Adding Integers
Objective Learn how to add positive and
negative integers.
You will learn how to add, subtract, multiply, and divide
integers.
Rules for addition of negative numbers
- Adding a negative number to a positive number gives the
same result as subtracting the corresponding positive
number.
- When we subtract a larger number from a smaller one, the
result will be a negative number.
5 - 7 = -2
- Therefore, when we have an addition problem involving one
positive number and one negative number, we may end up
with a negative answer.
5 + (-7) |
= 5 - 7 |
|
14 + (-21) |
= 14 - 21 |
|
= -2 |
|
|
= -7 |
- If we add two negative numbers, then the result is the
opposite of what we would get if we added the
corresponding positive numbers:
You should try to figure out why these rules are true. The
following may help you:
Why are these rules for addition true?
You can represent positive numbers by collections of + 1
blocks and negative numbers by collections of - 1 blocks.
![](./articles_imgs/12/pic1.GIF)
These collections of blocks represent - 6 and 8, respectively.
We can also combine collections of positive (+1) and negative (-1)
blocks. Think about this as adding a positive and a negative
number.
![](./articles_imgs/12/pic2.GIF)
This collection represents 8 + ( - 6).
Remember that we are thinking of negative numbers as a
deficit, so the collections we just drew should be thought of as
a collection of 8 (dollars, say) together with a deficit of 6
dollars. Of course, the 8 dollars could be used to pay off the
deficit of 6 dollars, and there would be 2 dollars left over.
This can be pictured using blocks if we allow ourselves to cancel
a negative (-1) block and a positive (+1) block.
![](./articles_imgs/12/pic3.GIF)
In the picture, we cancelled each negative (-1) block with a
positive (+1) block and removed both blocks from the picture.
Whenever we have both negative (-1) and positive (+1) blocks, we
should cancel unlike blocks in pairs until we have a collection
that consists only of blocks of a single color (kind).
Now consider 6 + ( - 8).
![](./articles_imgs/12/pic4.GIF)
When we cancel pairs of positive (+1) and negative (-1)
blocks, we are left with two negative (-1) blocks.
![](./articles_imgs/12/pic5.GIF)
This means that we have a deficit of 2, so 6 + ( -8) = -2.
Example 4
Compute ( - 4) + ( - 6).
Solution
Draw -4 as a collection of four negative (-1) blocks and -6 as
a collection of 6 negative (-1) blocks. When we merge them to
add, we will get 10 negative (-1) blocks.
![](./articles_imgs/12/pic6.GIF)
Therefore, ( - 4) + ( - 6) = -10.
Here is a very important special case of these rules of
addition.
Adding Opposites When we add a positive
number and its opposite negative number, we always get zero.
Example 5
Demonstrate that 5 + ( - 5) = 0.
Solution
Represent 5 + ( - 5) with 5 positive (+1) blocks and 5
negative (-1) blocks. The opposite blocks (-1 and +1) cancel in
pairs. We are left with no blocks at all, so the result is 0.
![](./articles_imgs/12/pic7.GIF)
Negative Numbers in Algebra
When working with variables, one can use the Distributive
Property together with the rules about adding and subtracting
integers to simplify expressions and equations.
Example 6
Simplify the expression 4x + ( - 3)x.
Solution
Use the distributive property.
4x + ( - 3)x |
= [4 + ( -3)] x |
|
= [1] x |
|
= x |
As with actual numbers, one can think of each x and - x as
units that cancel out, so -3x cancels out 3x .
![](./articles_imgs/12/pic8.GIF)
|