Operations on Real Numbers

Double Negative Signs

A negative of a negative is a positive, for example

    -(-7)  =  7 

and in general 

-(-x) = x

 

Addition Involving Positive and Negative Numbers

To add two negative numbers add then without the negatives and placing a negative in front of the answer.  To subtract two numbers when one is positive and the other is negative, subtract the smaller from the larger and if the larger is negative, the result is also negative.  To subtract two negative numbers, just recall that minus a negative is the same as a plus.

Some Useful Rules of Addition and Subtraction

  1. -a + (-b)  =  -(a + b)

  2. a - b  =  -(b - a)

  3. -a - (-b)  =  -a + b


Algebraic Examples

  1. -3 + 2 :     3 - 2 = 1         since 3 > 2 the answer is -1.

  2.  -4 + 10:     10 - 4 = 6    since 10 > 4 the answer is 6.

  3. -3 - 2:  same sign so that 3 + 2 = 5, since they are both negative the answer is -5.

  4. -1 - 10  =  -11.

  5. 4 - 6 :  6 - 4  =  2 since 6 > 4 the answer is -2.

 Multiplication and Division

Multiplication and Division Rules:  

  1. An even number of negatives gives a positive.

  2. An odd number of negatives gives a negative.

  3. We can never divide by 0.

  4. 0/(non zero) = 0.

  5. (-a)/b = a/(-b) = -(a/b) (The negative sign can be placed on the numerator, the denominator or to the left.)


Examples

        (-3)(-6)  =  18,         (-4)(8)  =  -32

                -12                    0                     7
                           =  3                 =  0               =  undefined                                              
                 -4                    10                     0

Exercises:  Simplify  (Hold the mouse on the yellow rectangle for the solution).

  1.      -3 + 4
                                            -1/2
            -2

  2.      5 - (-3)  
                      + -4                0
            2

  3. |3 - 4| - 4                           -3

  4. -2(|-4| - |-3|)                       -2

  5.       -1            1
                 +                         -1/6
           2            3  

  6.     2/5 - 1/3
                                            1/2
        1/3 - 1/5

  7.      6x
                                            -2x
          -3

  8. (-5)(5z)                             -25z

  9. -4(-3x + 2)                        12x - 8

  10. -(x - y)                              -x + y

Applications

Example:

Suppose that you own a casino and want to monitor the average winnings of your patrons.  The four players at your table had the following results:  player 1 won $4, player 2 lost $2, player 3 lost $7 and player 4 broke even. What was the average earnings?

Solution:

To find the average, we add up all the numbers and divide by 4:

               4 + (-2) + (-7) + 0            2 + (-7)            -5
                                                  =                   =            or   $-1.25
                            4                              4                4

The average winnings of your patrons was $-1.25, hence they lost an average of $1.25.


Exercise  

Suppose your bank balance is $250 and your Visa balance is $750.  What is your net worth?

        -500

 

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