Lecture 10
Inverse Operation

Definitions:

– is called the subtraction symbol.

– ≡ Inverse of addition or the opposite of adding (see lecture 1 for Addition)

Subtraction is the inverse of Addition. Subtraction “undoes” Addition.

When subtracting, do the opposite of what would be done when adding.

For instance:

0 + 1 = 1 , hence 0 - 1 = -1.

0 + ( -1 ) = -1, hence 0 - ( -1 ) = 1.

1 + 1 = 2, hence 1 - 1 = 0.

1 + ( -1 ) = 0 , hence 1 - ( -1 ) = 2.

2 + 1 = 3, hence 2 - 1 = 1.

2 + ( -1 ) = , hence 2 - ( -1 ) = 3.

Objective: To understand that subtracting a Negative is the same as adding Positive.

Examples:

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

-2 - ( - 1 ) = -2 + 1 by Definition of Inverse Operation

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

= -1 + 0 by Definition of Zero

= -1 by Additive Identity

Hence, - 2 - ( -1 ) = - 1.

2. Show: - 5 - ( -1 ) = - 4

-5 - ( - 1 ) = -5 + 1 by Definition of Inverse Operation

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

= - 1 + ( -1 ) + ( -1 ) + ( -1 ) + 0 by Definition of Zero

= - 1 + ( -1 ) + ( -1 ) + ( -1 ) by Additive Identity

= -4 by Definition of Number

Hence, - 5 - ( -1 ) = - 4.

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

-5 - ( - 2 ) = -3 + 2 by Definition of Inverse Operation

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

= - 1 + ( -1 ) + 0 + 1 by Definition of Zero

= - 1 + ( -1 ) + 1 by Additive Identity

= - 1 + 0 by Definition of Zero

= - 1 by Additive Identity

Hence, - 3 - ( -2 ) = 1.

Exercises:

Use the Definition of Inverse Operation, the Distribution of Number, and the Additive Identity, to complete the following:

1. Show: - 4 - ( -1 ) = - 3

-4 - ( - 1 ) = -4 + 1 by Definition of Inverse Operation

= by Definition of Number

= - 1 + ( -1 ) + ( -1 ) + 0 by Definition of Zero

= by Additive Identity

= by Definition of Number

Hence, - 4 - ( -1 ) = - 3.

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

-3 - ( - 1 ) = -3 + 1 ________________________

= - 1 + ( -1 ) + ( -1 ) + 1 ________________________

= - 1 + ( -1 ) + ( -1 ) + 0 ________________________

= - 1 + ( -1 ) + ( -1 ) ________________________

= -3 ________________________

Hence, - 3 - ( -1 ) = - 2.

3. Show: - 5 - ( -2 ) = - 3

-5 - ( - 2 ) = -5 + 2 by Definition of Inverse _______

= - 1 + ( -1 ) + ( -1 ) + ( -1 ) + ( -1 ) + 1 + 1 by Definition of _____________

= - 1 + ( -1 ) + ( -1 ) + ( -1 ) + 0 + 1 by Definition of _____________

= - 1 + ( -1 ) + ( -1 ) + ( -1 ) + 1 by Additive _________________

= - 1 + ( -1 ) + ( -1 ) + 0 by Definition of Zero

= - 1 + ( -1 ) + ( -1 ) by _____________________

= -3 _______________________

Hence, - 5 - ( -2 ) = - 3.

4. Show: 3 - ( -2 ) = 5

3 - ( - 2 ) = 3 + 2 by Definition of Inverse Operation

= 1 + 1 + 1 + 1 + 1 by Definition of Number

= 5 by Definition of Number

Hence, 3 - ( -2 ) = 5.

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

2 - ( - 2 ) = by Definition of Inverse Operation

= by Definition of Number

= by Definition of Number

Hence, 2 - ( -2 ) = 4.

Solutions:

1. Show: - 4 - ( -1 ) = - 3

-4 - ( - 1 ) = -4 + 1 by Definition of Inverse Operation

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

= - 1 + ( -1 ) + ( -1 ) + 0 by Definition of Zero

= - 1 + ( -1 ) + ( -1 ) by Additive Identity

= -3 by Definition of Number

Hence, - 4 - ( -1 ) = - 3.

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

-3 - ( - 1 ) = -3 + 1 by Definition of Inverse Operation

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

= - 1 + ( -1 ) + ( -1 ) + 0 by Definition of Zero

= - 1 + ( -1 ) + ( -1 ) by Additive Identity

= -3 by Definition of Number

Hence, - 3 - ( -1 ) = - 2

3. Show: - 5 - ( -2 ) = - 3

-5 - ( - 1 ) = -5 + 2 by Definition of Inverse Operation

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

= - 1 + ( -1 ) + ( -1 ) + ( -1 ) + 0 + 1 by Definition of Zero

= - 1 + ( -1 ) + ( -1 ) + ( -1 ) + 1 by Additive Identity

= - 1 + ( -1 ) + ( -1 ) + 0 by Definition of Zero

= - 1 + ( -1 ) + ( -1 ) by Additive Identity

= -3 by Definition of Number

Hence, - 5 - ( -2 ) = - 3.

4. Show: 3 - ( -2 ) = 5

3 - ( - 2 ) = 3 + 2 by Definition of Inverse Operation

= 1 + 1 + 1 + 1 + 1 by Definition of Number

= 5 by Definition of Number

Hence, 3 - ( -2 ) = 5.

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

2 - ( - 2 ) = 2 + 2 by Definition of Inverse Operation

= 1 + 1 + 1 + 1 by Definition of Number

= 4 by Definition of Number

Hence, 2 - ( -2 ) = 4.