Main content

### Course: 7th grade > Unit 3

Lesson 4: Subtracting negative numbers fluently# Adding the opposite with integer chips

Use integer chips to explore the conjecture that subtracting a number gives the same result as adding the opposite of the number. Created by Sal Khan.

## Want to join the conversation?

- my teacher is making me watch these even though i know how to do it(7 votes)
- you can ask her to test you about all this(2 votes)

- Im not great at math yet i am acing this with 100% on everything. Um please explain. :p(5 votes)
- I think that the problems are always easy in the beginning of the unit. It gets more complicated at the end. Maybe this is a really-easy-to-understand-unit.(2 votes)

- Why did you add two positive chips when the question was both negative? I mean it would be much easier to just subtract 3 from 5 to find the answers.(4 votes)
- why would you wright -1 as -8? Just turn seven to a negetive and add it to the -1.... And with -3 - (-5) Just turn the two minus signs to adition you obviosly cant subract a negetive EX:

-3 - (-5) keep the three it looks like this

-3 + 5 =2 easy.....(3 votes) - why is it so hard(2 votes)
- Because some math is hard(3 votes)

- how do you subtract(2 votes)
- Guys Now I'm confused with brackets of negative numbers

I thought you should add a bracket to seperate only when There are two symbol which represent different meanings

But at3:07, Sal just add an unnecessary bracket on -3+5

Why is that?(0 votes)- he probably did it because he was trying to explain how the inverse operations can be used to make equivalent expressions or something like that.(5 votes)

- Imagine if everything you learned, you would just never forget it? That would be so amazing. Thank you Sal!(2 votes)
- This would make more sense if they explained how to do it without integer chips.(2 votes)
- I just heard a lot of mumble jumble that I didn't understand at all.(1 vote)

## Video transcript

- [Instructor] So let's
use integer chips again to start exploring a little bit more when we deal with negative numbers. So let's say we wanted to compute what negative one minus seven is. See if you can pause this video and figure that out using integer chips. Well, let's do this together. So we're starting with negative one. We could represent negative
one as just one integer, one negative integer chip, but we need to subtract seven,
positive seven from that. We have no positive integer chips here, so, and we need to have at least seven positive integer chips in
order to subtract seven. So how could we get some
positive integer chips? Well, we can just add pairs of negative and positive integer chips. So if I add one negative integer chip and I add one positive
integer chip, just like that, this is still negative one over here because these two integer chips are going to cancel each other out. So let me just do that seven times. So let me just do this. So that's two, three, four, five, six and seven. And then I just have to
add the corresponding positive integer chips three, four, five, six and seven. So notice, what I just wrote, this is just another way
of writing negative one. But I wrote it this way
because I can actually subtract out positive seven now from this. So now let's subtract out positive seven. So subtract out one, two, three, four, five, six, seven. And then what are we left with? Well, we're left with all of
this business right over here. And what is that? That's one, two, three, four, five, six, seven, eight. This is equal to negative eight. Now that's interesting by itself, but you might notice something. What I have left over
when I take a negative one and I subtract positive seven from that, I'm left with essentially the equivalent of negative one and negative seven. So another way of
writing what we just have left over here is I have negative one is that one negative integer
chip right over there. And then I have negative seven, these seven negative integer
chips right over there. So you could also view
this as the same thing as negative one plus negative seven. And so this makes us
think about something. Is it true that if I subtract a positive, that's the same thing
as adding the inverse of that positive, adding, in
the case of a positive seven, in the case of subtracting
a positive seven, that's gonna be the same thing
as adding a negative seven? Interesting. And let's see actually if it works the other way around. So let's see what happens
when we subtract a negative. So if we have negative
three minus negative five, maybe, maybe this is the
same thing as negative three plus the opposite of negative five, which would be positive five. Let's see if these two things actually amount to be the same thing. So let's just start with
this first one up here. We're gonna start with negative three, so that gives us three
negative integer chips. So negative one, negative
two, negative three. Now if we wanna subtract
out negative five, if we wanna take away five
negative integer chips, well we need more negative
integer chips here. We need at least two more
negative integer chips. So if we have two more
negative integer chips, we're not changing the value of that if we have two more
positive integer chips. What I have depicted here
is still negative three because that and that cancel out. And so this is still the
number negative three being represented, but
I added these two pairs because now I can subtract out
five negative integer chips. That's what negative five represents. These top four negative integer
chips, there's five of 'em, I can take 'em all away. That's subtracting out a negative five. And what am I left with? What I'm left with is just these
two positive integer chips. So this is going to be
equal to positive two. Well, that's interesting
because that's kind of feeling very similar to what we have here. If we start with negative
three, so negative one, negative two, negative three,
and I add a positive five, so five positive integer chips, one, two, three, four and five. Well, we already know that
that cancels with that, that cancels with that,
that cancels with that. This is the equivalent of positive two. And what I just did here on both sides, this isn't a proof that
this will always work, but hopefully this gives you an intuition that it does seem to work. And I will tell you that without
giving you the full proof that it actually does always work, that it is actually the case
that if you subtract a number, it's the same thing as adding
the opposite of that number. If you subtract a number
it's the same thing as adding the opposite of that number.