Addition as movement: starting position plus movement gives target position

0:00Here are two questions. They sound completely different, and they have the same answer. First question. You are standing on Floor plus one, and you press the up button twice. Which floor do you end on? Second question. You press the up button once, and then you press it twice more. What did all your pressing come to? One of those is about a place. The other is about a journey.

0:27And yet both of them get written down in exactly the same way: plus one, plus two. That is not a coincidence, and by the end of this video you will know why. Here is the idea this whole video runs on. One sentence, worth reading slowly. The floor you start on, plus the movement you make, gives the floor you end on. Three things, and they are not the same kind of thing. Look at the middle one.

0:58The floor you start on is a place. You can stand on it. The floor you end on is a place too. But the movement is not a place at all. It is how far you travel, and which way. Notice what has happened to addition. It is no longer two piles pushed together. It is a position, and something done to it. Let us run it once, slowly, on the building.

1:25You are at the food court. The food court is Floor plus one, so plus one is where we begin. Now press the up button twice. That is a movement of plus two: two floors, upward. Watch the lift. Up one. Up two. You are at the comic shop, and the comic shop is Floor plus three. Written out: plus one, plus plus two, equals plus three. The first number was a floor. The second was never a floor at all.

1:58Now the case that breaks the old picture of addition. Start at the ice cream floor, which is plus four. Press the down button three times. That is a movement of minus three. Three floors, downward. Down one. Down two. Down three. You are at the food court, Floor plus one. So plus four, plus minus three, equals plus one. And plus one is smaller than the plus four you started from.

2:28You added something and the answer got smaller. If addition means heaping things up, that is impossible. If it means moving, it is obvious: the movement pointed downward. Let us run four of them quickly, and watch the same relation each time. Plus one, plus plus four. Start at the food court, go up four, land on sports. Plus five. Now the same two numbers the other way round. Plus four, plus plus one. Start at ice cream, go up one, and you land on sports again.

3:06Different journeys entirely, and the same floor at the end. Hold on to that; we come back to it. Plus four, plus minus three, we have already done. Plus one. Minus one, plus plus two. Start at the toy shop, one below the entrance, go up two, and you come out at the art centre. Plus one. Two more, and they are the ones people expect to be difficult. Minus one, plus plus one. Start at the toy shop, go up one floor, and you are at the entrance. Zero.

3:42Nothing special happened. You walked up one flight of stairs and arrived somewhere real. Now zero, plus minus two. You are standing at the entrance, and you press the down button twice. You are at the video games floor. Minus two. Nothing was taken from anything, and there was never any question of taking two away from nothing. Zero is a floor. It behaves like every other floor here, and needs no special rule of its own.

4:16Now turn the question round, because this is where it gets useful. Suppose the dinosaurs are what you want. Floor minus five, right at the bottom. You are at the art centre, plus two. What do you press? Count the gap. From plus two down to minus five is seven floors, and the direction is down. So the movement is minus seven. Start somewhere else and the movement changes, but the way you find it does not. From the entrance it is minus five. From the top it is minus eleven, the longest ride this lift gives.

4:56In every case the movement is the floor you want, minus the floor you are on. And from the dinosaurs themselves it is zero: you press nothing, you are already there. Here is somebody making a mistake, and the mistake is instructive. They are at the toy shop and they want to go down two floors. By accident, they press the up button twice. That is a movement of plus two. Then, to fix it, they press down three times. A movement of minus three.

5:31So what did those two runs of presses come to, added together? Plus two, plus minus three. The two movements combine into one movement of minus one. One floor down. So they end one floor below the toy shop, on the video games floor, which is Floor minus two. The movement was minus one. The floor is minus two. Two different numbers, and mixing them up is the easiest mistake in this topic.

6:03Also worth noticing: the person did not actually get what they wanted. They wanted to go down two, and they went down one. To fix it properly they needed four presses of the down button, not three. But look at what we just did. We added two movements together, without any floor being involved at all. Which means the same written expression is doing two different jobs, and now we can chain them.

6:32Plus four, plus minus three, plus minus two. Start at ice cream. Go up nothing, down three to the food court, down two more, and you are at the toy shop. Minus one. Or read it the other way: three movements, plus four and minus three and minus two, combining into one single movement of minus one. Same numbers. Same answer. Now back to the thing I asked you to hold on to.

7:03Plus one then plus four landed on sports. Plus four then plus one landed on sports as well. Why must that happen? Not because of a rule about swapping numbers. Look at the arrows. Two movements joined end to end reach out a certain distance in a certain direction. Turn them round and join them the other way, and the far end lands in the same place. The route is different. You pass different floors on the way, and carrying something heavy you would notice. But the finish is the same.

7:40So the order of your presses changes your journey and never changes your destination. That is a fact about arrows, not a fact about symbols. Now let us take the relation somewhere no lift in a building could go. This is a mine. A shaft runs down the middle, and the levels are marked in metres rather than floors. The surface is zero. Above it, levels at forty and a hundred metres and higher. Below it, minus fifty, minus eighty, and further down still.

8:16Start at plus forty metres and rise sixty. That comes to a hundred, and there is a level marked at plus one hundred. You have arrived somewhere with a name. Now start at minus ninety and drop fifty five more. Minus one hundred and forty five, and there is no marked level there at all. You are between two levels, in the rock. That matters. The relation does not need a labelled stopping place in order to be true. Nothing has changed but the unit and the size of the numbers.

8:53So let us take the building away completely. Here is a scale with zero in the middle, marks every hundred, and arrowheads at both ends. No floors, no shops. Just positions and movements. Minus one hundred and twenty five, plus minus thirty, is minus one hundred and fifty five. Plus eighty, plus plus one hundred and fifty, is plus two hundred and thirty. Neither answer lands on a mark. They do not need to. The marks are there to help you see, not to be looked up.

9:29And one last one, because it kills the old habit for good. You are at plus forty, and you want to reach minus two hundred. What is the movement? Minus two hundred and forty. You add a number to plus forty, and the answer is two hundred below zero. What addition cannot yet tell you is which of two negative numbers is the larger. That is next.