Lecture 9a

Definitions:

– is called the negative symbol.

– ≡ Opposite of positive (see lecture 5)

Negative: - 5 is the opposite of 5.

Note: “By Definition of Negative”, - ( - 5 ) = 5, since the opposite of negative five is positive five.

For instance:

- ( - 1 ) = ( 1 )

- ( - 4 ) = ( 4 )

Hence – ( – 5 ) means, “The opposite of the opposite of 5”, which is 5.

Objective: To show that a Negative Integer times Negative Integer equals the product of its opposites or Positive Integer.

Examples:

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

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

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

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

= 3 + 3 by Definition of a Negative

= 2 * [ 3 ] by Definition of Multiplication

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

Exercises:

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

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

- 2 * ( - 2 ) = - [ ( - 2 ) + ( - 2 ) ] by Definition of __________________

= - ( - 2 ) + ( - ( - 2 ) ) by Distribution of _________________

= 2 + 2 by Definition of a Negative

= 2 * ( 2 ) by _____________________________

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

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

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

= 3 by Distribution of ___________________

= 1 ( - 3 ) by Definition of ___________________

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

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

- 3 * ( - 2 ) = - ( ( - 2 ) + ( - 2 ) + ( - 2 ) ) _________________________

= - ( - 2 ) + ( - ( - 2 ) ) + ( - ( - 2 ) ) _________________________

= 2 + 2 + 2 _________________________

= 3 * ( 2 ) _________________________

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

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

- 2 * ( - 4 ) =

=

=

=

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

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 a Negative

= 2 * ( 2 ) by Definition of Multiplication

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

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

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

= 3 by Distribution of a Negative

= 1 ( 3 ) by Definition of Multiplication

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

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

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

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

= 2 + 2 + 2 by Definition of Negative Sign

= 3 * ( 2 ) by Definition of Multiplication

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