Adding and Subtracting Negative Numbers

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":

...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:

...or, which is the same thing:

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?

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:

...so:

This works the same way for bigger numbers (and is much simpler than trying to draw the picture). Since:

...then:

(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".)

What is the rule for adding two negative numbers?

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.

What is the rule for subtracting a negative number?

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:

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

Why is "minus of a minus" equal to a plus?

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:

(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:

To give a specific example of the above:

Another model says:

Or think about driving a car:

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 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?"

(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

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

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

What are the rules for adding and subtracting with negative numbers?

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.