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 4So –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 5So –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, soStart at –2, down 4So –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, soStart at 2, up 3So 2 – (–3) = 5.
When subtracting negative numbers, you can think of the two minus signs canceling each other out to create a positive.