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## 7th grade

### Course: 7th grade > Unit 3

Lesson 4: Subtracting negative numbers fluently# Adding the opposite with number lines

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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- i’ve always been told i can’t

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## Video transcript

- [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.