Adding the opposite with number lines | 7th grade | Khan Academy

So, this number line diagram here, it looks like I'm adding or subtracting two numbers. I'm starting with what looks like a positive nine. I'm starting at 0 and going nine units to the right, so that's a positive nine.

To that, it looks like I might be adding or subtracting something. I'll have this arrow that starts exactly at the tip of that nine, and then it goes, let's see, one, two, three, four units to the left. So, I could think about this arrow right over here as just subtracting 4. So it could just be nine minus four, which is, of course, equal to where we end up, which is equal to this 5 right over here.

So, that's one way to represent what is going on as an addition or subtraction equation. But what's another way? We could start with the nine, and is there anything that we could add to the nine that would get us the same result? Well, we know when we add negative numbers, we also move that many units to the left.

So, you could also view this purple arrow right over here instead of viewing it as subtracting a positive four; you could view it as adding a negative four—adding a negative four. Now, this is interesting because in this case, when I subtract a positive number, it's the same thing as adding the opposite of it.

Now, will that work the other way around? If I subtract a negative number, is that the same thing as adding the opposite? Let's say we had negative 5 minus negative 3. What is that going to be equal to? And let's think about this with integer chips.

So, we could start with five negative integer chips: one, two, three, four, and five. And now I'm going to take away three of those negative integer chips, so I'm going to take away one, two, three of them. And so what am I left with? I am left over here with negative 2.

So, if we think about it on a number line, let me do that right over here. So if I have a number line, and let's say that this is 0 here, and I'm going to go negative one, two, three, four, five, so negative 5. So, this is negative 5 right over here.

I can represent— I can start at zero and I can go five units to the left, and I know when I subtract negative 3, I'm going to end up at negative 2. So, I know I'm going to end up— I'm gonna end up with this in a different color— I'm gonna end up right over here.

So, subtracting negative 3 needs to be the same thing as moving three units to the right. If I was adding negative three, I'd go three units to the left, but subtracting negative 3 must be the same thing as going three units to the right.

Well, what also is the same thing as going three units to the right? Well, that's the same thing as adding three. So, this is going to be the same thing. So, negative 5 minus negative 3 is actually the same thing as negative five plus three.

This purple arrow right over here, it could— it's— you could view it as subtracting negative three, or you could view it as adding the opposite of negative three. So, it looks like when you subtract a number, it's the same thing as adding the opposite. When you subtract a number, it's the same thing as adding the opposite.

Now, that's really useful because now we can even think about doing things with things that aren't even integers. We can start thinking about just rational numbers, for example, negative fractions.

So, if we take that principle we just came up with, and if we say, okay, well, what's 3 minus negative two-fifths? Well, if we take that principle, that's subtracting a number is the same thing as adding its opposite. Subtracting negative two-fifths is the same thing as adding the opposite of negative two-fifths, so it's adding two-fifths.

Well, we have seen this type of thing before, and we could even do that on a number line. Let me do that right over here. So, let's say that this is zero, one, two, three, four, and five. So, we're starting here with a 3 in either case. So, that's three units to the right, a positive three, and so subtracting a negative two-fifths, we're saying is the same thing as adding two-fifths.

So, adding two-fifths, we are just moving two-fifths to the right, so it's one-fifth and two-fifths. So, we're just going to do that. Subtracting a number is the same thing as adding its opposite.