Explore the claim that subtracting a number is the same as adding its opposite using number lines. We move to the right on the number line when we add a positive number. We move to the left on the number line both when we subtract a positive number and when we add a negative number. So which direction do we draw the arrow when we subtract a negative number on the number line? Created by Sal Khan.
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- [Instructor] So this number lined 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 zero and going nine units to the right. So that's a positive nine. And to that, it looks like I might be adding or subtracting something. And we 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 four, so it could just be nine minus four, which is of course equal to where we end up; which is equal to this five 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 five minus negative three. 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 gonna take away one, two, three of them. And so what am I left with? I am left over here with negative two. 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 zero here and I'm gonna go negative one, two, three, four, five. So negative five, so this negative five right over here, I can represent, I can start at zero, and I could go five units to the left. And I know when I subtract negative three, I'm gonna end up at negative two. So I know I'm gonna end up, I'm gonna end up, I'll do this in a different color. I'm gonna end up right over here. So subtracting negative three 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 three 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 five minus negative three is actually the same thing as negative five plus three. This purple arrow right over here, 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 three minus negative two fifths? Well, if we take that principle that 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 fives. Well, we have seen this type of thing before and we could even do that on a number line. Lemme 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 three 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.