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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.
9 + (-4) = 9 - 4
  = 5
  • 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:
(-2) + (-3) = -2 -3
  = -5

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.

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.

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.

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).

When we cancel pairs of positive (+1) and negative (-1) blocks, we are left with two negative (-1) blocks.

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.

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.

 

 

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 .

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