The additive inverse, and how it turns every subtraction into an addition

0:00Someone is standing at the entrance of the building, and presses the wrong button. They press plus three by accident. The lift goes up three floors, and now they are standing on Floor plus three. But they did not want to go anywhere at all. They wanted to stay exactly where they were, at the entrance. So what do they press to fix it? Minus three. Down three floors, back past the food court, back to the entrance, back to zero.

0:31And here is the pair of presses, written down together. Plus three, plus minus three, is zero. Two presses that between them do nothing at all. That looks like nothing much. It is the most useful fact in this whole chapter. Every number has a partner like that. A number which, added to it, brings you back to zero. And the partner has a name. It is called the inverse.

1:00The inverse of plus three is minus three, because plus three plus minus three is zero. Now notice what that definition is about. It is not about what the symbol looks like. It is about what the pair adds up to. That distinction is going to matter in about a minute, so it is worth saying once more. The definition is about what the two of them add to, not about the sign on either.

1:30So try it the other way round, starting below the entrance instead of above it. Press minus four by mistake, and you are on Floor minus four. What gets you back? Plus four. And the pair reads minus four, plus plus four, is zero — the same shape as before, with the two signs swapped over. Which means the inverse of minus four is plus four. And that tells you the relation runs both ways. If one number is the inverse of another, then that other one is the inverse of the first.

2:07They are partners. Neither of them is the original, and neither of them is the copy. Here are six numbers. Write down the partner of each one. Plus four goes with minus four. Minus four goes with plus four. Minus three goes with plus three. Then zero. Hold on to that one for a moment. Plus two goes with minus two. And minus one goes with plus one. Now look at the third one. The inverse of minus three is plus three, and it has a plus sign in front of it.

2:44So an inverse is not a number with a minus stuck on the front. Try that rule and it fails on every negative number there is. Now zero, which is the awkward one on that list, and it is worth stopping on. What do you have to add to zero to get back to zero? Zero. You are already there. The movement you need is no movement at all. So zero is its own partner, and it is the only number in the whole system that is.

3:17And that is not an exception, and it is not a rule somebody added afterwards to tidy things up. Put zero into the definition, and zero is exactly what comes back out. Every other number is somewhere, and has to travel to get home. Zero is already home. There are two ways of saying what an inverse is, and they turn out to be the same sentence. The first is the one we started with. You made a movement, and the inverse undoes it.

3:51The second one is about position. You are standing on a floor, and the inverse is the movement that gets you home. From Floor plus four, press minus four. From Floor minus two, press plus two. Both of them land at the entrance. So the journey home from any floor is that floor's own inverse. Undo a movement, or come back from a place. It is the same number either way.

4:20Now a loose end, left lying around earlier. A journey from two hundred metres below the surface up to two thousand above it came to plus two thousand two hundred. That was written as a subtraction. Plus two thousand, minus minus two hundred. And the addition, plus two thousand plus plus two hundred, came to exactly the same number. Look at the two second numbers. Minus two hundred, and plus two hundred. Those two are inverses.

4:53One matching pair is not a proof. But it is a very good clue, and we can go and get the reason now. Take a journey, and break it in one particular place. Start at two. Finish at minus three. Read straight off the line, that is a movement of minus five. Five places to the left, and you are there. Now do it the other way. First travel from two back to zero. That leg is minus two, which is two's own inverse.

5:26Then from zero out to minus three. That leg is just minus three, because you set off from nothing. Minus two, plus minus three, is minus five. The same answer. And look at what is written down: there is no subtraction sign anywhere in it. That was not a lucky choice of numbers, and it was not a lucky choice of breaking point either. Every journey can be broken at zero, because every number has a partner that gets it there.

6:00So here it is, in general, for any two numbers at all. A minus B is A, plus the inverse of B. Watch what moves and what does not. The first number does not change at all. The operation flips from a minus to a plus, and the second number flips to its partner. Nothing else moves. And that is the end of subtraction as something you have to know how to do. From here it is a way of speaking.

6:31It runs backwards as well, and that is worth ten seconds of your time, because it says something the forward version does not. A plus B is A minus the inverse of B. So any addition at all can be written out as a subtraction instead, if you ever wanted one. Plus eight plus plus two is the same thing as plus eight minus minus two. Both of them come to plus ten.

6:59Nobody actually wants that. But it tells you something worth knowing: these two operations are not really two operations. They are one operation, and a choice about how you write the second number. Four of them to run through, and you already have everything you need for all four. Plus seven minus plus five. The partner of plus five is minus five, so it becomes plus seven plus minus five. Plus two.

7:31Minus three minus plus eight. That becomes minus three plus minus eight. Minus eleven. Plus eight minus minus two. The partner of minus two is plus two, so plus eight plus plus two. Plus ten. Plus six minus minus nine. Plus six plus plus nine. Plus fifteen. And there is the answer to the clue. The last two got bigger, because their second number was negative and its partner is positive.

8:04One last thing, and it is about the name. You will meet this idea again, called the additive inverse. It is the same idea. The longer name just says which operation it is the inverse for. Addition. The inverse of seven is minus seven. The inverse of minus five hundred and forty three is plus five hundred and forty three. You would not want to draw that second one on a line, and you do not have to. By now this is arithmetic, not geography.

8:39And more than thirteen hundred years ago, a mathematician called Brahmagupta wrote down the rules for subtracting with negative numbers. Every one of them is this, said differently.