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Adding and Subtracting Negative Numbers

When people start working with negative numbers a lot of confusion comes up because of the way numbers can be moved around. If you want to rearrange the numbers in an expression that has negative numbers, the negative sign has to move with its number. So,

-5 + 3

can become

3 + (-5)

but not

-3 + 5

In the 3 + (-5) we took the entire -5 and switched it with the 3. In the -3 + 5 case we just moved the numbers and left the negative sign where it was. That's incorrect.

So how about

5 - 2

The trick with this situation is to realize that the numbers involved are +5 and -2. If you wanted to reverse them the result would be

-2 + 5     not     2 - 5

This brings us to another point. If you have a situation like

3 + (-5)

You can rewrite it as

3 - 5

In other words

3 + (-5) = 3 - 5

Give that some thought. I see a lot of students get very confused when a teacher pulls a negative out of a pair of parentheses and turns what looked like addition into subtraction. The table below summarizes some correct and incorrect rearrangements.

Original Right Wrong
-10 + 7 7 + (-10)   or   7 - 10 -7 + 10
3 - 2 -2 + 3 2 - 3
-5 + (-3) -5 - 3

With those rules in mind we can talk about adding and subtracting negatives. When adding and subtracting negative numbers the rule is: If both numbers have the same sign then add the numbers together and the answer has the same sign as the numbers. If they have different signs, then subtract the two numbers and the answer has the sign of the bigger number. So, for example:

3 + 5 = 8 both numbers are positive so the result is positive
-3 - 5 = -8 both numbers are negative so to get the result we add the number parts, 8 and 5, and the result is negative because the numbers are negative
-3 + 5 = 2 The numbers involved here are 3 and 5. If you subtract them you get 2 and the answer is positive because the 5 is greater than 3 and it's positive.
3 · -5 = 15 one number is positive and the other is negative so the result is negative

Notice in the last two examples that it didn't matter if the negative number came first or second. As long as there's one of each, a positive number and a negative number, the result is negative.

Division works the same way:

-20 / -5 = 4 both numbers are negative so the result is positive
20 / 5 = 4 both numbers are positive so the result is positive
-20 / 5 = -4 one number is positive and the other is negative so the result is negative
20 / -5 = 4 one number is positive and the other is negative so the result is negative