Lecture 9

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:

1. 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:

1. 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 )

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

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

= ( - 3 ) by Distribution _____________

= 1 ( - 3 ) by Definition of _____________

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

3. 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 ).

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 *

= by Distribution of Negative Sign

= by Definition of a Number

= by Associative Property of Addition

= by Associative Property of Addition

= 2 * ( - 3 ) by Definition of Multiplication

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

5. 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 ).