 Another thing we might do is to remove a set from a set. The mathematical abstraction of this process is subtraction. For example, we might want to subtract 1, 2, 5, base 6 minus 1, 1, base 6. So remember, arithmetic is bookkeeping, and so this 1, 2, 5, base 6 means that we have 1 large, 2 medium, and 5 small. Now we want to remove 1 medium, 1 small. And if we do that, this leaves us with 1 large, 1 medium, 4 small, or 1, 1, 4, base 6. What about what's usually called borrowing? Well, what if we try to find 1, 0, 0, base 5 minus 2, 4, base 5? So we have 1 large, and we want to remove 2 medium and 4 small. But we don't have enough mediums or smalls. We could borrow, but since we never return what we borrow, we should say that we are trading. Remember, how you speak influences how you think. So let's try that subtraction. We have 1 large, and we want to remove 2 medium and 4 small. Now since this is base 5, we can trade 1 large for 5 medium. We should try to keep things organized and put the mediums in the correct place. And we can trade 1 medium for 5 small. And now this gives us enough mediums and smalls to remove what we want. So removing 2 medium and 4 small leaves 2 medium and 1 small, and so we can write. And as with addition, everything you need to know about subtraction is based on these 2 ideas, removing units when they're available, and trading to make more units available. All else is about making the process more efficient.