Lecture 5

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:

1. 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

2. 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

3. 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 ).