Integers: Operations with Signed Numbers

Before you do ANY computation, determine the OPERATION!
Then follow the instructions for THAT operation.


Addition
Subtraction
Multiplication or Division
Practice Exercises

 
ADDITION

Do the numbers have the SAME SIGN?
YES - Same Signs:
Find the SUM:
  NO - Different signs:
Find the DIFFERENCE:
(-3) + (-6) = (-9)

(+4) + (+5) = (+9)
  (+5) + (-7) = (-2)

(-4) + (+6) = (+2)

Either way:   Keep the sign of the LARGER* number.

* "LARGER" is used here as a quick (but mathematically imprecise) way to describe the integer with the greater Absolute Value (ie. distance from zero). In each of the examples above, the SECOND integer has a greater Absolute Value.


 
SUBTRACTION

First, change the SUBTRACTION problem to an ADDITION problem;

Then, follow the rules (above) for solving the new ADDITION problem.

(-6) - (+2) =

First, copy the problem exactly.

1. The first number stays the same.

2. Change the operation.

3. Switch the NEXT SIGN.

4. Follow the rules for addition.

 

(-6) - (+2) =

(-6)

(-6) +

(-6) + (-2)

(-6) + (-2) = (-8)

Subtract means:      
Add the opposite.

(+2) - (-6) =
(+2) + (+6) = (+8)

Subtract means:      
Add the opposite.

(-7) - (-3) =
(-7) + (+3) = (-4)
Subtract means:      
Add the opposite.

(+4) - (+9) =
(+4) + (-9) = (-5)

 
MULTICATION OR DIVISION

First, DO the multiplication or division.

Then determine the sign:

Count the number of negative signs....

Are there an EVEN number of negative signs?
YES (an EVEN number of negative signs) the answer is POSITIVE
NO (an ODD number of negative signs) the answer is NEGATIVE

First, copy the problem exactly.

  (-2) * (-4) * (-6) =

DO the multiplication or division.

  |2| * |4| * |6| = |48|

Count the number of negative signs....
Determine the sign of the answer:

 

(-2) * (-4) * (-6) =

Are there an EVEN number of negatives?
If YES, the answer is POSITIVE
otherwise, the answer is negative.

  A total of THREE NEGATIVES
Three is NOT EVEN   (it's odd).
So the answer is NEGATIVE

-48


(4) ÷ (2) * (6) = 12
A total of ZERO NEGATIVES
Zero IS EVEN .
So the answer is POSITIVE


 
(4) ÷ (-2) * (6) = -12
A total of ONE NEGATIVE
One is NOT EVEN (it's odd).
So the answer is NEGATIVE


 
(-4) ÷ (2) * (-6) = 12
A total of TWO NEGATIVES
TWO IS EVEN .
So the answer is POSITIVE



Another way of thinking of it:

The Party in Mathland

Have you ever been to a party like this?

Everyone is happy and having a good time (they are ALL POSITIVE). Suddenly, who should appear but the GROUCH (ONE NEGATIVE)! The grouch goes around complaining to everyone about the food, the music, the room temperature, the other people....

What happens to the party? Everyone feels a lot less happy... the party may be doomed!!

ONE NEGATIVE MAKES EVERYTHING NEGATIVE

But wait... is that another guest arriving?

What if another grouch (A SECOND NEGATIVE) appears? The two negative grouches pair up and gripe and moan to each other about what a horrible party it is and how miserable they are!! But look!! They are starting to smile; they're beginning to have a good time, themselves!!

PAIRS OF NEGATIVES BECOME POSITIVE

Now that the two grouches are together the rest of the people (who were really positive all along) become happy once again. The party is saved!!


The moral of the story is that (at least in math, when multiplying or dividing) the number of positives don't matter, but watch out for those negatives!!

To determine whether the outcome will be positive or negative, count the number of negatives: If there are an even number of negatives -and you can put them in pairs- the answer will be positive, if not... it'll be negative:

Negatives in PAIRS are POSITIVE;
NOT in pairs, they're NEGATIVE.


Practice Exercises
(with instant feedback)



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