In order to extend your knowledge so that you can subtract a larger number from a smaller one, you need think of addition and subtraction not as accumulating or liquidating of quantities, but rather the movement right (for +) and left (for -) on a number line.
If you want to do something like 1234 - 4321, what you are really doing is starting at 1234 on a number line, then moving 4321 units to the left. Obviously you are going to go past 0. The answer is going to tell you where you end up, i.e. how many units away from 0, and in which direction (a + for to the right, and a - for to the left).