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The Rules Don't Apply

Do you remember all of those rules we told you about for addition? We told you about the commutative law, associative law, and addition identities. When you look at the big picture, none of those laws work for subtraction. You can't reorder, shuffle, or regroup subtraction problems in the same way as addition. You need to be very careful with subtraction.

20 - 8 - 6 = 6
8 - 6 - 20 = -18 (you cannot rearrange and get the same answer)

(20 - 8) - 6 = 6
20 - (8 - 6) = 18 (you cannot regroup and get the same answer)

Some Rules Do Apply

Some of the rules still work for subtraction. The order of operations that you use still works. Look for parentheses and work inside of those blocks first. Look at how these problems come out with very different answers if you ignore the parentheses.

(20 - 8) - (6 - 4) = 12 - 2 = 10
20 - 8 - 6 - 4 = 2

What About Mixing Operations?

Some of your homework problems will have addition and subtraction. Can you use any of the addition laws to solve those problems or make things easier? Yes. Let's look at an example...

5 + 6 - 8 - 2 + 9 -1 = ?
So how can we work this out? Can we move things around? Yes, but only the values with addition symbols.
5 + 6 - 8 - 2 + 9 -1 = 9
5 + 6 + 9 - 8 - 2 - 1 = 9
Same answer. We only moved the values that were being added. We left the values to be subtracted in the same order. The laws don't apply to subtraction.

Blowing Your Mind

We want to introduce a big idea in math right now. We have explained that you cannot use any of the addition identities or laws for subtraction problems. Take a moment and think about a subtraction problem and what it really is. For simple integers, when you subtract, you are actually adding negative numbers. So 3 - 2 is the same thing as 3 + (-2). That fact means that every subtraction problem is actually an addition problem in disguise.

Example:
3 - 2 = 1
3 + (-2) = 1

18 - 13 = 5
18 + (-13) = 5

20 - 6 - 5 - 2 = 7
20 + (-6) + (-5) + (-2) = 7

The cool idea is that once you have made an addition problem, you can shuffle things and group things as you did before. Since all of our operations are addition, all of the laws work again. Just as a note for you: we use the parentheses to make it easier to see the negative numbers when they are in a problem.

20 + (-6) + (-5) + (-2) = 7
(-2) + (-6) + 20 + (-5) = 7
(-8) + 15 = 7
15 + (-8) = 7
15 - 8 = 7 (we switched the negative value back into a subtraction problem)

This won't make all of your problems easier, but it is an important idea to remember. The laws, identities, and axioms in math can all be used as little tricks when you start solving more difficult math problems. Never forget that the rules in math are your friends. They will always help to give you a path that will lead to the correct answer.

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