Addition and Subtraction with Negative Numbers
The great secret to adding and subtracting negative numbers is to turn every problem into a series of ups and downs on the number line. When you know how to do this, you find that all these problems are quite simple.
Don’t worry about memorizing every little bit of this procedure. Instead, just follow along so you get a sense of how negative numbers fit onto the number line.
Starting with a negative number
When you’re adding and subtracting on the number line, starting with a negative number isn’t much different from starting with a positive number. For example, suppose you want to calculate –3 + 4. Using the up and down rules, you get the following:
Start at –3, up 4
So –3 + 4 = 1.
Similarly, suppose you want to calculate –2 – 5. Again, the up and down rules help you out. You’re subtracting, so move to the left:
Start at –2, down 5
So –2 – 5 = –7.
Adding a negative number
Suppose you want to calculate –2 + –4. You already know to start at –2, but where do you go from there? Here’s the up and down rule for adding a negative number:
Adding a negative number is the same as subtracting a positive number — go down on the number line.
By this rule, –2 + –4 is the same as –2 – 4, so
Start at –2, down 4
So –2 + (–4) = –6.
Note: The problem –2 + –4 can also be written as –2 + (–4). Some people prefer to use this convention so that two operation symbols (– and +) are not side by side. Don’t let it trip you up. The problem is the same.
If you rewrite a subtraction problem as an addition problem — for instance, rewriting 3 – 7 as 3 + (–7) — you can use the commutative and associative properties of addition. Just remember to keep the negative sign attached to the number when you rearrange: (–7) + 3.
Subtracting a negative number
The last rule you need to know is how to subtract a negative number. For example, suppose you want to calculate 2 – (–3). Here’s the up and down rule:
Subtracting a negative number is the same as adding a positive number — go up on the number line.
This rule tells you that 2 – (–3) is the same as 2 + 3, so
Start at 2, up 3
So 2 – (–3) = 5.
When subtracting negative numbers, you can think of the two minus signs canceling each other out to create a positive.