Chapter 1. The Real Number System

Transcription

1 Chapter The Real Number System Section.. All whole numbers are also integers.. {0,,, }. {...,, 0,,, }. {, 0,, } 9. Ø. {,, 6, }. Any integer n can be written as n.. True. True 9. False, since 0. is not a natural number, but it is a positive number.. False. Negative numbers are rational.. True. Zero. Whole numbers 9. The number 0. is a terminating decimal, whereas 0. is a repeating decimal....; terminating 6 0. ; repeating. 0. ; repeating (c), 0,, (d),, 0,, (e) None 9. (a), 9 (b), 9 (c), 9 (d), 9, (e). (a), 6... (b), 6 (c), 6 (d),, 0., 6,. (e) (a), (b) 0,,. > 0, 0 <. 0, 0

2 Chapter : The Real Number System SSM: Elementary Algebra. >, <. n, n 9. y, y 6. False 6. True 6. True 6. False < 6... >. <. > π 9. {..., 9,, }. {0, }. Ø. () 9(). 9. <.. <.. > > % December and January combined account for more than of the fatal fires. 9. (a) False (b) True (c) True (d) False (a) 9 0. (b) (c) 0. 9 Section.. Addends are the numbers in a sum; factors are the numbers in a product.. (c), divisor. (f), minuend. (g), exponent 9. (b), subtrahend. (),, ()(). A natural number exponent indicates repeated multiplication t t t t t t

3 SSM: Elementary Algebra Chapter : The Real Number System 9.. yyyyy y. x , The symbol is used for subtraction, as in 6 ; for negative numbers, as in ; and for opposites, as in t.... ( ). ( ) > ( ) 6. 6 < () ( ) 0 9. ( ) ( ) ()( ) 9. ( + ) ( + ) ( ). [ ( 6 )] [ ( )] [] ( ) ,. 6 (6) ,., 9. 6, 9. 6(..90) 6(0.). There were 9 more home runs hit by an American League team than by a National League team.

4 Chapter : The Real Number System SSM: Elementary Algebra % Presidential debates, newspaper stories, and T.V. news have percentages greater than %. 0. No; a person may have chosen more than one category. 0. a x, so a ( x) x 0. ( + ) 0. ( ) Section.. In a sum, the addends can be added in any order. In a product, the factors can be multiplied in any order.. ( + 9) + + (9 + ) ; Associative Property of Addition. ( ) ( ) 0; Associative Property of. Associative Property of Addition 9. Property of Additive Inverses. Additive Identity Property. Distributive Property. Commutative Property of. For addition, the number 0 is the additive identity. 9. c c. x + x 0. x + y (x + y). x, x 0 x. (6 + x) (x + 6) 9. (a) Associative Property of Addition (b) Commutative Property of Addition (c) Associative Property of Addition (d) Distributive Property (e) Commutative Property of. (a) Associative Property of (b) Commutative Property of (c) Associative Property of. (d), 0 n 0. (e), n. (b), a a, a 0 9. n n(0 + ) 0n + n. (x + ) x + 6. (y ) y. x( + y) x + xy. a(x ) ax a 9. x + y (x + y). y z (y z). ax + ay a(x + y). x + 6 (x + ). a 6 (a ) 9. 0 (00 + ) (00 ) (9 + ) 6. ( ) (0 + ) 69. (a) 00 ( ) 00 % of the families eat together on or weekdays.

5 SSM: Elementary Algebra Chapter : The Real Number System (b) + % of the families eat together on weekdays ( + + ). 9 (00 ) 00. (x) x; Associative Property of. + ( )y ( + y); Distributive Property Section.. Add the absolute values and use the sign of the addends.. + ( ) ( ) ( ) ( ) ( ) ( ) ( 0) ( ) + ( ) 6 + ( ). + [6 + ( 9)] + ( ) ( ) ( ) + ( ) ( ) [( ) + ] 6+ + ( ) + ( ) (.) (.) (.) ( ) ( ) + ( ) + ( ) (.) ( ) (.) ( ) ( ) ( ) + ( ) 6. + ( ) + ( ) ( 0.9) 0.. The Commutative Property of Addition allows us to reorder, and the Associative Property of Addition allows us to regroup.. (a) Commutative Property of Addition (b) Associative Property of Addition. (a) Commutative Property of Addition (b) Associative Property of Addition (c) Property of Additive Inverses (d) Additive Identity Property ( + ) + () +

6 6 Chapter : The Real Number System SSM: Elementary Algebra ( ) ( ) ( ) ( + ) ? +? 9.? + ( ) + 9? + 9? 9. ( 9) A yard gain ( ) + + ( ) + ( ) + + ( ) + + ( ) pounds 0. a + b is positive 0. a + ( b) is negative 0. Since it takes three steps to move one step forward, it will take () steps to move steps forward. Section.. Equivalent fractions are fractions with the same decimal value A fraction can be reduced if there is a common factor in the numerator and denominator cannot be reduced Although is a common denominator, is the least common denominator

7 SSM: Elementary Algebra Chapter : The Real Number System ( ) ( ) + ( ) + ( )

8 Chapter : The Real Number System SSM: Elementary Algebra ( )

9 SSM: Elementary Algebra Chapter : The Real Number System 9. (a) (b) , 000 0, 000 0, , 000 0, 000, , 000 0, 000 0, , , , , , , , 000 In these four occupations, there are fatalities per 00,000 workers a a a a. (ii) If b and d 6, then the LCD is, not.

10 0 Chapter : The Real Number System SSM: Elementary Algebra Section.6. Change the minus sign to a plus sign and change to.. + ( ). + ( ) 0. 9 ( ) ( ) ( ). 0 ( ) 0 +. ( ) +. + ( ) 9. 6 ( ) 6 +. ( ) ( 6) 0. ( ) +. Taking the opposite of a difference changes the order in which the numbers are subtracted. 9. ( ) ( ) +. [ ( 6)] ( ) 0. (a 9) 9 a. 0 + ( 0) + ( ) + ( ) 0. 0 ( 6 9) 0 [ 6 + ( 9)] 0 ( ) ( ) ( 0) ( ) ( + 0) + ( ) 9 + ( ) + ( 9). 6 [( ) + 0 ( 9)] 6 [( ) ] ( ) ( 9) ( ) + ( ).. ( 6.9) (.) (.). 9.?? 6.?? 6 6. ( ) + 6. ( 6)

11 SSM: Elementary Algebra Chapter : The Real Number System ( ) ( ) (.) ( ) (a) Definition of Subtraction (b) Associative Property of Addition. (a) Property of the Opposite of a Difference (b) Definition of Subtraction (c) Associative Property of Addition (d) Property of Additive Inverses (e) Additive Identity Property. () 9. ( ) 9 9. ( 6) ( ) , 0 ( 90) 9, , 9 feet (.) + (.) inches below average 99., 00 +, , , 90, miles 0 0. (a)a b is negative (b) b a is positive 0. ( ) 0. ( ) Section.. In both cases, multiply the absolute values of the numbers. For like signs, the product is positive; for unlike signs, the product is negative.. ( ). () 0. ( )( ) 9. 6( ) ( ).. (0)( ) 0 9. (.)(.) 6...(.)..??.??. Because there are an even number of negative factors, the product is positive. 9. ( )( )( ). ( )( )()(). ( )( )()()( 6) 6

12 Chapter : The Real Number System SSM: Elementary Algebra. ( ). ( ) 6 9. ( ). The Property of states that multiplying a number by results in the opposite of the number (.6)(.9).. 0. ( )() ( ) ( )( )(0)()( ) ( ) 9( 6) ( ) ( ) 6( 0) ( ) ( )( ) ( 6) ( ) ( ) ( ) ( ). 9(0.6).. ( ). (a) Commutative Property of (b) Associative Property of. (a) Associative Property of (b) Property of 9. +, 6,. + ( ), ( ),. + ( 6), ( 6), 6. +, (),.?( )? 9. (?)? 6 9. (. 9) $ ( ) + ( ) + () + () + ( ) + ( ) (a) Brazil: ( ). China: ( ). Germany: ( ). Mexico: ( ). Russia: ( ).

13 SSM: Elementary Algebra Chapter : The Real Number System U.S.: ( ). Zimbabwe: ( ). (b) United States 9. a b c is negative 99. a a is positive 0. (6 ) 0 Section.. We call the dividend, the divisor, and 6 the quotient.. ( ).. 0 ( ) 9. 6 ( ) is undefined ( ) ( ). [ 6 ( )]. 6. ( 6 ) ( ) (. ) x x or ; Property of Signs in y y Quotients.. 9. y y y or. Invert the divisor and multiply.. (a) (b). (a) (b). (a) (b) None

14 Chapter : The Real Number System SSM: Elementary Algebra (. ) ( 0) ( 6) ( 6) ( ) is undefined. ( 9) 9 + ( ) +. ( ) 9. ( ) ( ) 9 ( ) ( + ) 9 ( ) 6 ( ) (. ). 0 0 ( 0) 0. (a) Rule for Fractions (b) Distributive Property (c) Rule for Fractions. (a) Rule for Fractions (b) Associative Property of (c) Property of Inverses (d) Multiplicative Identity Property.? ( )? 9.? ( ) 0? 0 9. Average daily change 9

15 SSM: Elementary Algebra Chapter : The Real Number System 9. (a) 0,00 +, ,660 0, 00 0 (b), 660 9, , 660 0, , , , Army: 0,6 +,60 6,06 Navy:,99 +,9,0 Marines: 690 +,0 09 Air Force:,06 +,09 6,, 000, Army:, 000 +, 000 6, 000 6, 000, Navy:, 000 +, 000 6, Marines: , 000, 000 Air Force:, 000 +, 000 6, a b a b is negative is positive Chapter Review Exercises. {,,, 0}. (a) 0.6, terminating (b) 06., repeating. (a) {0,,, } (b) {...,, 0,, }

16 6 Chapter : The Real Number System SSM: Elementary Algebra. (a) True (b) True (c) False (d) True 9. (a) (b) Division (c) Subtraction (d) Addition. b b b b b [ ( 0)] (0) 0. (a) + + (b) > (c) 0 < 9. Property of Multiplicative Inverses. Commutative Property of Addition. Property of Additive Inverses. 6 (x + ) 6x +. Additive Identity, ( ) (.) ( ) ( ). ( 9 + ) + ( + ) ( ) + ( ) + ( ) 9. (ii); and LCD is The number n must contain a factor of or. 9. Change minus to plus and change to.. ( ) +. ( ) + 9. ( 9) ( ) ( ) ( 6) ( ) 6. ( )( )( )( 6) ( ) ( ) ( )( 6) 6. ( 6) ( )( 6) ( ) ( ) ( )

17 SSM: Elementary Algebra Chapter : The Real Number System. ( ) ( ) 9 + ( ) 9 9 Looking Ahead. ( x ) x( ) x. ( y) ( ) y y. ( ) ( ). ( ) + < ( ) + < + < 9. ( ) ( ) + > + > >. a a Chapter Test. (a) True (b) True (c) False. Since π is an irrational number, π is also irrational. (d) True. n or n (b) Property of (c) Property of Additive Inverses (d) Commutative Property of 9. The Multiplicative Inverse Property states that the product of a nonzero number and its reciprocal is always and () ;,. (a) 9.. (b) ( ) (c) ( ). (a) Associative Property of Addition