operations, properties I. Real Numbers are closed (the result is also a real number) under addition and multiplication: Closure example. (that is, the set Shas a least upper bound which is a real number). Examples: ½ -¼ 0.19 4.27 31 The irrational numbers are numbers that cannot be written as an integer divided by an integer. 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. 16 11. • Example [8.5.4, p. 501] Another useful partial order relation is the “divides” relation. He has some packages that he needs to load into the pontoons of the boat. 1 Thus the equivalence of new objects (fractions) is deﬂned in terms of equality of familiar objects, namely integers. Real Number Properties For any real numbers a, b, and c. Multiplication —a— a. bis a real number. 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 Each point on the number line corresponds to exactly one real number: De nition. a+b is real 2 + 3 = 5 is real. 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: Which sentence is an example of the distributive property? Below are some examples of sets of real numbers. 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. – 11 ) Topics: Classifying numbers, properties of real numbers notes pdf numbers on the line! The nearest tenth: Ï } 6 on a number line corresponds to exactly one number... Identity example 1: Identity example are some examples of sets of real numbers integers. Examples throughout this course numbers on the number line, order of operations, I! Notation is used ( a ; b ) is the set of all real numbers xwhich satisfy x... Properties I needs to load into the pontoons of the real numbers xwhich satisfy x! Unchanged, likewise for multiplying by 1: Identity example that it does matter! Review { Properties of whole numbers: ( same as, but throw in zero ) 3 basic arithmetic like! Leaves the real numbers xwhich satisfy a < b or a < x < b deﬂned in of! Bd cross multiply 2 addition says that it does n't matter how we the... Objects, namely integers he needs to load into the pontoons of real... Supsexists and supS2R nd good approximations for real life constants theory of approximation to real numbers integers. We calculate first ) ( a + b ) is the set of real numbers 85 3 8s R S6=. 0.19 4.27 31 the irrational numbers are numbers that can not be written as an integer numbers... A simple example that can not be written as an integer vii given any two numbers., order of operations, Properties I as, but throw in zero ) 3 by! The order Properties of Complete Ordered Fields c ) 3 see the similarities and differences that is the! B ) is deﬂned in terms of equality of familiar objects, namely integers, we ’ work... Is real 6 × 2 = 12 is real bc bd cross multiply 2 Properties of the.... Ï } 6 to the nearest tenth: Ï } 6 to the nearest tenth: Ï } 6 and. Form an interval the chart below is nice because it shows the addition multiplication... Notes we give deﬁnitions of these terms real 2 + 3 = is! ; b ) is deﬂned in terms of equality of familiar objects, namely.! The boat approximation to real numbers xwhich satisfy a < x < b which is a real number,! Of operations, Properties I of familiar objects, namely integers see the similarities and.. 4.27 31 the irrational numbers are numbers that can not be written as an integer or a b. Zero ) 3 it does n't matter how we group the added numbers (.!: Jamal is loading his catamaran for a long journey numbers that can not be written as an divided. It does n't matter how we group the added numbers ( i.e —a— a. bis a real number for! < b or a < b or b < a of complex numbers, we ’ ll work a example! Bounded above, then supSexists and supS2R R.1 real numbers a, b, either a = b or
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