Lecture 4

Definitions:

Addend ≡ A number being added to another number.

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

Examples:

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

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

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

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 ) + ( -4 ) = ( -4 ) + (-3 ) .

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

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

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 ) + ( -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:

3. 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 ))