Definitions:

* is called the times symbol.

* ≡ Times

Times: The value to the left of * indicates the number of addends to sum of the value to the right of *.

For instance:

1 * 1 = ( 1 )

1 * 2 = ( 2 )

1 * 3 = ( 3 )

2 * 1 = ( 1 + 1 )

2 * 2 = ( 2 + 2 )

2 * 3 = ( 3 + 3)

3 * 1 = ( 1 + 1 + 1 ) In English, “Three times one equals the sum of three addends of one.”

3 * 2 = ( 2 + 2 + 2 ) In English, “Three times two equals the sum of three addends of two.”

3 * 3 = ( 3 + 3 + 3 ) In English, “Three times three equals the sum of three addends of three.”

Definitions:

* is also called the multiplication symbol.

* ≡ Multiplication

Multiplication: The value to the right of * indicates how many addends of the value to the left to sum .

For instance:

1 * 1 = ( 1 )

1 * 2 = ( 1 + 1 )

1 * 3 = ( 1 + 1 + 1 )

2 * 1 = ( 2 )

2 * 2 = ( 2 + 2 )

2 * 3 = ( 2 + 2 + 2)

3 * 1 = ( 3 ) In English, “Three multiplied by one equals three.”

3 * 2 = ( 3 + 3 ) In English, “Three multiplied by two equals the sum of two addends of three.”

3 * 3 = ( 3 + 3 + 3 ) In English, “Three multiplied by three equals the sum of three addends of three.”

3 * 4 = ( 3 + 3 + 3 + 3 ) In English, “Three multiplied by four equals the sum of four addends of three.”

Objective: To show that both Times and Multiplication are Commutative.

Examples:

Use the definition of *, the definition of numbers, and the Associative Property of Addition to complete the following:

1. Show: 2 * 3 = 3 * 2

2 * 3 = 3 + 3 by Definition of *

= ( 1 + 1 + 1 ) + ( 1 + 1 + 1 ) by Definition of Number 3

= ( 1 + 1 + 1 + 1 +1 + 1 ) by Associative Property of Addition

= ( 1 + 1 ) + ( 1 + 1 ) + ( 1 + 1 ) by Associative Property of Addition

= 2 + 2 + 2 by Definition of Number 2

= 3 * 2 by Definition of *

Hence, 2 * 3 = 3 * 2

2. Show: 2 *( 1 + 1 ) = 4

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

= ( 1 + 1 + 1 + 1 ) by Associative Property of Addition

= 4 by Definition of *

Hence, 2 *( 1 + 1 ) = 4.

Exercises:

Use the definition of *, the definition of numbers, and the Associative Property of Addition to complete the following:

1. Show: 2 * 4 = 4 * 2

2 * 4 =

Hence,

2. Show: 3 *( 1 + 1 ) = 6

3 *( 1 + 1 ) = by

= by

Hence,

3. Show: 3 *( ( - 1 ) + ( - 1 ) ) = - 6

3 *( ( - 1 ) + ( - 1 ) ) = by

= by

Hence,

4. Show: 3 *( - 3 ) = - 9

3 *( - 3 ) = by

= by

Hence,

5. Show: 6 *( - 1 ) = - 6

6 *( - 1 ) = by

= by

Hence,

6. Show: 4 *( -2 ) = - 8

4 *( -2 ) = by

= by

Hence,

Selected Solutions:

1. Show: 2 * 4 = 4 * 2

2 * 4 = 4 + 4 by Definition of *

= ( 1 + 1 + 1 + 1 ) + ( 1 + 1 + 1 + 1 ) by Definition of Number 4

= ( 1 + 1 + 1 + 1 +1 + 1 + 1 + 1 ) by Associative Property of Addition

= ( 1 + 1 ) + ( 1 + 1 ) + ( 1 + 1 ) + ( 1 + 1 ) by Associative Property of Addition

= 2 + 2 + 2 + 2 by Definition of Number 2

= 4 * 2 by Definition of *

2. Show: 3 *( 1 + 1 ) = 6

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

= ( 1 + 1 + 1 + 1 + 1 + 1 ) by Associative Property of Addition

= 6 by Definition of Number 6.

Hence, 3 *( 1 + 1 ) = 6.