Numbering the floors below the ground: why zero needs another side

0:00Here is a question, and it takes a whole video to answer it properly. You know the counting numbers. One, two, three, and on for ever. You know zero. And you know there are fractions tucked in between the whole numbers. And you have almost certainly seen this picture: a line with zero at the left, then one, then two, then three, marching off to the right. So, the question. Is there anything on the other side of zero?

0:31Not should there be. Is there? And if there is, what on earth would you call it? Look hard at that picture, because something is missing from it. There is an arrowhead on the right-hand end. That arrowhead means something: it says this goes on for ever, that way. There is no arrowhead on the left. The picture simply stops, at zero, dead. A path that starts somewhere and runs on for ever in one direction only has a name. It is called a ray.

1:04So what you have been shown all this time is not a line at all. It is half of one. And nobody ever said why it stops. It stops there because, so far, nothing you have met has needed the other half. So let me give you something that needs it. A building. It is a building of fun, and it has a great many floors. Walk in at the entrance and you are in the welcome hall. Go up, and there is a food court, then an art centre, then a comic shop, then ice cream, then sports, and right at the top, space.

1:42Six floors above the level you walked in on. But this building also goes down. Below the entrance there is a toy shop, then video games, then a cinema, then a ghost house, and at the very bottom, dinosaurs. Five floors below the level you walked in on. Twelve floors in all, and every single one of them is full of people. There is a lift, and here is the strange thing about it.

2:12It does not have twelve buttons, one for each floor. It has two. One is marked with a plus. The other is marked with a minus. Press the plus and you go up exactly one floor. Press the minus and you go down exactly one. That is the entire control panel. Two buttons, for a building of twelve floors. Which means every journey anyone makes here is a run of presses — and a run of presses is going to need writing down somehow.

2:47Suppose you press the plus three times in a row. Up one, up one, up one. Rather than write all three, write it once, like this. Plus three. And four presses of the minus gets written as minus four. Now run it backwards. You want to rise four floors — what do you press? The plus, four times. Which is plus four. You want to drop three floors. The minus, three times. Minus three.

3:16And notice what has quietly happened. That plus and that minus are not telling you to add or to subtract anything at all. They are telling you which way. Now for the naming, and it begins with a decision that somebody has to make. Someone has to pick one floor and call it zero. In this building they pick the entrance — the welcome hall. That floor is now Floor 0.

3:45Be very clear about why, because this is the part everybody gets wrong. It is not zero because there is nothing there. The welcome hall is the busiest floor in the building. It is zero because it is the level everything else will be measured from. It was chosen. Choose a different floor instead and every number in the rest of this video changes. The building would not change at all. Only the labels on it.

4:17So walk upwards, one floor at a time, naming as you go. From Floor 0, one press of the plus takes you to the food court. So the food court is Floor plus one. One more press and you are at the art centre. Floor plus two. Then the comic shop, plus three. Ice cream, plus four. Sports, plus five. And space, right at the top, plus six. Nothing surprising has happened yet. These are the counting numbers you already had, with a plus written in front of each one.

4:54Now go the other way, and this is where it gets interesting. From Floor 0, one press of the minus takes you to the toy shop. So what is the toy shop's number? The honest instinct is to call it Floor 1. It is one floor away from the entrance, after all. But the food court is already Floor 1. Two different floors with the same name — and a lift button marked 1 would have no way of knowing which one you meant.

5:25So the toy shop is Floor minus one. Video games is minus two. The cinema, minus three. The ghost house, minus four. And the dinosaurs, minus five. That minus is not doing arithmetic. It is painted on a door. Put two of those labels side by side. Plus two, and minus two. The two is doing exactly the same job in both of them. It says: two floors away from the entrance.

5:58The sign in front is doing all of the remaining work. It says which side. A number written with a plus in front of it is called a positive number. A number written with a minus in front is called a negative number. And that is the whole definition. Not big or small, not real or imaginary. Just which sign is written in front. It is also the sign that keeps every floor's name its own. Rub the signs out and ten of these twelve floors would be sharing a name with some other floor.

6:36Here is something worth noticing about minus three. It is the number on the cinema door. That is a place. You can stand on it and buy a ticket. But minus three is also the answer to how far did you move, when you press the minus three times from the entrance. That is a movement — an arrow, three floors long, pointing down. The same symbol, doing two completely different jobs.

7:05And the two agree only because of where Floor 0 was put. Starting from zero, the floor you arrive at is exactly the movement that got you there. From the ghost house up to sports is nine floors. From the dinosaurs up to space is eleven, and eleven floors is the longest ride this lift can give you. One number in this building is still missing its sign. Floor 0.

7:35Should it be plus zero, or minus zero? Think about what a sign actually says here. It says which side of the reference level a floor sits on. But Floor 0 is the reference level. It is not above the entrance and it is not below the entrance. It is the entrance. It has no side of itself to be on. And there is a second reason, just as good. Plus zero and minus zero would be two different names for one floor — which is precisely the thing the signs were brought in to stop.

8:11So zero gets no sign at all. That is forced on us. It is not a decision anybody made. Look at what we have now. The picture that used to stop dead at zero has been extended. There are numbers on the other side, they have proper names, and each of those names is doing two jobs at once. This building runs from minus five at the bottom all the way to plus six at the top.

8:41Which lets you ask a question you could not have asked ten minutes ago. Does it stop there? And a bigger one. We can name these numbers now. Can we actually calculate with them? What is minus four, plus six? Is minus three larger or smaller than minus two? Nothing said so far settles either of those. That is what comes next.