Chapter1 - Pre-Calculus Chapter 1A Chapter 1A Real Numbers...

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© : Pre - Calculus - Chapter 1A Chapter 1A - Real Numbers Properties of Real Numbers Real numbers are used in almost every human endeavor. Whenever we need to quantify objects, we use numbers. Cooking recipes, prices, interest rates, blood pressure, height, age, voltage, and wind velocity are a few of the everyday objects that are quantified by real numbers. As we know, the two operations of addition ! ! " and multiplication !#" are defined for real numbers. In other words, for any two real numbers a and b , the sum a ! b and the product a # b are uniquely defined real numbers. Two special real numbers are zero (0) and one (1). These operations satisfy: Properties of real numbers: Commutative: a ! b " b ! a ab " ba Example : 7 ! 3 " 3 ! 7 " 10 5 # 6 " 6 # 5 " 30 Associative: a ! ! b ! c " " ! a ! b " ! c a ! bc " " ! ab " c Example : 3 ! ! 4 ! 7 " " ! 3 ! 4 " ! 7 2 # ! 5 # 3 " " ! 2 # 5 " # 3 3 ! 11 " 7 ! 7 " 14 2 # 15 " 10 # 3 " 30 Identity: a ! 0 " 0 ! a " a a # 1 " 1 # a " a Example : 8 ! 0 " 0 ! 8 " 8 11 # 1 " 1 # 11 " 11 For each real number a , there is a real number, denoted by ! a , called the negative of a , for which Inverse: a ! ! ! a " " 0 " ! ! a " ! a Example : 7 ! ! ! 7 " " 0 Subtraction, denoted by a ! b , is defined as follows: a ! b " a ! ! ! b " For each a " 0, there is a real number, denoted by 1 a or 1/ a or a ! 1 , called the reciprocal of a , for which Inverse: a 1 a " 1 " 1 a a Example: 7 # 1 7 " 1 7 # 7 " 1 Division, denoted by a ! b or a b or a / b , where b " 0, is defined as follows: a ! b " a b " a / b " a ! 1 b " " ab ! 1 Finally, there is a property which relates addition and multiplication: Distributive: a ! b ! c " " ab ! ac ! a ! b " c " ac ! bc Example: ! 4 # ! 3 ! 2 " " ! ! 4 # 3 " ! ! ! 4 # 2 " ! 4 # 5 " ! 12 ! ! ! 8 " " ! 20 © : Pre - Calculus
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© : Pre - Calculus - Chapter 1A Types of Real Numbers Integers The real numbers are classified into several categories. The positive integers, also called counting numbers, are 1,2,3,4, # The negative integers, # , ! 4, ! 3, ! 2, ! 1 are the negatives (or additive inverses) of the positive integers. An integer is either a positive integer, a negative integer, or zero. Rationals . The ratio a b of any two integers a and b , where b " 0, is called a rational number. Common fractions are rationals. If b " 1 then the number is also an integer, so integers are also rational numbers. Examples: 4 5 , 121 54 , ! 16 31 , ! 7 Irrationals . Real numbers that are not rational are called irrational. Examples of irrational numbers include " , 2 , and so on. One distinction between rational numbers and irrational numbers is that rational numbers have repeating or terminating decimal expansions, whereas irrational numbers do not. Summary The diagram below shows the relationship among these types of numbers, all of which are a part of the set of real numbers: That is, whole numbers are a subset of the integers, which are a subset of the rational numbers. The irrational numbers are disjoint from all of these and the real numbers are comprised of the union of the set of irrational numbers and rational numbers. Real Numbers Irrational Numbers R a t i o n a l N u m b e r s I n t e g e r s Whole Numbers © : Pre - Calculus
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© : Pre - Calculus - Chapter 1A Number Lines and Absolute Value A number line is a method of picturing the set of real numbers. Each point on the number line corresponds to exactly one real number, as in the picture below: -5 -4 -3 -2 -1 0 1 2 3 4 5 P 3/2 O !
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