 The additive inverse of a written this way is the quantity where a plus the additive inverse is equal to 0 The whole number 0 1 2 3 and so forth and their additive inverses these things form the integers We sometimes refer to the integers as signed numbers and we say that a number like 7 is positive and You might ask well, where's the sign? And so we could write this as plus 7 Meanwhile a number like the additive inverse of 3 is Negative where we have the sign in front of it, which is read as a negative sign And here's an important idea how you speak influences how you think In previous courses you probably read something like this as negative 3 And if we think about this as a signed number, it is the signed number negative 3 But remember the important property of this number is that it is an additive inverse if we add it to a we get zero and So it's helpful to read this as the additive inverse of a and not as negative a It can be helpful to have some sort of picture of what's going on when we're working with signed numbers And so we can represent the additive inverse using what's called a chip model Why is it called a chip model? Well, that's because we have positive chips representing positive amounts and negative chips representing negative amounts and What's important here is that a positive and a negative chip can cancel each other out if we put them together we get Nothing Let's see how that works. Let's add 5 plus 3 Now since this is an addition, we're putting two things together 5 well, that's 5 positive chips and 3 that's 3 more positive chips and Altogether we have 8 positive chips and so we can write our sum 5 plus 3 equals 8 How about additive inverse of 5 plus additive inverse of 3? Again since this is an addition. We're putting two things together here additive inverse of 5. Well, that's 5 negative chips Additive inverse of 3 is 3 more negative chips and Altogether we have 8 negative chips So let's compare and generalize we found that 5 plus 3 is equal to 8 and The additive inverse of 5 plus the additive inverse of 3 is the additive inverse of 8 and The thing to notice here is that the numbers are actually the same in both cases here We have a 5 3 and an 8 and here we have a 5 a 3 and an 8 and the only difference between the two is that here We're dealing with the additive inverses and this suggests the following if I'm taking any real numbers If I want to add to Additive inverses, well, I'll just add the numbers and take the additive inverse of the sum So for example additive inverse 15 plus additive inverse of 7 Well, since they're both additive inverses. I can add the 15 of the 7 together and Take the additive inverse of the sum. What if I add 5 and the additive inverse of 3? So 5 we can represent by 5 positive chips and Then we'll add 3 negative chips and The important thing about the chips is that a positive and a negative chip can combine and cancel each other out And so we can remove a pair of positive and negative chips another pair and Another pair and so 5 plus additive inverse 3 is 2 What if we just do the subtraction 5 minus 3 we could model subtraction by taking things away And so we have 5 positive chips From which we'll remove 3 chips and we'll have 2 chips left over And again, let's compare and generalize we found 5 plus the additive inverse of 3 is 2 But we also found that 5 minus 3 is 2 And again, if you look we have the same numbers in both cases except we're doing something slightly different And this leads to a general rule for any real numbers a Minus b is the same as a plus the additive inverse of b So for example, if I want to find 18 plus the additive inverse of 7 That's the same as 18 minus 7 which is going to be 11