The rules for adding and subtracting negative numbers 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.

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Now, if you're adding a negative number, you can regard this is pretty much the same as when you were subtracting a positive number, 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*.

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 going five units backwards":

← *swipe* to view full image →

...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 room necessary in order to complete this subtraction. View the subtraction as adding a negative 9; that is, draw an arrow from zero to five, and then a "minus" arrow that goes nine units backwards:

← *swipe* to view full image →

...or, which is the same thing:

← *swipe* to view full image →

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 a minus number, 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 number (that is, the 9) that is bigger than the one that you'd started with (that is, the 5). 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

(nrich has a nice pictorial illustration of positives and negatives. Scroll about one-third of the way down their lesson page to get to their "Counters Model".)

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Adding two negative numbers is easy: you're just adding two "minus" arrows (that is, two arrows pointing to the left along the number line), so it's just like regular addition with positive numbers, but in the opposite direction. For instance, 4 +4 64 =4 10, and −44 −4 6 =4 −44 +4 (−6) =4 −10.

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When you subtract a "minus" (that is, a negative) number from another number, you are actually adding a positive number to that other number. For instance, if you have −9 − (−3), that second number represents two reversals. Thinking in terms of our arrows on our number line, we started with a −3, being an arrow of length three and pointing to the left, and we subtracted it from the other number, thereby reversing the arrow's direction so that it's now an arrow of length three but pointing to the right.

Subtracting a negative number (that is, minus-ing a minus number) turns into its equivalent, which is adding a positive number (that is, plus-ing a plus number). In a sense, the two minuses "cancel" each other off; think of drawing a vertical line through both "minus" signs, turning them into "plus" signs:

1 **−** (**−**5) becomes 1 **+** (**+**5) = 6

(In practice, yes, you will be drawing an actual vertical line through the "minus" signs to create the "plus" signs.)

This "minus of a minus is a plus" thing is actually a fairly important concept and, if you're asking why it works this way, then 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 some years ago:

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.

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.

(The article by nrich uses a hot-air balloon with puffs of air and sandbags to accomplish the same thing.)

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 of the above:

a family of four in their minivan gets home, safe and sound: a good thing

a drunk driver is veering all over the road in a stolen car: a bad thing

the family of four is killed by the drunk driver (he flees without a scratch): a bad thing

the drunk driver is caught and locked up before he hurts anybody: a good thing

Another model says:

When you're feeling down, listening to sad music can make you feel better.

Or think about driving a car:

Suppose "adding" means "driving in 'Drive'", "subtracting" means "driving in 'Reverse'", "positive" means "facing the right way for trafic on your side of the street", and "negative" means "facing the wrong way". To add a positive, you're driving with traffic. To subtract a positive, you're reversing into the car behind you. To add a negative, you're in "Drive" but, since you're facing the wrong way, you still plow into the car behind you. But if you subtract a negative, then you're facing the wrong way, but that's okay, because you're driving in "Reverse", so you're going with the flow of traffic. (Don't do this in real life! No cop is gonna buy the "but it's for my math class" excuse.)

The analogies above aren't technical explanations or proofs, but I hope they make the "minus of a minus is a plus" thing seem a bit more reasonable.

For some 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. (Thank you, Professor Mazumdar!)

(For an amusing and informative rant on the topic, try this "Math with Bad Drawings" article, which rejects "two negatives make a positive" in favor of "the opposite of the opposite of a thing is the thing itself".)

If somebody asks you, "Why does the minus of minus have to be a plus", you can (if you're wanting to be contrary) ask in reply, "Well, what *else* could it be?"

Suppose −(−1) = −1 instead of +1. Then:

0 = 0(−1)

= (1 − 1)(−1) ←(A)

= (1 + (−1))(−1)

= (1)(−1) + (−1)(−1) ←(B)

= −1 + (−1) ←(C)

= −2

But zero (where we started) does not equal 2 (where we ended up)! What happened?

I started with the number zero. At step (A), I turned zero into 1 − 1, which equals zero. At step (B), I applied the Distributive Property. At step (C), I applied the (wrong) assumption that the minus of a minus should be another minus (rather than a plus). The result of that assumption was a "proof" that zero equals something that is very much *not* zero. Since the only questionable part was step (C), and since using (C) led to a contradiction (namely, saying that zero equals two), then (C) must be wrong. Ergo, minus of a minus has to be a plus.

(This is, by the way, an example of a "proof by contradiction", wherein you assume something that is the opposite of what is true [in this case, assuming that the minus of a minus is another minus], show that this assumption leads to a contradiction of known good information [namely, that zero equals two], and thereby prove that the assumption was false, so the opposite is true [namely, that the minus of a minus is a plus].)

A note to native-English speakers: Don't use the "a double negative is a positive" thing, because that's a thing pretty much only in ("proper") English.

We've seen how to work with two numbers. What if we're adding and subtracting lots of numbers? The main thing to be careful of is that signs stay with their numbers.

- Simplify 18 − (−16) − 3 − (−5) + 2

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

In my working above, I used the fact that the minus of a minus is a plus to convert −(−16) to +16 and −(−5) to +5. To help me keep the signs with their numbers, I also converted −3 to +(−3). That way, when I moved the −3, I would be sure that the "minus" sign moved with it.

- Simplify −43 − (−19) − 21 + 25

−43 − (−19) − 21 + 25

= −43 + 19 − 21 + 25

= (−43) + 19 + (−21) + 25 *

= (−43) + (−21) + 19 + 25 *

= (−64) + 44

= 44 + (−64)

= 44 − 64

= −20

I got that last value by noting that 64 − 44 = 20, that 64 is larger than 44, and that 64 has the "minus" sign. so my answer is:

−20

To get from the first starred line above to the second, I moved terms around to get all the plusses together and all the minusses together. It's okay for me to move the terms around, as long as I move their signs around *with* their numbers. When you're just starting out, you may find it helpful to convert "minus [some number]" to "plus (the minus of [that number])", which can make it easier to keep track of where those "minus" signs need to go.

- Simplify 84 + (−99) + 44 − (−18) − 43

I'll start by converting the minus of minusses to plusses, and the minusses to plusses of minusses. This will help me keep the signs straight:

84 + (−99) + 44 − (−18) − 43

= 84 + (−99) + 44 + 18 + (−43)

= 84 + 44 + 18 + (−99) + (−43)

= 146 + (−142)

= 146 − 142

= 4

The take-aways from this page are the following rules for adding and subtracting with negative numbers:

- If you're adding two negative numbers, then add in the usual way, remembering to put a "minus" sign on the result. Example: −2 + (−3) = −5
- If you're adding a positive number and a negative number, subtract the smaller number (that is, the number that's closer to zero) from the larger number (that is, the number that's further from zero), and use the sign for whatever was the larger number on your answer. Example: +2 + −3 = −(3 − 2) = −(1) = −1
- If you're subtracting a negative number, then cancel the "minus" signs to convert to adding a positive. Example: −2 − (−3) = −2 + 3 = +1

As long as you're careful about moving signs *with* their numbers, you should be okay.

URL: https://www.purplemath.com/modules/negative2.htm

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