Adding and Subtracting Negatives

A winter's morning starts at -2 degrees and warms up by five; a bank account with 10 pounds in it has 15 taken out. Below-zero temperatures, money you owe, floors under the ground, points behind in a game — the moment life dips below zero, you are adding and subtracting negatives.

Once the number line stretches into the negative numbers, adding and subtracting still work exactly the same way — you just keep moving along the line, even past zero. Nothing new to memorise: the line simply carries on to the left of 0.

Adding moves right. Subtracting moves left. That single rule never changes. So if you start at 2 and subtract 5, you take five steps left — straight through zero and out the other side:

2 - 5 = -3

And starting in the negatives is fine too. From -1, adding 4 means four steps right, which carries you back up past zero:

(-1) + 4 = 3

Notice that subtracting no longer always makes things smaller in the way you might expect — 2 - 5 is below where you started, but it has landed in the negatives. Stepping left of zero is allowed; that is the whole point of having a negative side.

Press play, then replay it: each time the marker starts somewhere new and hops for a calculation that crosses zero, reading each number aloud — minus signs and all.

See it: a hop across zero

Here is one calculation drawn as a single arched hop. The dot starts on the first number; the arrow shows the jump — to the right for adding, to the left for subtracting — and lands on the answer. Watch what happens as the arrow sails over 0 into the blue (negative) side. Press Refresh for a brand-new jump every time.

A thermometer is a number line standing up

sun Temperature is the friendliest way to feel negative numbers. Warm days are above zero; freezing nights drop below. Adding heat moves you up the scale; losing heat moves you down — exactly like moving right and left on the number line, only turned on its side.

Suppose it is -2 degrees at dawn and the sun warms it by 6 degrees. The temperature rises, so we add — six steps up from -2:

(-2) + 6 = 4

Now imagine the evening cools by 7 degrees from 4. A fall subtracts, taking us seven steps down — through zero and into a frosty 4 - 7 = -3 degrees.

On a thermometer, the numbers get bigger as you climb and smaller as you drop — and -3 is lower than -1, so it is colder. That can feel upside down at first: with negatives, the number with the bigger digit (3 beats 1) is the smaller, chillier one. The further left of zero you go, the less you have.

Three worked examples

Read each one as a journey along the line: start, then step.

3 - 5: start at 3, step 5 to the left — 2, 1, 0, then past zero to −1, −2. You land on 3 - 5 = -2
(-2) + 6: start at −2, step 6 to the right — −1, 0, then up to 1, 2, 3, 4. You land on (-2) + 6 = 4
4 - (-3): this one has two minus signs touching. Taking away a negative turns you around so you step to the right instead — 3 steps up from 4. You land on 4 - (-3) = 4 + 3 = 7

Subtracting a negative

There is one more move to learn: subtracting a negative. Subtracting normally means "step left", but a negative amount points the other way — so the two reversals cancel and you end up stepping right. Subtracting a negative turns into adding:

3 - (-2) = 3 + 2 = 5

That is the famous rule: two minuses make a plus. Whenever you see a minus sign right next to a negative, swap the pair for a single plus and carry on.

coin Money makes this feel obvious. A negative amount is a debt — money you owe. If someone takes away a 3-coin debt for you, you are 3 coins better off. Removing a debt is the same as being handed the coins: 4 - (-3) = 7.

Yes. A minus sign means "the opposite of", and the opposite of the opposite is right back where you started. The opposite of 5 is -5; the opposite of -5 is 5 again. Each minus flips you to the other side of zero, so two flips land you home.

The sign traps that catch everyone:

Subtracting a negative ADDS. Don't read 4 - (-3) as 4 - 3 = 1. The two signs touching make a plus: 4 - (-3) = 4 + 3 = 7.
Two signs together: -\,- makes +, while a plus next to a minus (+\,- or -\,+) stays a minus.
Subtracting does not always make things smaller — but it never tricks you if you just step the right way: minus steps left, and a second minus turns that step around.

Khan Academy works through these moves here: