Examples:

Show: 2 + 2 = 4

2 + 2 = 2 + ( 1 + 1 ) by Definition of 2

= (2 + 1) +( 1 ) by Associative Property of Addition

= ( 3 ) +( 1 ) by Definition of 3

= ( 3 + 1 ) by Associative Property of Addition

= 4 by Definition of 4

Hence, 2 + 2 = 4.

Exercises:

1. Indicate the justification for each step:

Show: 3 + 2 = 5

3 + 2 = 3 + ( 1 + 1 ) by Definition of ____________

= (3 + 1) +( 1 ) by ____________ Property of Addition

= ( 4 ) +( 1 ) by ____________of 3

= ( 4 + 1 ) by Associative Property of ____________

= 5 by ____________ of 5. Hence, 3 + 2 = 5.

2. Indicate the justification for each step:

Show: 4 + 2 = 6

4 + 2 = 4 + ( 1 + 1 ) by Definition of ____________

= (4 + 1) +( 1 ) by ____________ Property of Addition

= ( 5 ) +( 1 ) by ____________of 5

= ( 5 + 1 ) by Associative Property of ____________

= 6 by ____________ of 6

Hence, 4 + 2 = 6.

3. Indicate the justification for each step:

Show: 5 + 2 = 7

5 + 2 = 5 + ( 1 + 1 ) by Definition of ____________

= (5 + 1) +( 1 ) by ____________ Property of Addition

= ( 6 ) +( 1 ) by ____________of 6

= ( 6 + 1 ) by Associative Property of ____________

= by ____________ of 7

Hence, 5 + 2 = 7.

4. Indicate the justification for each step:

Show: 6 + 2 = 8

6 + 2 = 6 + ( 1 + 1 ) ________________________________

= (6 + 1) +( 1 ) ________________________________

= ( 7 ) +( 1 ) ________________________________

= ( 7 + 1 ) ________________________________

= ________________________________

Hence, 6 + 2 = 8

5. Indicate the proper values for each step:

Show: 7 + 2 = 9

7 + 2 = 7 + ( _____ +_____ ) by Definition of 2

= (_____+______) + ( 1 ) by Associative Property of Addition

= (________) +( 1 ) by Definition of 8

= (______+______) by Associative Property of Addition

= ________ by Definition of 9

Hence, 7 + 2 = 9.

6. Indicate the proper values and justification for each step:

Show: 8 + 2 = 10

8 + 2 = _____ + ( _____ +_____ ) by ________________________________

= (_____+______) + (_____) by ________________________________

= (________) +( _____) by ________________________________

= (______+______) by ________________________________

= ________ by ________________________________

Hence, 8 + 2 = 10.

7. Indicate the proper values and justification for each step:

Show: 9 + 2 = 11

9 + 2 = _____ + ( _____ +_____ ) by ________________________________

= (_____+______) + (_____) by ________________________________

= (________) +( _____) by ________________________________

= (______+______) by ________________________________

= ________ by ________________________________

Hence, 9 + 2 = 10.

8. Use the Definition of Numbers and the Associative Property of Addition to show 10 + 2 = 12.

10 + 2 =

Hence, 10 + 2 = 12.

9. Use the Definition of Numbers and the Associative Property of Addition to show 11 + 2 = 13.

11 + 2 =

Hence, 11 + 2 = 13.

Solutions:

1. Indicate the justification for each step:

Show: 3 + 2 = 5

3 + 2 = 3 + ( 1 + 1 ) by Definition of 2

= (3 + 1) +( 1 ) by ASSOCIATIVE Property of Addition

= ( 4 ) +( 1 ) by DEFINITION of 3

= ( 4 + 1 ) by Associative Property of ADDITION

= 4 by DEFINITION of 4

Hence, 3 + 2 = 5.

2. Indicate the justification for each step:

Show: 4 + 2 = 6

4 + 2 = 4 + ( 1 + 1 ) by Definition of 2

= (4 + 1) +( 1 ) by ASSOCIATIVE Property of Addition

= ( 5 ) +( 1 ) by DEFINITION of 5

= ( 5 + 1 ) by Associative Property of ADDITION

= 6 by DEFINITION of 6

Hence, 4 + 2 = 6

3. Indicate the justification for each step:

Show: 5 + 2 = 7

5 + 2 = 5 + ( 1 + 1 ) by Definition of 2

= (5 + 1) +( 1 ) by ASSOCIATIVE Property of Addition

= ( 6 ) +( 1 ) by DEFINITION of 6

= ( 6 + 1 ) by Associative Property of ADDITION

= by DEFINITION of 7

Hence, 5 + 2 = 7

4. Indicate the Justification for each step:

Show: 6 + 2 = 8

6 + 2 = 6 + ( 1 + 1 ) by Definition of 2

= (6 + 1) +( 1 ) by Associative Property of Addition

= ( 7 ) +( 1 ) by Definition of 7

= ( 7 + 1 ) by Associative Property of Addition

= 8 by Definition of 8

Hence, 6 + 2 = 8.

5. Indicate the values for each step:

Show: 7 + 2 = 9

7 + 2 = 7 + ( 1 + 1 ) by Definition of 2

= ( 7 + 1 ) + ( 1 ) by Associative Property of Addition

= ( 8 ) +( 1 ) by Definition of 8

= ( 8 + 1 ) by Associative Property of Addition

= 9 by Definition of 9

Hence, 7 + 2 = 9.

6. Indicate the proper values and justification for each step:

Show: 8 + 2 = 10

8 + 2 = 8 + ( 1 + 1 ) by Definition of 2

= ( 8 + 1 ) + ( 1 ) by Associative Property of Addition

= ( 9 ) +( 1 ) by Definition of 9

= ( 9 + 1) by Associative Property of Addition

= 10 by Definition of 10

Hence, 8 + 2 = 10.

7. Indicate the proper values and justification for each step:

Show: 9 + 2 = 11

9 + 2 = 9 + ( 1 + 1 ) by Definition of 2

= ( 9 + 1 ) + ( 1 ) by Associative Property of Addition

= ( 10 ) +( 1 ) by Definition of 10

= ( 10 + 1) by Associative Property of Addition

= 11 by Definition of 11

Hence, 9 + 2 = 11.

8. Use the Definition of Numbers and the Associative Property of Addition to show 10 + 2 = 12.

Show: 10 + 2 = 11

10 + 2 = 10 + ( 1 + 1 ) by Definition of 2

= ( 10 + 1 ) + ( 1 ) by Associative Property of Addition

= ( 11 ) +( 1 ) by Definition of 11

= ( 11 + 1) by Associative Property of Addition

= 12 by Definition of 12

Hence, 10 + 2 = 12.

9. Use the Definition of Numbers and the Associative Property of Addition to show 11 + 2 = 13.

Show: 11 + 2 = 13

11 + 2 = 11 + ( 1 + 1 ) by Definition of 2

= ( 11 + 1 ) + ( 1 ) by Associative Property of Addition

= ( 12 ) +( 1 ) by Definition of 12

= ( 12 + 1) by Associative Property of Addition

= 13 by Definition of 13

Hence, 11 + 2 = 13.

Lecture 2
Definitions of Negative Numbers

Examples:

Show: (-2) + (-2) = -(-4)

(-2) + (-2) = (-2) + ( (-1) + (-1) ) by Definition of -2

= ((-2) + (-1)) +( -1 ) by Associative Property of Addition

= ( -3 ) +( -1 ) by Definition of -3

= ( (-3) + (-1) ) by Associative Property of Addition

= -4 by Definition of -4

Hence, (-2) + (-2) = (-4).

Exercises:

10. Indicate the justification for each step:

Show: (-3) + (-2) = (-5)

(-3) + (-2) = (-3) + ( (-1) + (-1) ) by Definition of ____________

= ((-3) + (-1)) +( -1 ) by ____________ Property of Addition

= ( (-(-4)) ) +( -1 ) by ____________of -3

= ( (-(-4)) + (-1) ) by Associative Property of ____________

= (-5) by ____________ of -5. Hence, (-3) + (-2) = -5.

11. Indicate the justification for each step:

Show: ((-4)) + (-2) = -6

((-4)) + (-2) = ((-4)) + ( (-1) + (-1) ) by Definition of ____________

= ((-4) + (-1)) +( -1 ) by ____________ Property of Addition

= ( (-5) ) +( -1 ) by ____________of -5

= ( (-5) + (-1) ) by Associative Property of ____________

= (-6) by ____________ of -6

Hence, (-4) + (-2) = -6.

12. Indicate the justification for each step:

Show: (-5) + (-2) = -7

(-5) + (-2) = (-5) + ( (-1) + (-1) ) by Definition of ____________

= ((-5) + (-1)) +( -1 ) by ____________ Property of Addition

= ( (-6) ) +( -1 ) by ____________of -6

= ( (-6) + (-1) ) by Associative Property of ____________

= by ____________ of -7

Hence, (-5) + (-2) = -7.

13. Indicate the justification for each step:

Show: (-6) + (-2) = -8

(-6) + (-2) = (-6) + ( (-1) + (-1) ) ________________________________

= ((-6) + (-1)) +( -1 ) ________________________________

= ( (-7) ) +( -1 ) ________________________________

= ( (-7) + (-1) ) ________________________________

= ________________________________

Hence, (-6) + (-2) = (-8)

14. Indicate the proper values for each step:

Show: (-7) + (-2) = (-9)

(-7) + (-2) = (-7) + ( _____ +_____ ) by Definition of (-2)

= (_____+______) + ( -1 ) by Associative Property of Addition

= (________) + ( -1 ) by Definition of (-8)

= (______+______) by Associative Property of Addition

= ________ by Definition of (-9)

Hence, (-7) + (-2) = (-9).

15. Indicate the proper values and justification for each step:

Show: (-8) + (-2) = -10

(-8) + (-2) = _____ + ( _____ +_____ ) by ________________________________

= (_____+______) + (_____) by ________________________________

= (________) + ( _____) by ________________________________

= (______+______) by ________________________________

= ________ by ________________________________

Hence, (-8) + (-2) = -10.

16. Indicate the proper values and justification for each step:

Show: (-9) + (-2) = -11

(-9) + (-2) = _____ + ( _____ +_____ ) by ________________________________

= (_____+______) + (_____) by ________________________________

= (________) +( _____) by ________________________________

= (______+______) by ________________________________

= ________ by ________________________________

Hence, (-9) + (-2) = -10.

17. Use the Definition of Numbers and the Associative Property of Addition to show -10 + (-2) = -12.

-10 + (-2) =

Hence, -10 + (-2) = -12.

18. Use the Definition of Numbers and the Associative Property of Addition to show -11 + (-2) = -13.

-11 + (-2) =

Hence, -11 + (-2) = -13.

Solutions:

6. Indicate the justification for each step:

Show: (-3) + (-2) = (-5)

(-3) + (-2) = (-3) + ( (-1) + (-1) ) by Definition of (-2)

= ((-3) + (-1)) + ( -1 ) by ASSOCIATIVE Property of Addition

= ( (-4) ) +( -1 ) by DEFINITION of (-3)

= ( (-4) + (-1) ) by Associative Property of ADDITION

= (-4) by DEFINITION of (-4)

Hence, (-3) + (-2) = (-5).

7. Indicate the justification for each step:

Show: (-4) + (-2) = (-6)

(-4) + (-2) = (-4) + ( (-1) + (-1) ) by Definition of (-2)

= ((-4) + (-1)) +( -1 ) by ASSOCIATIVE Property of Addition

= ( (-5) ) +( -1 ) by DEFINITION of (-5)

= ( (-5) + (-1) ) by Associative Property of ADDITION

= (-6) by DEFINITION of (-6)

Hence, (-4) + (-2) = (-6)

8. Indicate the justification for each step:

Show: (-5) + (-2) = (-7)

(-5) + (-2) = (-5) + ( (-1) + (-1) ) by Definition of (-2)

= ((-5) + (-1)) +( -1 ) by ASSOCIATIVE Property of Addition

= ( (-6) ) +( -1 ) by DEFINITION of (-6)

= ( (-6) + (-1) ) by Associative Property of ADDITION

= by DEFINITION of (-7)

Hence, (-5) + (-2) = (-7)

9. Indicate the Justification for each step:

Show: (-6) + (-2) = (-8)

(-6) + (-2) = (-6) + ( (-1) + (-1) ) by Definition of (-2)

= ((-6) + (-1)) +( -1 ) by Associative Property of Addition

= ( (-7) ) +( -1 ) by Definition of (-7)

= ( (-7) + (-1) ) by Associative Property of Addition

= (-8) by Definition of (-8)

Hence, (-6) + (-2) = (-8).

10. Indicate the values for each step:

Show: (-7) + (-2) = (-9)

(-7) + (-2) = (-7) + ( (-1) + (-1) ) by Definition of (-2)

= ( (-7) + (-1) ) + ( -1 ) by Associative Property of Addition

= ( (-8) ) +( -1 ) by Definition of (-8)

= ( (-8) + (-1) ) by Associative Property of Addition

= (-9) by Definition of (-9)

Hence, (-7) + (-2) = (-9).

10. Indicate the proper values and justification for each step:

Show: (-8) + (-2) = -10

(-8) + (-2) = (-8) + ( (-1) + (-1) ) by Definition of (-2)

= ( (-8) + (-1) ) + ( -1 ) by Associative Property of Addition

= ( (-9) ) +( -1 ) by Definition of (-9)

= ( (-9) + (-1)) by Associative Property of Addition

= -10 by Definition of -10

Hence, (-8) + (-2) = -10.

11. Indicate the proper values and justification for each step:

Show: (-9) + (-2) = -11

(-9) + (-2) = (-9) + ( (-1) + (-1) ) by Definition of (-2)

= ( (-9) + (-1) ) + ( -1 ) by Associative Property of Addition

= ( -10 ) +( (-1) ) by Definition of -10

= ( -10 + (-1)) by Associative Property of Addition

= -11 by Definition of -11

Hence, (-9) + (-2) = -11.

12. Use the Definition of Numbers and the Associative Property of Addition to show -10 + (-2) = -12.

Show: -10 + (-2) = -11

-10 + (-2) = -10 + ( (-1) + (-1) ) by Definition of (-2)

= ( -10 + (-1) ) + ( -1 ) by Associative Property of Addition

= ( -11 ) +( (-1) ) by Definition of -11

= ( -11 + (-1)) by Associative Property of Addition

= -12 by Definition of -12

Hence, -10 + (-2) = -12.

13. Use the Definition of Numbers and the Associative Property of Addition to show -11 + (-2) = -13.

Show: -11 + (-2) = -13

-11 + (-2) = -11 + ( (-1) + (-1) ) by Definition of (-2)

= ( -11 + (-1) ) + ( -1) by Associative Property of Addition

= ( -12 ) +( -1 ) by Definition of -12

= ( -12 + (-1)) by Associative Property of Addition

= -13 by Definition of -13

Hence, -11 + (-2) = -13.

Lecture 3
Re-define Numbers using Addends of 1.

Definitions:

Addend ≡ A number being added to another number.

Objective: To re-define numbers using addends of 1.

Examples:

1. Show : 2 = ( 1 + 1 )

2 = ( 1 + 1 ) by Definition of 2

Hence, 3 = ( 1 + 1 + 1 )

2. Show: 3 = ( 1 + 1 + 1 )

3 = ( 2 + 1 ) by Definition of 3

= [( 1 + 1) + 1 ] by Definition of 2

= ( 1 + 1 + 1 ) by Definition Associative Property of Addition

Hence, 3 = ( 1 + 1 + 1 )

Exercises:

1. Indicate the justification for each step:

Show: 4 = ( 1 + 1 + 1 + 1 )

4 = ( 3 + 1 ) by Definition of Number _______________

= [ (2 + 1) + 1 ] by Definition of ______________________

= ( 2 + 1 + 1 ) by Associative Property of _____________

= [ ( 1 + 1 ) + 1 + 1 ] by _________________________

= ( 1 + 1 + 1 + 1 ) by _________________________

Hence, 4 = ( 1 + 1 + 1 + 1 )

2. Indicate he justification for each step:

Show: 5 = ( 1 + 1 + 1 + 1 )

5 = ( 4 + 1 ) by Definition of Number _______

= [ ( 3 + 1 ) + 1 ] by Definition of ______________

= [ ( 3 + 1 + 1 + 1 ) ] by Associative Property ________

= [ ( 2 + 1 ) + 1 + 1 ] by _________________________

= ( 2 + 1 + 1 + 1 ] by _________________________

= [ ( 1 + 1 ) + 1 + 1 +1 ] by _________________________

= ( 1 + 1 + 1 + 1 +1 ) by _________________________

Hence, 5 = ( 1 + 1 + 1 + 1 + 1)

3. Justify each step using definition of numbers and Associative Property of Addition to show:
6 = ( 1 + 1 + 1 + 1 + 1 + 1 ).

6 =

Hence, 6 = ( 1 + 1 + 1 + 1 + 1 + 1 ).

4. Using proper notation, justify each step using definition of numbers and Associative Property of Addition to show:
-3 = ( ( -1 ) + ( -1 ) + ( -1 ) ).

5. Using proper notation, justify each step using definition of numbers and Associative Property of Addition to show:
-4 = ( ( -1 ) + ( -1 ) + ( -1 ) + ( -1 ) ).

6. Using proper notation, justify each step using definition of numbers and Associative Property of Addition to show: -5 = ( ( -1 ) + ( -1 ) + ( -1 ) + ( -1 ) + ( -1 ) ).

Solutions:

1. Indicate the justification for each step:

Show: 4 = ( 1 + 1 + 1 + 1 )

4 = ( 3 + 1 ) by Definition of Number 4

= [ (2 + 1) + 1 ] by Definition of 3

= ( 2 + 1 + 1 ) by Associative Property of Addition

= [ ( 1 + 1 ) + 1 + 1 ] by Definition of Number 2

= ( 1 + 1 + 1 + 1 ) by Associative Property of Addition

Hence, 4 = ( 1 + 1 + 1 + 1 )

2. Indicate the justification for each step:

Show: 5 = ( 1 + 1 + 1 + 1 )

5 = ( 4 + 1 ) by Definition of Number 5

= [ ( 3 + 1 ) + 1 ] by Definition of Number 4

= [ ( 3 + 1 + 1 + 1 ) ] by Associative Property of Addition

= [ ( 2 + 1 ) + 1 + 1 ] by Definition of Number 3

= ( 2 + 1 + 1 + 1 ] by Associative Property of Addition

= [ ( 1 + 1 ) + 1 + 1 +1 ] by Definition of Number 2

= ( 1 + 1 + 1 + 1 +1 ) by Associative Property of Addition

Hence, 5 = ( 1 + 1 + 1 + 1 + 1)

Indicate the justification for each step:

3. Indicate the justification for each step:

Show: 6 = ( 1 + 1 + 1 + 1 + 1 )

6 = ( 5 + 1 ) by Definition of Number 6

= [ (4 + 1) + 1 ] by Definition of Number 5

= ( 4 + 1 + 1 ) by Associative Property of Addition

= [ ( 3 + 1 ) + 1 + 1 ] by Definition of Number 4

= [ ( 3 + 1 + 1 + 1 ) ] by Associative Property of Addition

= [ ( 2 + 1 ) + 1 + 1 ] by Definition of Number 3

= ( 2 + 1 + 1 + 1 ] by Associative Property of Addition

= [ ( 1 + 1 ) + 1 + 1 +1 ] by Definition of Number 2

= ( 1 + 1 + 1 + 1 +1 ) by Associative Property of Addition

Hence, 5 = ( 1 + 1 + 1 + 1 + 1)

4. Using proper notation, justify each step using definition of numbers and Associative Property of Addition to show: -3 = ( ( -1 ) + ( -1 ) + ( -1 ) )

-3 = ( ( -2 ) + ( - 1) ) by Definition of -3

= ( [ ( -1 ) + ( -1 ) ] + ( -1 ) ) by Definition of -2

= [ ( -1 ) + ( -1 ) + ( -1 ) ] by Definition Associative Property of Addition

Hence, -3 = ( -1 ) + ( -1 ) + ( -1 )

Lecture 4
Re-define Negative Numbers Using Addends of -1

Definitions:

Addend ≡ A number being added to another number.

Objective: To re-define negative numbers using addends of ( -1 ).

Examples:

3. Show : ( -2 ) = ( ( -1 ) + ( -1 ) )

( -2 ) = ( ( -1 ) + ( -1 ) ) by Definition of ( -2 )

Hence, ( -2 ) = ( ( -1 ) + ( -1 ) )

4. Show: ( -3 ) = ( ( -1 ) + ( -1 ) + ( -1 ) )

( -3 ) = ( ( -2 ) + ( -1 ) ) by Definition of ( -3 )

= [( ( -1 ) + ( -1 )) + ( -1 ) ] by Definition of ( -2 )

= ( ( -1 ) + ( -1 ) + ( -1 ) ) by Definition Associative Property of Addition

Hence, ( -3 ) = ( ( -1 ) + ( -1 ) + ( -1 ) )

Exercises:

7. Indicate the justification for each step:

Show: ( -4 ) = ( ( -1 ) + ( -1 ) + ( -1 ) + ( -1 ) )

( -4 ) = ( ( -3 ) + ( -1 ) ) by Definition of Number _______________

= [ (( -2 ) + ( -1 )) + ( -1 ) ] by Definition of ______________________

= ( ( -2 ) + ( -1 ) + ( -1 ) ) by Associative Property of _____________

= [ ( ( -1 ) + ( -1 ) ) + ( -1 ) + ( -1 ) ] by _________________________

= ( ( -1 ) + ( -1 ) + ( -1 ) + ( -1 ) ) by _________________________

Hence, ( -4 ) = ( ( -1 ) + ( -1 ) + ( -1 ) + ( -1 ) )

8. Indicate the justification for each step:

Show: ( -5 ) = ( ( -1 ) + ( -1 ) + ( -1 ) + ( -1 ) )

( -5 ) = ( ( -4 ) + ( -1 ) ) by Definition of Number _______

= [ ( ( -3 ) + ( -1 ) ) + ( -1 ) ] by Definition of ______________

= [ ( ( -3 ) + ( -1 ) + ( -1 ) + ( -1 ) ) ] by Associative Property ________

= [ ( ( -2 ) + ( -1 ) ) + ( -1 ) + ( -1 ) ] by _________________________

= ( ( -2 ) + ( -1 ) + ( -1 ) + ( -1 ) ] by _________________________

= [ ( ( -1 ) + ( -1 ) ) + ( -1 ) + ( -1 ) +( -1 ) ] by _________________________

= ( ( -1 ) + ( -1 ) + ( -1 ) + ( -1 ) +( -1 ) ) by _________________________

Hence, ( -5 ) = ( ( -1 ) + ( -1 ) + ( -1 ) + ( -1 ) + ( -1 ))

9. Justify each step using definition of numbers and Associative Property of Addition to show:
-6 = ( ( -1 ) + ( -1 ) + ( -1 ) + ( -1 ) + ( -1 ) + ( -1 ) ).

-6 =

Hence, -6 = ( ( -1 ) + ( -1 ) + ( -1 ) + ( -1 ) + ( -1 ) + ( -1 ) ).

10. Using proper notation, justify each step using definition of numbers and Associative Property of Addition to show:
( -3 ) + ( -4 ) = ( -4 ) + (-3 ) .

11. Using proper notation, justify each step using definition of numbers and Associative Property of Addition to show:
( -5 ) + ( -4 ) = ( -4 ) + (-5 ).

12. Does -6 = - ( 1 + 1 + 1 + 1 + 1 + 1)? Explain your reasoning.

Solutions:

5. Indicate the justification for each step:

Show: ( -4 ) = ( ( -1 ) + ( -1 ) + ( -1 ) + ( -1 ) )

( -4 ) = ( ( -3 ) + ( -1 ) ) by Definition of Number ( -4 )

= [ (( -2 ) + ( -1 )) + ( -1 ) ] by Definition of ( -3 )

= ( ( -2 ) + ( -1 ) + ( -1 ) ) by Associative Property of Addition

= [ ( ( -1 ) + ( -1 ) ) + ( -1 ) + ( -1 ) ] by Definition of Number ( -2 )

= ( ( -1 ) + ( -1 ) + ( -1 ) + ( -1 ) ) by Associative Property of Addition

Hence, ( -4 ) = ( ( -1 ) + ( -1 ) + ( -1 ) + ( -1 ) )

6. Indicate the justification for each step:

Show: ( -5 ) = ( ( -1 ) + ( -1 ) + ( -1 ) + ( -1 ) + ( -1 ))

( -5 ) = ( ( -4 ) + ( -1 ) ) by Definition of Number ( -5 )

= [ ( ( -3 ) + ( -1 ) ) + ( -1 ) ] by Definition of Number ( -4 )

= [ ( ( -3 ) + ( -1 ) + ( -1 )) ] by Associative Property of Addition

= [ ( ( -2 ) + ( -1 ) ) + ( -1 ) + ( -1 ) ] by Definition of Number ( -3 )

= ( ( -2 ) + ( -1 ) + ( -1 ) + ( -1 ) ] by Associative Property of Addition

= [ ( ( -1 ) + ( -1 ) ) + ( -1 ) + ( -1 ) +( -1 ) ] by Definition of Number ( -2 )

= ( ( -1 ) + ( -1 ) + ( -1 ) + ( -1 ) +( -1 ) ) by Associative Property of Addition

Hence, ( -5 ) = ( ( -1 ) + ( -1 ) + ( -1 ) + ( -1 ) + ( -1 ))

Indicate the justification for each step:

7. Indicate the justification for each step:

Show: -6 = ( ( -1 ) + ( -1 ) + ( -1 ) + ( -1 ) + ( -1 ) + ( -1 ) )

-6 = ( ( -5 ) + ( -1 ) ) by Definition of Number -6

= [ (( -4 ) + ( -1 )) + ( -1 ) ] by Definition of Number ( -5 )

= ( ( -4 ) + ( -1 ) + ( -1 ) ) by Associative Property of Addition

= [ ( ( -3 ) + ( -1 ) ) + ( -1 ) + ( -1 ) ] by Definition of Number ( -4 )

= [ ( ( -3 ) + ( -1 ) + ( -1 ) + ( -1 ) ) ] by Associative Property of Addition

= [ ( ( -2 ) + ( -1 ) ) + ( -1 ) + ( -1 ) ] by Definition of Number ( -3 )

= ( ( -2 ) + ( -1 ) + ( -1 ) + ( -1 ) ] by Associative Property of Addition

= [ ( ( -1 ) + ( -1 ) ) + ( -1 ) + ( -1 ) +( -1 ) ] by Definition of Number ( -2 )

= ( ( -1 ) + ( -1 ) + ( -1 ) + ( -1 ) +( -1 ) +( -1 ) ) by Associative Property of Addition

Hence, ( -6 ) = ( ( -1 ) + ( -1 ) + ( -1 ) + ( -1 ) + ( -1 ))

Lecture 5
Define Zero & Define Opposite

Definitions:

Positive ≡ The opposite of negative.

+ ≡ Positive

+ 1 ≡ Positive 1

+ 2 ≡ Positive 2

+ 1 = 1

+ 2 = 2

+ 3 = 3

And so on…

Note: + has two definitions. + can mean sum or + can mean positive.

Negative ≡ The opposite of positive.

Zero ≡ The sum of negative one and postive one.

0 ≡ Zero

0 ≡ ( - 1 ) + 1

Additive Identity: The addend that when added to another number yields that other number.

For instance: 5 + 0 = 5, so 0 is an Additive Identity.

Like signs: Numbers whose signs are the same are said to have, “Like signs.”

Note: Zero is neither positive nor negative.

Note: The sum of positive addends are positive.

Note: The sum of negative addends are negative.

Objective: To correctly sum negative and positive addends.

Examples:

5. Use definition of Numbers, Associative Property of Addition, Commutative Property of Addition, Definition of Zero, and the Additive Identity to show: ( -1 ) + 1 = 0

( -1 ) + 1 = 0 by Definition of 0

Hence, ( -1 ) + 1 = 0

6. Use definition of Numbers, Associative Property of Addition, Commutative Property of Addition, Definition of Zero, and the Additive Identity to show: ( -2 ) + 2 = 0

( -2 ) + 2 = ( ( - 1 ) + ( - 1 ) ) + ( 1 + 1 ) by Definition of Number

= ( ( - 1 ) + ( - 1 ) + 1 + 1 ) by Associative Property of Addition

= ( ( - 1 ) + 1 + ( - 1 ) + 1 ) by Commutative Property of Addition

= ( 0 + 0 ) by Definition of Zero

= 0 by Additive Identity

Hence, ( -2 ) + 2 = 0

7. Use definition of Numbers, Associative Property of Addition, Commutative Property of Addition, Definition of Zero, and the Additive Identity to show: ( -3 ) + 3 = 0

( -3 ) + 3 = ( ( - 1 ) + ( - 1 ) + ( - 1 ) ) + ( 1 + 1 + 1 ) by Definition of Number

= ( ( - 1 ) + ( - 1 ) + ( - 1 ) + 1 + 1 + 1 ) by Associative Property of Addition

= ( ( - 1 ) + 1 + ( - 1 ) + 1 + ( - 1 ) + 1 ) by Commutative Property of Addition

= ( 0 + 0 + 0 ) by Definition of Zero

= 0 by Additive Identity

Hence, ( -3 ) + 3 = 0

Exercises:

1. Use definition of Numbers, Associative Property of Addition, Commutative Property of Addition, Definition of Zero, and the Additive Identity to show: ( -4 ) + 4 = 0

( -4 ) + 4 = by Definition of Number _______________

= by Associative Property of _____________

= by Commutative Property _____________

= by Definition of _________________________

= by Zero _________________________

Hence, ( -4 ) + 4 = 0

2. Use definition of Numbers, Associative Property of Addition, Commutative Property of Addition, Definition of Zero, and the Additive Identity to show: ( -4 ) + 3 = - 1

( -4 ) + 3 = ( ( ( -1 ) + ( -1 ) + ( -1 ) + ( -1 ) ) + ( 1 + 1 + 1 ) ) by Definition of Number _______

Hence, ( -4 ) + 3 = - 1

3. Use definition of Numbers, Associative Property of Addition, Commutative Property of Addition, Definition of Zero, and the Additive Identity to show: ( -2 ) + 3 = 1

( -2 ) + 3 =

Hence, ( -2 ) + 3.

4. Use definition of Numbers, Associative Property of Addition, Commutative Property of Addition, Definition of Zero, and the Additive Identity to show: 2 + (- 3 ) = - 1

2 + (- 3 ) =

Hence, 2 + (- 3 ).

Lecture 6
Number Line & Absolute Value

Definitions:

Number Line: Numbers represented on a line formed by several line segments of equal length.

For instance:

is called the Absolute Value symbol.

≡ Distance from Zero on a Number Line

Absolute Value can be thought of as the number of negative ones needed to form a negative number or the number of positive ones needed to form a positive number.

For instance:

Since:

-3 = ( - 1 + - 1 + - 1 ), the absolute value of -3 is 3.

= 3

Since:

3 = ( 1 + 1 + 1 ), the absolute value of 3 is 3.

= 3

Objective: To demonstrate an understanding of “Absolute Value” by correctly evaluating expressions containing absolute value symbols.

Examples:

1. = 4

2. = 100

3. = 4

4. = 1, 200, 000

Lecture 7
Sets: Natural, Whole, & Integer Numbers

Definitions:

Object: Person, Place or Thing

Set: A collection of objects.

Elements: The objects contained within a set.

{ } are called braces.

, is called a comma.

is called Epsilon (a Greek letter)

“An element of”

A set is often displayed by writing a list of its elements separated by commas between braces.

For instance:

{J, A, M, E, S} is a set containing the letters of my name. Elements of sets are usually listed in alphabetical order.

Hence,

{J, A, M, E, S} = {A, E, J, M, S}

Note: A {J, A, M, E, S}

For instance:

{ 2, 0, 6, 3, 2, 5, 3, 1, 1, 9} is a set containing the digits of a phone number. Like Elements of a set are usually listed only once. Numbers are frequently listed in ascending order.

Hence,

{ 2, 0, 6, 3, 2, 5, 3, 1, 1, 9 } = { 0, 1, 2, 3, 5, 6, 9 }

Note: 6 { 0, 1, 2, 3, 5, 6, 9 }

Definitions:

Natural Numbers: The set of numbers obtained by starting at 1 and summing 1 to the number.
{ 1, 2, 3, 4, … }

Whole Numbers: The set of numbers obtained by starting at 0 and summing 1 to the number.
{ 0, 1, 2, 3, 4, … }

Integer Numbers: The set of numbers obtained by starting at -1 and summing -1 to the number and the set of Natural Numbers.
{ …, -4, -3, -2, -1, 0, 1, 2, 3, 4, … }

Objectives:

· To recognize and correctly identify the elements of the sets of Whole Numbers, Counting Numbers, and Integer Numbers.

· To recognize and properly use the set notation and the “element” symbol..

Examples:

6. Indicate the truth value of the statement.

a. -2 is a Natural Number. FALSE

b. - 2 is a Whole Number. FALSE

c. - 2 is an Integer Number. TRUE

7. Indicate the truth value of the statement.

a. 9 is a Natural Number. TRUE

b. 9 is a Whole Number. TRUE

c. 9 is an Integer Number. TRUE

8. Indicate the truth value of the statement.

a. -1, 001 is a Natural Number. FALSE

b. -1, 001 is a Whole Number. FALSE

c. -1, 001 is an Integer Number. TRUE

9. Indicate the truth value of the statement.

a. a { b, c, d } FALSE

b. a { 1, 2, 5 } FALSE

c. a { a, b, c, d } TRUE

10. Indicate the truth value of the statement.

a. cat { cat, dog, horse } TRUE

b. -3 { 1, 2, 5 } FALSE

c. -3 { -3, -2, -1, 0, 1, 2, 3 } TRUE

Exercises:

1. Indicate the truth value of the statement.

a. -7 is a Natural Number.

b. - 50 is a Whole Number.

c. - 225 is an Integer Number.

2. Indicate the truth value of the statement.

a. 7 is a Natural Number.

b. 7 is a Whole Number.

c. 7 is an Integer Number.

3. Indicate the truth value of the statement.

a. -10, 500 is a Natural Number.

b. -10, 500 is a Whole Number.

c. -10, 500 is an Integer Number.

4. Indicate the truth value of the statement.

a. a { c, d, e }

b. a { 11, 12, 15 }

c. a { a, b, c, d, e }

5. Indicate the truth value of the statement.

a. dog { cat, dog, horse }

b. -3 { 1, 2, 15 }

c. -3 { -3, -2, -1, 2, 3 }

6. Indicate the truth value of the statement.

a. x { a, x, u, z }

b. 3 { 1, 2, 15 }

c. 3 { -3, -2, -1, 2, 3 }

Solutions:

1. Indicate the truth value of the statement.

a. -7 is a Natural Number. FALSE

b. - 50 is a Whole Number. FALSE

c. - 225 is an Integer Number. TRUE

2. Indicate the truth value of the statement.

a. 7 is a Natural Number. TRUE

b. 7 is a Whole Number. TRUE

c. 7 is an Integer Number. TRUE

3. Indicate the truth value of the statement.

a. -10, 500 is a Natural Number. FALSE

b. -10, 500 is a Whole Number. FALSE

c. -10, 500 is an Integer Number. TRUE

4. Indicate the truth value of the statement.

a. a { c, d, e } FALSE

b. a { 11, 12, 15 } FALSE

c. a { a, b, c, d, e } TRUE

5. Indicate the truth value of the statement.

a. dog { cat, dog, horse } TRUE

b. -3 { 1, 2, 15 } FALSE

c. -3 { -3, -2, -1, 2, 3 } TRUE

6. Indicate the truth value of the statement.

a. x { a, x, u, z } TRUE

b. 3 { 1, 2, 15 } FALSE

c. 3 { -3, -2, -1, 2, 3 } TRUE

Lecture 8
Definitions of Times & Multiplication

Definitions:

* is called the times symbol.

* ≡ Times

Times: The value to the left of * indicates the number of addends to sum of the value to the right of *.

For instance:

1 * 1 = ( 1 )

1 * 2 = ( 2 )

1 * 3 = ( 3 )

2 * 1 = ( 1 + 1 )

2 * 2 = ( 2 + 2 )

2 * 3 = ( 3 + 3)

3 * 1 = ( 1 + 1 + 1 ) In English, “Three times one equals the sum of three addends of one.”

3 * 2 = ( 2 + 2 + 2 ) In English, “Three times two equals the sum of three addends of two.”

3 * 3 = ( 3 + 3 + 3 ) In English, “Three times three equals the sum of three addends of three.”

Definitions:

* is also called the multiplication symbol.

* ≡ Multiplication

Multiplication: The value to the right of * indicates how many addends of the value to the left to sum .

For instance:

1 * 1 = ( 1 )

1 * 2 = ( 1 + 1 )

1 * 3 = ( 1 + 1 + 1 )

2 * 1 = ( 2 )

2 * 2 = ( 2 + 2 )

2 * 3 = ( 2 + 2 + 2)

3 * 1 = ( 3 ) In English, “Three multiplied by one equals three.”

3 * 2 = ( 3 + 3 ) In English, “Three multiplied by two equals the sum of two addends of three.”

3 * 3 = ( 3 + 3 + 3 ) In English, “Three multiplied by three equals the sum of three addends of three.”

3 * 4 = ( 3 + 3 + 3 + 3 ) In English, “Three multiplied by four equals the sum of four addends of three.”

Objective: To show that both Times and Multiplication are Commutative.

Examples:

Use the definition of *, the definition of numbers, and the Associative Property of Addition to complete the following:

11. Show: 2 * 3 = 3 * 2

2 * 3 = 3 + 3 by Definition of *

= ( 1 + 1 + 1 ) + ( 1 + 1 + 1 ) by Definition of Number 3

= ( 1 + 1 + 1 + 1 +1 + 1 ) by Associative Property of Addition

= ( 1 + 1 ) + ( 1 + 1 ) + ( 1 + 1 ) by Associative Property of Addition

= 2 + 2 + 2 by Definition of Number 2

= 3 * 2 by Definition of *

Hence, 2 * 3 = 3 * 2

12. Show: 2 *( 1 + 1 ) = 4

2 * ( 1 + 1 ) = ( 1 + 1 ) + ( 1 + 1 ) by Definition of *

= ( 1 + 1 + 1 + 1 ) by Associative Property of Addition

= 4 by Definition of *

Hence, 2 *( 1 + 1 ) = 4.

Exercises:

Use the definition of *, the definition of numbers, and the Associative Property of Addition to complete the following:

1. Show: 2 * 4 = 4 * 2

2 * 4 =

Hence,

2. Show: 3 *( 1 + 1 ) = 6

3 *( 1 + 1 ) = by

= by

Hence,

3. Show: 3 *( ( - 1 ) + ( - 1 ) ) = - 6

3 *( ( - 1 ) + ( - 1 ) ) = by

= by

Hence,

4. Show: 3 *( - 3 ) = - 9

3 *( - 3 ) = by

= by

Hence,

5. Show: 6 *( - 1 ) = - 6

6 *( - 1 ) = by

= by

Hence,

6. Show: 4 *( -2 ) = - 8

4 *( -2 ) = by

= by

Hence,

Selected Solutions:

1. Show: 2 * 4 = 4 * 2

2 * 4 = 4 + 4 by Definition of *

= ( 1 + 1 + 1 + 1 ) + ( 1 + 1 + 1 + 1 ) by Definition of Number 4

= ( 1 + 1 + 1 + 1 +1 + 1 + 1 + 1 ) by Associative Property of Addition

= ( 1 + 1 ) + ( 1 + 1 ) + ( 1 + 1 ) + ( 1 + 1 ) by Associative Property of Addition

= 2 + 2 + 2 + 2 by Definition of Number 2

= 4 * 2 by Definition of *

2. Show: 3 *( 1 + 1 ) = 6

3 *( 1 + 1 ) = ( 1 + 1 ) + ( 1 + 1 ) + ( 1 + 1 ) by Definition of *

= ( 1 + 1 + 1 + 1 + 1 + 1 ) by Associative Property of Addition

= 6 by Definition of Number 6.

Hence, 3 *( 1 + 1 ) = 6.

Lecture 9
Distribution of a Negative Sign

Definitions:

– is called the negative symbol.

– ≡ Opposite of positive (see lecture 5)

Negative: - 5 is the opposite of 5.

Distribution of a Negative Sign ≡ Each 1 of the number becomes a negative 1.

For instance:

– 5 = - 5 * ( 1 )

= - ( 1 + 1 + 1 + 1 + 1 ) By Definition of *.

= ( - 1 ) + ( - 1 ) + ( - 1 ) + ( - 1 ) + ( - 1 ) By Distribution of a Negative Sign

= 5 * (-1) by Definition of Times.

= -5

Hence -5 means, “The opposite of 5.” Since 5 means 5 ones, - 5 means the opposite of five ones, which is five opposite of ones, which is five negative ones, which is negative five.

Objective: To show that a Negative Integer times a Negative Integer equals a Positive Integer.

Objective: To show commutative property of multiplication holds for a Positive Integer times a Negative Integer.

Examples:

Use the Definition of *, the Distribution of Negative Sign, the Definition of Multiplication, and the Associative Property of Addition to complete the following:

13. Show: - 2 * ( 3 ) = 3 * ( - 2 )

- 2 * 3 = - ( 3 + 3 ) by Definition of *

= ( - 3 ) + ( - 3 ) by Distribution of Negative Sign

= [ ( - 1 ) + ( - 1 ) + ( - 1 ) ] + [ ( - 1 ) + ( - 1 ) + ( - 1 ) + ( - 1 ) ] by Definition of a Number

= [ ( - 1 ) + ( - 1 ) ] + [ ( - 1 ) + ( - 1 ) ] + [ ( - 1 ) + ( - 1 ) ] by Associative Property of Addition

= [ - 2 ] + [ - 2 ] + [ - 2 ] by Associative Property of Addition

= 3 * [ - 2 ] by Definition of Multiplication

Hence, - 2 * ( 3 ) = 3 * ( - 2 ).

Exercises:

Use the Definition of *, the Distribution of Negative Sign, the Definition of Multiplication, and the Associative Property of Addition to complete the following:

2. Show: - 2 * ( 2 ) = 2 *( - 2 )

- 2 * 2 = by Definition of *

= by Distribution of Negative Sign

= by Definition of a Number

Hence, - 2 * ( 2 ) = 2 *( - 2 )

14. Show: - 1 * ( 3 ) = 1 * ( - 3 )

- 1 * 3 = - ( 3 ) by Definition of _____________

= ( - 3 ) by Distribution _____________

= 1 ( - 3 ) by Definition of _____________

Hence, - 1 * ( 3 ) = 1 * ( - 3 ).

15. Show: - 3 * ( 2 ) = 3 * ( - 2 )

- 3 * 2 = - ( 2 + 2 + 2 ) by Definition of *

= by Distribution of Negative Sign

= 3 * ( - 2 ) by Definition of Multiplication

Hence, - 2 * ( 3 ) = 3 * ( - 2 ).

16. Show: - 3 * ( 2 ) = 2 * ( - 3 ) Note: This is showing the Commutative Property of Multiplication for a Positive and Negative Integer.

- 3 * 2 = - ( 2 + 2 + 2 ) by Definition of *

= by Distribution of Negative Sign

= by Definition of a Number

= by Associative Property of Addition

= 2 * ( - 3 ) by Definition of Multiplication

Hence, - 2 * ( 3 ) = 3 * ( - 2 ).

17. Show: - 3 * ( 4 ) = 4 * ( - 3 ) Note: This is showing the Commutative Property of Multiplication for a Positive and Negative Integer.

- 3 * 4 =

Hence, - 3 * ( 4 ) = 4 * ( - 3 ).

Selected Solutions:

1. Show: - 2 * ( 2 ) = 2 * ( - 2 )

- 2 * 2 = - ( 2 + 2 ) by Definition of *

= ( - 2 ) + ( - 2 ) by Distribution of Negative Sign

= 2 * [ - 2 ] by Definition of Multiplication

Hence, - 2 * ( 2 ) = 2 * ( - 2 ).

2. Show: - 1 * ( 3 ) = 3 * ( - 1 )

- 1 * 3 = - ( 3 ) by Definition of _____*________

= ( - 3 ) by Distribution _of Negative Sign

= 1 ( - 3 ) by Definition of Multiplication__

Hence, - 1 * ( 3 ) = 1 * ( - 3 ).

3. Show: - 3 * ( 2 ) = 2 * ( - 3 )

- 3 * 2 = - ( 2 + 2 + 2 ) by Definition of *

= ( - 2 ) + ( - 2 ) + ( - 2 ) by Distribution of Negative Sign

= 3 * ( - 2 ) by Definition of Multiplication

Hence, - 2 * ( 3 ) = 3 * ( - 2 ).

4. Show: - 3 * ( 2 ) = 2 * ( - 3 ) Note: This is showing the Commutative Property of Multiplication for a Positive and Negative Integer.

- 3 * 2 = - ( 2 + 2 + 2 ) by Definition of *

= ( - 2 ) + ( - 2 ) + ( - 2 ) by Distribution of Negative Sign

= ( - 2 ) + [ ( - 1 ) + ( - 1 ) ] + ( - 2 ) by Definition of a Number

= [ ( - 2 ) + ( - 1 ) ] + [ ( - 2 ) + ( - 1 ) ] by Associative Property of Addition

= [ - 3 ] + [ - 3 ] by Associative Property of Addition

= 2 * ( - 3 ) by Definition of Multiplication

Hence, - 2 * ( 3 ) = 2 * ( - 3 ).

5. Show: - 3 * ( 4 ) = 4 * ( - 3 ) Note: This is showing the Commutative Property of Multiplication for a Positive and Negative Integer.

- 3 * 4 = - ( 4 + 4 + 4 ) by Definition of *

= ( - 4 ) + ( - 4 ) + ( - 4 ) by Distribution of Negative Sign

= [ ( - 3 ) + ( -1 ) ] + [ ( - 3 ) + ( - 1 ) ] + [ ( - 3 ) + ( - 1 ) ] by Definition of a Number

= ( - 3 ) + ( - 3 ) + ( - 3 ) + [ ( - 1 ) + ( -1 ) + ( -1 ) ] by Associative Property of Addition

= ( - 3 ) + ( - 3 ) + ( - 3 ) + ( - 3 ) by Associative Property of Addition

= 4 * ( - 3 ) by Definition of Multiplication

Hence, - 3 * ( 4 ) = 4 * ( - 3 ).

Lecture 10
Inverse Operation of Addition

Definitions:

– is called the subtraction symbol.

– ≡ Inverse of addition or the opposite of adding (see lecture 1 for Addition)

Subtraction is the inverse of Addition. Subtraction “undoes” Addition.

When subtracting, do the opposite of what would be done when adding.

For instance:

0 + 1 = 1 , hence 0 - 1 = -1.

0 + ( -1 ) = -1, hence 0 - ( -1 ) = 1.

1 + 1 = 2, hence 1 - 1 = 0.

1 + ( -1 ) = 0 , hence 1 - ( -1 ) = 2.

2 + 1 = 3, hence 2 - 1 = 1.

2 + ( -1 ) = , hence 2 - ( -1 ) = 3.

Objective: To understand that subtracting a Negative is the same as adding Positive.

Examples:

18. Show: - 2 - ( -1 ) = - 1

-2 - ( - 1 ) = -2 + 1 by Definition of Inverse Operation

= - 1 + ( -1 ) + 1 by Definition of Number

= -1 + 0 by Definition of Zero

= -1 by Additive Identity

Hence, - 2 - ( -1 ) = - 1.

19. Show: - 5 - ( -1 ) = - 4

-5 - ( - 1 ) = -5 + 1 by Definition of Inverse Operation

= - 1 + ( -1 ) + ( -1 ) + ( -1 ) + ( -1 ) + 1 by Definition of Number

= - 1 + ( -1 ) + ( -1 ) + ( -1 ) + 0 by Definition of Zero

= - 1 + ( -1 ) + ( -1 ) + ( -1 ) by Additive Identity

= -4 by Definition of Number

Hence, - 5 - ( -1 ) = - 4.

20. Show: - 3 - ( -2 ) = - 1

-5 - ( - 2 ) = -3 + 2 by Definition of Inverse Operation

= - 1 + ( -1 ) + ( -1 ) + 1 + 1 by Definition of Number

= - 1 + ( -1 ) + 0 + 1 by Definition of Zero

= - 1 + ( -1 ) + 1 by Additive Identity

= - 1 + 0 by Definition of Zero

= - 1 by Additive Identity

Hence, - 3 - ( -2 ) = 1.

Exercises:

Use the Definition of Inverse Operation, the Distribution of Number, and the Additive Identity, to complete the following:

1. Show: - 4 - ( -1 ) = - 3

-4 - ( - 1 ) = -4 + 1 by Definition of Inverse Operation

= by Definition of Number

= - 1 + ( -1 ) + ( -1 ) + 0 by Definition of Zero

= by Additive Identity

= by Definition of Number

Hence, - 4 - ( -1 ) = - 3.

2. Show: - 3 - ( -1 ) = - 2

-3 - ( - 1 ) = -3 + 1 ________________________

= - 1 + ( -1 ) + ( -1 ) + 1 ________________________

= - 1 + ( -1 ) + ( -1 ) + 0 ________________________

= - 1 + ( -1 ) + ( -1 ) ________________________

= -3 ________________________

Hence, - 3 - ( -1 ) = - 2.

3. Show: - 5 - ( -2 ) = - 3

-5 - ( - 2 ) = -5 + 2 by Definition of Inverse _______

= - 1 + ( -1 ) + ( -1 ) + ( -1 ) + ( -1 ) + 1 + 1 by Definition of _____________

= - 1 + ( -1 ) + ( -1 ) + ( -1 ) + 0 + 1 by Definition of _____________

= - 1 + ( -1 ) + ( -1 ) + ( -1 ) + 1 by Additive _________________

= - 1 + ( -1 ) + ( -1 ) + 0 by Definition of Zero

= - 1 + ( -1 ) + ( -1 ) by _____________________

= -3 _______________________

Hence, - 5 - ( -2 ) = - 3.

21. Show: 3 - ( -2 ) = 5

3 - ( - 2 ) = 3 + 2 by Definition of Inverse Operation

= 1 + 1 + 1 + 1 + 1 by Definition of Number

= 5 by Definition of Number

Hence, 3 - ( -2 ) = 5.

22. Show: 2 - ( -2 ) = 4

2 - ( - 2 ) = by Definition of Inverse Operation

= by Definition of Number

Hence, 2 - ( -2 ) = 4.

Solutions:

1. Show: - 4 - ( -1 ) = - 3

-4 - ( - 1 ) = -4 + 1 by Definition of Inverse Operation

= - 1 + ( -1 ) + ( -1 ) + ( -1 ) + 1 by Definition of Number

= - 1 + ( -1 ) + ( -1 ) + 0 by Definition of Zero

= - 1 + ( -1 ) + ( -1 ) by Additive Identity

= -3 by Definition of Number

Hence, - 4 - ( -1 ) = - 3.

2. Show: - 3 - ( -1 ) = - 2

-3 - ( - 1 ) = -3 + 1 by Definition of Inverse Operation

= - 1 + ( -1 ) + ( -1 ) + 1 by Definition of Number

= - 1 + ( -1 ) + ( -1 ) + 0 by Definition of Zero

= - 1 + ( -1 ) + ( -1 ) by Additive Identity

= -3 by Definition of Number

Hence, - 3 - ( -1 ) = - 2

3. Show: - 5 - ( -2 ) = - 3

-5 - ( - 1 ) = -5 + 2 by Definition of Inverse Operation

= - 1 + ( -1 ) + ( -1 ) + ( -1 ) + ( -1 ) + 1 + 1 by Definition of Number

= - 1 + ( -1 ) + ( -1 ) + ( -1 ) + 0 + 1 by Definition of Zero

= - 1 + ( -1 ) + ( -1 ) + ( -1 ) + 1 by Additive Identity

= - 1 + ( -1 ) + ( -1 ) + 0 by Definition of Zero

= - 1 + ( -1 ) + ( -1 ) by Additive Identity

= -3 by Definition of Number

Hence, - 5 - ( -2 ) = - 3.

4. Show: 3 - ( -2 ) = 5

3 - ( - 2 ) = 3 + 2 by Definition of Inverse Operation

= 1 + 1 + 1 + 1 + 1 by Definition of Number

= 5 by Definition of Number

Hence, 3 - ( -2 ) = 5.

5. Show: 2 - ( -2 ) = 4

2 - ( - 2 ) = 2 + 2 by Definition of Inverse Operation

= 1 + 1 + 1 + 1 by Definition of Number

= 4 by Definition of Number

Hence, 2 - ( -2 ) = 4.

Lecture 11
Sets: Rational Numbers

Definitions:

Object: Person, Place or Thing

Set: A collection of objects.

Elements: The objects contained within a set.

{ } are called braces.

, is called a comma.

is called Epsilon (a Greek letter)

“An element of”

A set is often displayed by writing a list of its elements separated by commas between braces.

For instance:

{J, A, M, E, S} is a set containing the letters of my name. Elements of sets are usually listed in alphabetical order.

Hence,

{J, A, M, E, S} = {A, E, J, M, S}

Note: A {J, A, M, E, S}

For instance:

{ 2, 0, 6, 3, 2, 5, 3, 1, 1, 9} is a set containing the digits of a phone number. Like Elements of a set are usually listed only once. Numbers are frequently listed in ascending order.

Hence,

{ 2, 0, 6, 3, 2, 5, 3, 1, 1, 9 } = { 0, 1, 2, 3, 5, 6, 9 }

Note: 6 { 0, 1, 2, 3, 5, 6, 9 }

Definitions:

Natural Numbers: The set of numbers obtained by starting at 1 and summing 1 to the number.
{ 1, 2, 3, 4, … }

Whole Numbers: The set of numbers obtained by starting at 0 and summing 1 to the number.
{ 0, 1, 2, 3, 4, … }

Integer Numbers: The set of numbers obtained by starting at -1 and summing -1 to the number and the set of Natural Numbers.
{ …, -4, -3, -2, -1, 0, 1, 2, 3, 4, … }

Objectives:

· To recognize and correctly identify the elements of the sets of Whole Numbers, Counting Numbers, and Integer Numbers.

· To recognize and properly use the set notation and the “element” symbol..

Examples:

23. Indicate the truth value of the statement.

a. -2 is a Natural Number. FALSE

b. - 2 is a Whole Number. FALSE

c. - 2 is an Integer Number. TRUE

24. Indicate the truth value of the statement.

a. 9 is a Natural Number. TRUE

b. 9 is a Whole Number. TRUE

c. 9 is an Integer Number. TRUE

25. Indicate the truth value of the statement.

a. -1, 001 is a Natural Number. FALSE

b. -1, 001 is a Whole Number. FALSE

c. -1, 001 is an Integer Number. TRUE

26. Indicate the truth value of the statement.

a. a { b, c, d } FALSE

b. a { 1, 2, 5 } FALSE

c. a { a, b, c, d } TRUE

27. Indicate the truth value of the statement.

a. cat { cat, dog, horse } TRUE

b. -3 { 1, 2, 5 } FALSE

c. -3 { -3, -2, -1, 0, 1, 2, 3 } TRUE

Exercises:

7. Indicate the truth value of the statement.

a. -7 is a Natural Number.

b. - 50 is a Whole Number.

c. - 225 is an Integer Number.

8. Indicate the truth value of the statement.

a. 7 is a Natural Number.

b. 7 is a Whole Number.

c. 7 is an Integer Number.

9. Indicate the truth value of the statement.

a. -10, 500 is a Natural Number.

b. -10, 500 is a Whole Number.

c. -10, 500 is an Integer Number.

10. Indicate the truth value of the statement.

a. a { c, d, e }

b. a { 11, 12, 15 }

c. a { a, b, c, d, e }

11. Indicate the truth value of the statement.

a. dog { cat, dog, horse }

b. -3 { 1, 2, 15 }

c. -3 { -3, -2, -1, 2, 3 }

12. Indicate the truth value of the statement.

a. x { a, x, u, z }

b. 3 { 1, 2, 15 }

c. 3 { -3, -2, -1, 2, 3 }

Solutions:

7. Indicate the truth value of the statement.

a. -7 is a Natural Number. FALSE

b. - 50 is a Whole Number. FALSE

c. - 225 is an Integer Number. TRUE

8. Indicate the truth value of the statement.

a. 7 is a Natural Number. TRUE

b. 7 is a Whole Number. TRUE

c. 7 is an Integer Number. TRUE

9. Indicate the truth value of the statement.

a. -10, 500 is a Natural Number. FALSE

b. -10, 500 is a Whole Number. FALSE

c. -10, 500 is an Integer Number. TRUE

10. Indicate the truth value of the statement.

a. a { c, d, e } FALSE

b. a { 11, 12, 15 } FALSE

c. a { a, b, c, d, e } TRUE

11. Indicate the truth value of the statement.

a. dog { cat, dog, horse } TRUE

b. -3 { 1, 2, 15 } FALSE

c. -3 { -3, -2, -1, 2, 3 } TRUE

12. Indicate the truth value of the statement.

a. x { a, x, u, z } TRUE

b. 3 { 1, 2, 15 } FALSE

c. 3 { -3, -2, -1, 2, 3 } TRUE