Lecture 3
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 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 = ( ( -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 )
5. Indicate the values for each step: