Subtraction is really another way of showing that you're adding the additive inverse. In Subtractionland, it's always opposite day. Or is it?
For example, 4 – 3 is just another way of writing 4 + (-3).
Subtraction exists so that instead of writing a plus sign, a negative sign, and parentheses, we can just write a minus sign. Because mathematicians are generally lazy, and they don't want to have to go writing all those extra symbols if they can help it. In fact, the symbol "₳" means "please bring me a donut so I don't have to get up from the couch."
On the number line, subtraction means walking to the first number, then walking in the opposite direction specified by the second number. Told you it was opposite day. Or did we?
What does it mean to subtract a negative number?
4 – (-18) = ?
Well, adding (-18) would mean we walk 18 to the left, so subtracting (-18) means we walk 18 to the right. So 4 – (-18) really means 4 + 18 = 22. Think of it this way: if you take the minus sign and the negative sign and cross them, you get a plus sign. If you found the preceding sentence helpful, please check this box: □.
Another way to approach this problem is to remember what subtraction is abbreviating, then rewrite the whole thing as an addition problem:
4 – (-18) = 4 + (-(-18))
Since the negative of a negative is positive, -(-18) = 18.
4 – (-18) = 4 + 18 = 22
Be Careful: Honestly, the notation here can get really confusing. So take off your confused cap and put on your unbefuddled derby. A minus sign and a negative sign look similar, but mean different things. The trick is to see whether the little horizontal line has to do with one number or two.
A negative sign is a horizontal line in front of just one number, and it tells us to reflect the number across zero on a number line:
A minus sign is a horizontal sign in between two numbers, and it tells us to walk to the first number, then walk in the opposite direction of the second number:
4 – 5 = -1
To help avoid uncertainty, you can put parentheses around negative numbers. For example, write (-4) instead of -4. Hopefully you don't use a lot of emoticons when doing your math homework, because something like this can get awfully confusing:
(-:3 - 7;-0):-)
You can also make your negative signs smaller and higher up than minus signs. Don't put them so high up that you can't get them back down when you need them, though.
Be Careful: Subtraction does not commute! (It works from home.) This is because subtraction changes the direction we're walking on the number line for the second number only.
8 – 10 = -2
10 – 8 = 2
So 8 – 10 ≠ 10 – 8.
Subtraction is also not associative. The order in which we perform multiple subtractions changes the final answer.
(3 – 4) – 2 = -3
3 – (4 – 2) = 1
You're always going to want to perform subtraction from left to right: 3 – 4 – 2 = -3. This should be easy to remember, because that's also the direction in which you read, as well as the direction in which you open the chocolates on your Advent calendar. Or, if you're Jewish, the direction in which you light the candles of your menorah. Or, if you're Buddhist, the direction in which you align your chakras.
Another way to think about this is that subtraction is really just the addition of a negative. This way, you can rewrite the problem as:
3 + (-4) + (-2)
Now it's an addition problem, so it doesn't matter what order you add them in!
If we're subtracting a bigger number from a smaller number (for example, 13 – 25), one way to find the answer is to pretend the problem is structured in reverse: evaluate 25 – 13 to get 12. Since we know 13 – 25 will be to the left of zero, stick a negative sign onto the front of it to get -12. Make sure you use Gorilla Glue so that thing really stays on there.
The reason this method works is that whether you go 13 to the right and 25 to the left, or 25 to the right and 13 to the left, you'll end up the same distance from zero. To give the correct final answer, use common sense to figure out which side of zero the answer is on. If you're lacking in the common sense department, then maybe don't use this method. Also, don't rest your palm on one of the eyes of a stove when it's glowing orange.