If you're seeing this message, it means we're having trouble loading external resources for Khan Academy.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Watch the entire video

Adding/subtracting negative numbers

Adding and subtracting negative numbers
Back

Adding/subtracting negative numbers

Discussion and questions for this video
I am still trying to grasp the 3-(-3)=6 I know that you say it means you cancel the negatives and make it a positive but I just can't visualize why. It's not enough for me to just except that that is what you do. I need to really understand it. I cant figure out how to recreate it on the number line. Can anyone show 3-(-3) looks on a number line?
Someone explained it to me this way. This made more sense to me. Hope this might help others here.

"It goes back to what we understand of finding the difference between two numbers. You are finding the difference between 3 and -3. Draw a number line to see how many units these two numbers are apart. you will see that 6 units separate these two numbers. Thus, 3-(-3) has to be equal to 6 units of difference."
what happens if you get 7+-4???? does the minus still turn into a +?
no, it doesn't, you still have (+7)+(-4) = (+3), to 7 positive units you add 4 negative units, that substract themselves from this positive and you are left with 3 positive units...
cheers
So a second Minus means that its a plus?
Think of it as removing a subtraction. To remove a subtraction, you have to add. For example, if I start with 5 and subtract 3 from it, I end up with 2.

5 - 3 = 2

Now, if I want to get back to 5, I have to remove the subtraction I did earlier.

2 - -3 = 5

The double negative sign in this example could then be interpreted as "Start at 2, and remove a subtraction of 3".

2 "Start at 2,"
- "and remove"
-3 "a subtraction of 3"

Once again, to remove a subtraction, you have to add. So "remove a subtraction" becomes "add"

2 "Start at 2,"
+ "and add"
3 "3"

2 + 3 = 5
I still don't understand adding and subtracting negative numbers. I need some help could someone explain this to me or refer me to a site or book that will explain adding and subtracting negative numbers, please!
Adding and subtracting negative numbers is TRICKY, so I've spent the last couple of days reviewing the subject. Cbelle2000, your first example problem (i.e., -3+(-4)=-7) is solved correctly. (You wondered about how NEG + NEG is "supposed" to result in a positive. Not really. ANY TIME you ADD integers/numbers that share the SAME sign [+/-] their sum/answer will ALWAYS have the same sign as the original numbers [addends]. So, 3+4 [both positive] =7; -3+(-4) [both negative] =-7.)

SUBTRACTING two negatives, though, is a different matter, and this is where you went wrong. The answer to your second example problem (i.e., -3 - (-5)=-8) should be 2. Here is why (get out your scratch paper and follow along – doing is learning!):

We start with -3 and subtract -5 from it. The first number is called the minuend; the second number is the subtrahend. (Not necessarily important to remember, just interesting to know....) So, we have:

-3 - (-5) =

Think of negative signs as traffic signs that tell you to hit the brakes and then go in the OPPOSITE direction. So, because subtracting says "go DOWN the number line", when you see it followed by a number with a negative sign, that negative sign is saying, "Nope! Hit the brakes!! Go the other way! Go UP the number line!" In other words, ADD. Subtracting a negative number is the SAME as adding a positive number, so the problem NOW reads like this:

-3 + 5 =

Simple enough, right? We aren't done, though. What's next?

ADDING opposite signed integers together is very easy if you remember the following:

1) Ignore the negative sign. That's right, drop it! (For now....)

2) SUBTRACT the smaller number from the larger number.

3) The answer (called the DIFFERENCE) is now given a positive or negative sign, depending on which absolute number (in this case, 3 and 5) is bigger. So....

-3 + 5 becomes 5 - 3 and the answer is 2. The answer is also POSITIVE because before we converted to a final subtraction problem the number with the larger absolute value (see the video on absolute values) was positive 5, not negative 3.

So, here is our completed problem:

-3 - (-5) = 2!

Let's do that again, without the commentary, as you would on your homework sheet:

-3 - (-5) becomes

-3 + 5 becomes

5 - 3 = 2 *

*Remember, looking at the middle step where it is an addition problem, 5 has greater absolute value than -3, so the final answer, 2, is positive, like the 5!

What if the original minuend and subtrahend were reversed, though? The answer would be -2! Here's why:

-5 - (-3) becomes

-5 + 3 becomes

5 - 3 = 2 BUT

In the middle step, negative 5 has a greater absolute value than positive 3, so....

The answer is -2!

You can also think through NEG - NEG problems this way:

-3 - (-5) =

Say to yourself: "Self, the bigger absolute number, 5, minus 3 equals 2. Hold that thought!"

Now say to yourself: "Self, the second number in this problem, the subtrahend, is a negative 5, and if I went through all the trouble to do it on paper it would become a positive 5. Now, hold that thought!"

Finally, say to yourself: "SELF! The absolute value of positive 5 is greater than the absolute value of negative 3, so the answer is positive 2!"

Now,let's do it with the numbers reversed:

-5 - (-3) =

Say to yourself: "Self, the bigger absolute number, 5, minus 3 equals 2. Hold that thought!"

Now say to yourself: "Self, the second number in this problem, the subtrahend, is a negative 3, and if I went through all the trouble to do it on paper it would become a positive 3. Now, hold that thought!"

Finally, say to yourself: "SELF! The absolute value of negative 5 is greater than the absolute value of positive 3, so the answer is negative 2!"

...and there you go!! It becomes very simple when you understand the steps. Subtracting negative numbers is the trickiest, but the most fun, too, when you know what to do.

You can try this with any two negative numbers. Check your answers with a calculator until you are comfortable enough to do it without having to check. Just plug in your negative numbers:

NEG - NEG = becomes

NEG + POS = becomes

Larger absolute number - smaller absolute number =

The answer is signed either positive or negative depending on the larger absolute addend in the second step.

I hope this helps! (Sorry it was so LENGTHY!) Have fun!!!
isnt 3 - (-3) equal to -6? hows it = +6 i didnt get it?

two negative numbers added becomes a positive.
At 5:28 I too was confused as to the warrant for the minus symbol canceling out. I don't understand this. If it cancels itself out, isn't it redundant? And if so, why is it that way?
That's a great question, actually. To answer your question, I'll be blunt and simple: It is redundant. If you are going on a day trip to a park, and need $15 for admissions and want $15 for food, you probably won't say that you'll need 15 - -15 dollars, and this would be a good case of where you don't need to write it out like this. That doesn't mean that this stuff isn't useful, however.

You are probably wondering why this stuff is even important, if it's so redundant like that anyway. Well, it is more useful in algebra, when dealing with variables, and for many, many other purposes. I think you will probably see that with time. For example, what if you are subtracting a variable, and what if you don't know if the variable is positive or negative? Let's look at this:

y = 8 - x

I'm assuming you have a knowledge of basic algebra (I understand that this is a pre-algebra playlist). Basically, this says that the variable Y, will equal 8 minus whatever X equals. If X is 5, than Y will be 8 - 5, which equals 3, so Y is 3 when X is 5.

But anyway, you don't actually know what X will equal. If you put positive numbers in for X, you are just doing basic subtraction, but what will Y equal when X is, say, -2?

Here, we have Y = 8 - -2... Redundant, huh? But if we didn't know how to do this kind of subtraction, our equation y = 8 - x would only work with positive numbers for X. But, since we know how to subtract negative numbers, we know that Y = 10 when X = -2. But we would be scratching our heads wondering what to do with this if we didn't know how to do this kind of subtraction.

Or what if you are adding or subtracting different equations? (See the "systems of equations" in the algebra playlist). Here, you'll also run into such "redundant" math operations.

It is for reasons like this when you are dealing with variables that you need to worry about this stuff. It's because you don't know if the variables contain positive or negative numbers. It's an extremely foundational concept in algebra, and it is my belief that you would not be able to pass an algebra course without learning this, because there is so much going on with variables where you don't know if the variables contain positive or negative numbers. It will probably become more clear to you as you go on towards algebra. I think that you'll see that there are a huge number of applications to this stuff, even though it seems redundant at first.

I hope this helps!
does the minus is the plus and the plus is the minus i am confused. please help
Think of subtraction on a number line. A subtraction sign means to travel left. But a negative sign AFTER a minus sign tells you to reverse directions. In other words, go to the right, and going to the right on a number line is what we do when we add. So a minus sign and negative sign together have the same effect on the problem as a single + sign. It also works when you add a negative number. The + sign means travel to the right on the number line, but if it is followed by a negative sign, it again means to reverse directions, and in this case you would travel to the left on the number line, just like when doing subtraction. So a + sign and - sign is the same as having a single - sign.
If you have 3-(-3)=? Then why don't you just say that you change the minus sign to a plus and then the sign of the second number??? 3-(-3)=? is equal to 3+(+3)=6. Right??? Maybe I'm over thinking this....
Changing the double negative to a positive is the mechanism, however the understanding is in that when adding or subtracting negative values you are being asked to add or subtract the lack of something. If you lack 30 (-30) and I tell you to subtract the lack of 15: -30-(-15) you must add 15 to account for subtracting the lack (absolute value) of 15. Your lack decreases to -15. If i tell you to add a lack of 15: -30+(-15), your lack increases to -45.
Now I understand how to add and subtract and it has helped me in school. But, how do you subtract when you have three numbers? ex: 18-(-17)-(-11)
18-(-17)-(-11) = 18+17+11 = (18+17)+11 = 35+11 = 46
if you have 18-17-11 then the answer will be like this:
18-17-11 = (18-17)-11 = 1-11 = -10