Subtraction as the movement that gets you from start to target

0:00You already know two different things by the name subtraction, and you have probably never had to tell them apart. Here is the first. Ten books on a shelf. Take four away. Six are left. And here is the second. One purse has ten. The other has six. How much does the second one need, to match the first? Four. And you would write that as ten minus six, without hesitating.

0:28Both are called subtraction, and on ordinary numbers they never disagree. But only one of them can come with us to the other side of zero. So we have to find out which. Start with taking away, because it is the one nearly everybody pictures first. Taking away needs something to be there, and needs it to be the kind of thing you can have less of. Four books off a shelf of ten. Four apples out of a basket. That works.

1:00Now try it in the building. You are on Floor plus five, and you want Floor plus one. Take away four floors. Take them away from what? Nothing has been removed. The building has exactly the twelve floors it had before, and every one of them still has people in it. You went somewhere. That is not a subtraction of stuff. It is a journey. So taking away has to stay behind.

1:30Which leaves the other one, and it turns out to be exactly the right shape. Go back to the two purses. Six, and ten. The question was: what is missing? Write it as a question about addition. Six, plus something, makes ten. And the answer to that question is the same number that ten minus six gives you. Four. That is the whole trick. A subtraction is a question about what has to be added.

2:00And notice what it is asking. Not how much is there. How much has to change. Before we take that anywhere new, let us check it does not break anything old. Fifteen minus five. Read the new way: five, plus what, makes fifteen? Ten. Same answer. One hundred minus ten. Ten, plus what, makes one hundred? Ninety. Same answer. Seventy four minus thirty four. Thirty four, plus what, makes seventy four? Forty. Same answer.

2:34That is not luck, and it is not three examples. On every pair of whole numbers where taking away already worked, the two readings give the identical answer. Which is what makes this a replacement rather than a different subject. Nothing you already knew has been contradicted. So take it into the building, where taking away could not go. You know from before: the floor you start on, plus the movement you make, gives the floor you end on.

3:08Now suppose it is the movement you do not know. You know where you are and you know where you want to be. Start, plus what, gives target? That is a missing-addend question, and we just decided those are subtractions. So: the movement you need is the target floor, minus the starting floor. Target first. Start second. That order is not a convention and it is not something to memorise — in a moment you will see it fall out of the addition.

3:41Try it. You are at the art centre, and you want to get to sports. The art centre is Floor plus two. Sports is Floor plus five. You do not need any arithmetic to work out the press. Just count: three floors up. Now write down what you did. Target, minus start. Plus five, minus plus two. Which is plus three. The subtraction did not tell you what to do. You already knew. The subtraction is the record of it.

4:13And look at the arrow on the shaft. Its tail is on the start, its head is on the target, and that head is the whole reason the answer has a sign. Here is the derivation, and it takes three lines. Line one, the relation you already have. Starting floor, plus movement, gives target floor. Line two. We want the movement on its own, so ask what has to be added to the starting floor to reach the target floor.

4:44Line three. That is a missing addend, and a missing addend is a subtraction. Movement equals target floor minus starting floor. That is all. No new rule was introduced anywhere in those three lines. It is the addition you already had, asked backwards. Which matters, because a formula handed to you is a thing to be remembered, and a formula you watched come out of something else is a thing you can rebuild.

5:17Now three journeys, chosen because between them they break every habit you have about subtraction. First. You are at video games, minus two, and you want the toy shop, minus one. Both below the entrance. Minus one, minus minus two. Count it on the shaft: one floor up. Plus one. Second. You are at the comic shop, plus three, and you want the toy shop, minus one. Now you cross the entrance, going down.

5:49Minus one, minus plus three. Four floors down. Minus four. A negative answer, and nothing has gone wrong — it is a journey downward. Third. From video games, minus two, up to the art centre, plus two. Crossing the entrance the other way. Plus four. Now put two of these side by side, because this is the part people get wrong. Plus one, minus plus four. That is minus three.

6:20Plus four, minus plus one. That is plus three. Same two numbers. Different answers. And that is not a mistake in one of them. Look at the arrows. Same length, opposite directions. One is the journey out and the other is the journey home. So you do not put the bigger number first. You put the target first, because the question tells you which floor you are trying to reach.

6:48One more pair, and it does something that will look impossible. Plus four, minus minus three. Start at the cinema, minus three. Finish at the ice cream floor, plus four. Count it. Up three to the entrance, then up four more. Seven floors up. Plus seven. Plus seven is larger than the plus four we started this sum with. A subtraction made the answer bigger. That happens exactly when the second number is negative, and the picture says why without any rule at all. If you are starting below the entrance and finishing above it, you press up.

7:29And also: minus two, minus minus two, is zero. Not minus four. The start and the target are the same floor, so there is no press to make. The relation does not care about buildings. Take it down a mine. Here the levels are metres. Plus forty, minus minus fifty. From fifty metres below the surface up to forty above it. Ninety metres. Plus ninety. And ninety metres is not a level anybody has labelled — it is the size of the climb, not a place.

8:06Now something no shaft could hold. You are two hundred metres below the surface, and you want to be two thousand above it. Break it at the surface. Up two hundred metres to get to zero. Then up two thousand more. Two thousand two hundred metres. Break it anywhere else you like and the total does not move. The legs change; the journey does not. So look at what that last one actually said.

8:36Plus two thousand, minus minus two hundred, came to plus two thousand two hundred. Now look at this addition. Plus two thousand, plus plus two hundred. Two thousand two hundred. The same number. A subtraction and an addition, with the same two numbers in them, giving the same answer. That was not a fluke of those numbers. Try it with any pair you like and it keeps happening. Taking a negative away does what adding the positive does.

9:08Nothing said in this video explains why. That is exactly where we go next.