How do you deal with adding and subtracting negatives? The process works similarly to adding and subtracting positive numbers. When you'd added a positive number, you'd moved to the right on the number line. When you'd subtracted a positive number, you'd moved to the left.
Now, if you're adding a negative, you can regard this is pretty much the same as when you were subtracting a positive, if you view "adding a negative" as adding to the left. That is, by plus-ing a minus, you're adding in the other direction. In the same vein, if you subtract a negative (that is, if you minus a minus), you're subtracting in the other direction; that is, you'll be subtracting by moving to the right.
For instance:
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Let's return to the first example from the previous page: "9 − 5" can also be written as "9 + (−5)". Graphically, it would be drawn as "an arrow from zero to nine, and then a 'negative' arrow five units long":
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...and you get "9 + (−5) = 4".
Now look back at that subtraction you couldn't do: 5 − 9. Because you now have negative numbers off to the left of zero, you also now have the "space" to complete this subtraction. View the subtraction as adding a negative 9; that is, draw an arrow from zero to five, and then a "negative" arrow nine units long:
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...or, which is the same thing:
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Then 5 − 9 = 5 + (−9) = −4.
Of course, this method of counting off your answer on a number line won't work so well if you're dealing with larger numbers. For instance, think about doing "465 − 739". You certainly don't want to use a number line for this. However, since 739 is larger than 465, you know that the answer to "465 − 739" has to be negative, because "minus 739" will take you somewhere to the left of zero. But how do you figure out which negative number is the answer?
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Look again at "5 − 9". You know now that the answer will be negative, because you're subtracting a bigger number than you'd started with (the nine is bigger than the five). The easiest way of dealing with this is to do the subtraction "normally" (with the smaller number being subtracted from the larger number), and then put a "minus" sign on the answer: 9 − 5 = 4, so 5 − 9 = −4. This works the same way for bigger numbers (and is much simpler than trying to draw the picture): since 739 − 465 = 274, then 465 − 739 = −274.
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Adding two negative numbers is easy: you're just adding two "negative" arrows, so it's just like "regular" addition, but in the opposite direction. For instance, 4 + 6 = 10, and −4 − 6 = −4 + (−6) = −10. But what about when you have lots of both positive and negative numbers?
Probably the simplest thing to do is convert everything to addition, group the positives together and the negatives together, combine, and simplify. It looks like this:
18 − (−16) − 3 − (−5) + 2
= 18 + 16 − 3 + 5 + 2
= 18 + 16 + (−3) + 5 + 2
= 18 + 16 + 5 + 2 + (−3)
= 41 + (−3)
= 41 − 3
= 38
"Whoa! Wait a minute!" I hear you say. "How do you go from '− (−16)' to '+16' in your first step? How did the 'minus of minus 16' turn into a 'plus 16'?"
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This is actually a fairly important concept, and, if you're asking, I'm assuming that your teacher's explanation didn't make much sense to you. So I won't give you a "proper" mathematical explanation of this "the minus of a minus is a plus" rule. Instead, here's a mental picture that I ran across years ago in an algebra newsgroup:
Imagine that you're cooking some kind of stew in a big pot, but you're not cooking on a stove. Instead, you control the temperature of the stew with magic cubes. These cubes come in two types: hot cubes and cold cubes.
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If you add a hot cube (add a positive number) to the pot, the temperature of the stew goes up. If you add a cold cube (add a negative number), the temperature goes down. If you remove a hot cube (subtract a positive number), the temperature goes down. And if you remove a cold cube (subtract a negative number), the temperature goes UP! That is, subtracting a negative is the same as adding a positive.
Now suppose you have some double cubes and some triple cubes. If you add three double-hot cubes (add three-times-positive-two), the temperature goes up by six. And if you remove two triple-cold cubes (subtract two-times-negative-three), you get the same result. That is, −2(−3) = + 6.
Here's another analogy that I've seen. Letting "good" be "positive" and "bad" be "negative", you could say:
good things happening to good people: a good thing
good things happening to bad people: a bad thing
bad things happening to good people: a bad thing
bad things happening to bad people: a good thing
To give a specific example:
the family of four in the minivan gets home, safe and sound: a good thing
the drunk driver in the stolen car veering all over the road doesn't get caught and stopped: a bad thing
the family of four is killed by the drunk driver, while the drunk flees the scene without a scratch: a bad thing
the drunk driver is caught and locked up before he hurts anybody: a good thing
The analogies above aren't technical explanations or proofs, but I hope they make the "minus of a minus is a plus" and "minus times minus is plus" rules seem a bit more reasonable.
For whatever reason, it seems helpful to use the terms "plus" and "minus" instead, of "add, "subtract", "positive", and "negative". So, for instance, instead of saying "subtracting a negative", you'd say "minus-ing a minus". I have no idea why this is so helpful, but I do know that this verbal technique helped negatives "click" with me, too.
Let's look at a few more examples:
−43 − (−19) − 21 + 25
= −43 + 19 − 21 + 25
= (−43) + 19 + (−21) + 25 *
= (−43) + (−21) + 19 + 25 *
= (−64) + 44
= 44 + (−64)
Technically, I can move the numbers around the way I did between the two starred steps above only after I have converted everything to addition. I cannot reverse a subtraction, I can only reverse an addition; only addition is commutative. In practical terms, this means that I can move the numbers around only if I also move their signs with them. If I move only the numbers and not their signs, I will have changed the values and will end up with the wrong answer. Continuing...
44 + (−64) = 44 − 64
Since 64 − 44 = 20, then 44 − 64 = −20.
84 + (−99) + 44 − (−18) − 43
= 84 + (−99) + 44 + 18 + (−43)
= 84 + 44 + 18 + (−99) + (−43)
= 146 + (−142)
= 146 − 142
= 4
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