Class : Numbers Exercise B 1. Compare the following pairs of rational numbers: 1 1 i First take the LCM of. LCM = 96 Therefore: 1 = 96 Hence we see that < 6 96 96 1 1 1 1 = 6 96 1 or we can say that < 1 First take the LCM of 1. LCM = 19 Therefore: = 10 19 Hence we see that 10 > 19 10 9 0 1 = 1 10 First take the LCM of 1. LCM = 1 Therefore: = 1 1 Hence we see that > 9 1 1 = 9 1 or we can say that > 1 1 or we can say that > 1 First take the LCM of 0. LCM = 10 1
Therefore: = 10 Hence we see that > 10 10 9 = 0 10 or we can say that > 9 0. Arrange in ascending order:,,, 1 6 9 1 1 LCM of 6, 9, 1, 1 = 6 The fractions can be written as 0,,, 6 6 6 6 6 Therefore the order would be 6 < < 0 < 6 6 6 6 1 < < < 1 9 6 1, 9,, 1 6 1 Note: a = a b b LCM of, 1, 6, 1 = The fractions can be written as 60,, 0, 9 Therefore the order would be 0 < 60 < < 9 6 < < 9 1 < 1 i, 1, 1 6, Note: a b = a b
LCM of, 6 = 6 The fractions can be written as 1,, 1, 16 6 6 6 6 Therefore the order would be 16 6 < 1 6 < 1 6 < 6 < 1 6 < < 1 1,,, 9 1 Note: a = a b b LCM of, 6 = 6 The fractions can be written as 1,, 1, 16 6 6 6 6 Therefore the order would be 16 6 < 1 6 < 1 6 < 6 < 1 6 < < 1. Represent each of these numbers on a Number Line: i 6 Divide the unit length between 0 1 in 6 equal parts then mark 6. 1 Divide the unit length between in equal parts then mark 1 Divide the unit length between 0-1 in equal parts then mark
1 Divide the unit length between - - in equal parts then mark 1 v. Divide the unit length between - in equal parts then mark. Find the additive inverse of: 9 i 1 9 1 + a = 0 Therefore the additive inverse is 9 1 1 1 + a = 0 Therefore the additive inverse is 1. Find the sum: + a = 0 Therefore the additive inverse is Therefore the additive inverse is 9 v. v + a = 0 Therefore the additive inverse is 1 1 + a = 0 Therefore the additive inverse is 1 9 9 + a = 0 + 6 = 1 1 1 1
i + = 1 = 1 1 1 1 + = + 1 1 60 = 19 0 (Note: LCM of 1 1 is 60) v. + = 1 1 1 6 = 6 + + + = 6 (Note: LCM of 1 1 is 6) (Note: LCM of 6,,, is ) v + + + = 1 9 6 (Note: LCM of,, 1, 9 is 6) 6. Subtract: from 6 i 6 = 6 6 = 1 6 from ( ) = + 1 = 9 from = = 9 9 6 from ( 6 ) = 16 6 + 6 = 9. The sum of two rational numbers is 9. If one of them is 1 6 then find the other.
1 6 + a = 9 a = 9 1 6 = 1. What number should be added to + a = 1 a = 1 + = 1 to get 1 9. What number should be subtracted from to get a = a = = 9. Find the products: 9 1 = 9 1 = i 9 1 = 9 1 =. Find the quotient: 1 1 = 1 1 = = 1 6 16 1 = 16 1 = 1 1 i 1 ( 16) = 1 1 16 = 9 ( 1 ) = 9 = 16 1 1. The product of two rational numbers is -. If one of the number is, then find the other. 6
a b = b = b = 1. By what number must 1 6 1 a = 6 9 1 6 1 a = 9 a = 1 6 9 = 16 be divided to get 9? 1. Find a rational number between each of the following pairs of rational numbers. 1 1 First take the LCM of 1 which is. Convert the numbers with as the denominator. Hence we get 9 0 Therefore the rational numbers between 16 1 are Therefore the rational numbers between 1 are 1 1 i First take the LCM of which is. Convert the numbers with as the denominator. Hence we get 1 0 Therefore the rational numbers between are 0 0 or 1 0
1 1 1 1 1 Therefore the rational numbers between 1 1. Find three rational numbers between: 1 1 Therefore the rational numbers between 1 are are 1 1 or 6 1 0 16 0 1 0 6 1 0 1 1 1 Therefore the rational numbers between 1 0 0 1 0 1 0 1 0 1 are 16. Find rational numbers between: 9 1 The rational numbers are 1 91, 9, 9, 9, 9 91, 6, 1,, 19 0 0 1 1 The rational numbers are
1,,,, 0 0 0 0 0 1,,, 1, 0 0 1. Determine whether the numbers are rational or irrational: 1 : Rational vi : Irrational 0.6 : Rational i 169 = 1 : Rational 1-6 = 6-6 = 0 : Rational v. 0.1 : Irrational v 1 1 : Rational v 9 = 9 = : Irrational ix. 1 = : Rational x. π : Irrational x 1 = 1 Rational x 1.1 : Rational xi 0.9 : Irrational x 0.09 = : Rational 1 xv. : Irrational 1. Skipped 1. 19. State whether True or False: Every real number is either rational or irrational: True Every real number can be represented on a number line: True i There exists integer which is not a rational number: False There exist a point on a number line which do not represent any real number: False v. An infinite number of rational numbers can be inserted between any two rational numbers: True v The multiplicative inverse of any rational number a is 1 a : False 0. Fill in the blanks 0 is a rational number that is its own additive inverse. 0 is a rational number that does not have a multiplicative inverse. i 1-1 are two rational numbers which are equal it their own reciprocal. The product of a rational number with its reciprocal is 1. v. The reciprocal of a negative number is negative. 9
v The multiplicative inverse of a rational number is 1, a 0 is a. a v Number of irrational number between any two rational number is infinite. 1. Arrange in ascending order, 1,, 6, First take everything within under root sign. That way we can compare the numbers easily. The numbers would then be 19, 60,,, 6 Now arrange in ascending order, 60, 6,, 19 6, 1,,,, 1,, 1, First take everything within under root sign. That way we can compare the numbers easily. The numbers would then be 0, 1, 1, 169, 1 Now arrange in ascending order 0, 1, 1, 1, 169, 1,,, 1. Write the rationalizing factors of the following: = Therefore rationalizing factor is i 6 6 = 1 Therefore rationalizing factor is + ( + )( ) = Therefore rationalizing factor is ( ) + ( + )( ) = Therefore rationalizing factor is ( + ) v.
( )( + ) = 1 Therefore rationalizing factor is ( + ) v ( )( + ) = 1 Therefore rationalizing factor is ( + ). Rationalize the denominator of each of the following: 6 = 6 = i v. = 6 + = ( ) + 6+ = (6 ) 6 6+ ( + ) = ( + ) ( + ) 19 = ( ) v + + + = (+ ) 9 v vi = ( ) + 6 6 = ( 6) + 6 6. Insert rational numbers between: 1 6,,, 9,
First take everything within the root sign. So we need to find rational numbers between Hence the numbers are, 9,,, 1 i. we can make the numbers as square root. So we need to find rational numbers between 6. Hence the numbers are.1,.,.,.,.. State True or False: + = : False i + = : True = : True (9 + ) + ( ) is a rational number: True (the value is 1 which is a rational number) v. ( )( ) is irrational number: False (the value is 16 which is a rational number) v ( )( 1 + ) is a rational number: True (the value is 6 which is a v rational number) + + + rational number) Is a rational number. True (the value is which is a 1