 One of the useful features about mathematics is that it is reversible. Remember we can represent signed numbers using our chip model, where we have positive numbers represented by positive chips, negative numbers represented by negative chips, and the important feature is that the positive and negative chips can cancel each other out. But let's think about this. If we can combine a positive and negative chip to form nothing, we should be able to form a positive and negative chip from nothing. And as it turns out, this will be very useful for us. So let's consider a problem like 3 minus 5. So we have 3 positive chips and want to remove 5 positive chips. But the problem is we don't have 5 positive chips. We need more positive chips. Since we want to be able to remove 5 positive chips, we need 2 more positive chips. So we'll make more at the cost of making negative chips as well. And now we're ready to remove 5 positive chips. And that leaves us with 2 negative chips. Now let's consider. We found 3 minus 5 is the additive inverse of 2. But let's consider another subtraction that uses the same numbers, not necessarily with the same signs. And that subtraction would be 5 minus 3, which is equal to 2. And if we compare our 2 subtractions, we see that again they have the same numbers, 5, 3, and 2. But 2 things have changed. We switch the order and we change the sign of the result. And this leads to an important theorem. For real numbers, a minus b is the same as the additive inverse of b minus a. So if I want to find 8 minus 15, the way we might approach it is this. I don't know what 8 minus 15 is. You can't subtract a larger number from a smaller number. But our theorem says you can reverse the order of the subtraction as long as you introduce an additive inverse. So instead of finding 8 minus 15, I'll find the additive inverse of 15 minus 8. And I know how to find 15 minus 8. That's 7. And at this point, we really don't need those parentheses. And so my result is the additive inverse of 7, or about 5 minus the additive inverse of 3. So we start out with 5 positive chips. It's a subtraction, so we need to remove 3 negative chips, but we don't have any to remove. No problem. We can make them from nothing as long as we get a positive chip at the same time. So let's make 3 negative chips from nothing getting 3 positive chips at the same time. And now I can remove 3 negative chips, which leaves me with 8 positive chips. And again, it's helpful to find the comparison. We found 5 minus the additive inverse of 3 is 8. And we also found that 5 plus 3 is 8. And we have the same numbers and the same result. And the only difference, that in one case we're subtracting an additive inverse. And so this leads to the following idea. A minus the additive inverse of B is the same as A plus B. So if I want to find 12 minus the additive inverse of 12, well, that's 12 plus the non-additive inverse of 12, which is 24.