so indeed we have verified that subtracting a negative is the same thing as adding the opposite of that negative which is positive.
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 7 is. See if you can pause this video and figure that out using integer chips. Well, let’s do this together.
We’re starting with negative one, which we could represent as just one negative integer chip. But we need to subtract positive 7 from that, and we have no positive integer chips here. So, how could we get some positive integer chips? Well, we can just add pairs of negative and positive 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 will cancel each other out. So let me just do that seven times.
So that’s two, three, four, five, six, and seven. 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 7 now from this.
So now let’s subtract out positive 7. 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. When we take negative one and subtract positive seven from that, we’re 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 as 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 going to 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. If we have negative 3 minus negative 5, maybe this is the same thing as negative 3 plus the opposite of negative 5, which would be positive 5. 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 going to start with negative 3, so that gives us three negative integer chips. So negative one, negative two, negative three. Now if we want to subtract out negative 5, if we want to 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 3, because that and that cancel out. And so this is still the number negative 3 being represented, but I added these two pairs because now I can subtract out five negative integer chips. That’s what negative 5 represents, these top four negative interchips. There’s five of them. I can take them all away, that’s subtracting out a negative 5. 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 2.
So indeed, we have verified that subtracting a negative is the same thing as adding the opposite of that negative, which is positive. It appears that subtracting a number is equivalent to adding its opposite. Without giving the full proof, it seems that this is true. For example, starting with -3 and adding 5 positive integers (1, 2, 3, 4 and 5) results in a total of 2. This is not a proof that this will always work, but rather an example that gives intuition that it does.