Laying the integers out in order, and why −8 is less than −2

0:00Four people are in the building, one in each of four different shops. One is in the art centre, one in the sports centre, one in the cinema, and one in the toy shop. Only the first is told to you directly. The art centre is Floor plus two. The other three you read off the building. Sports is plus five. The cinema is minus three. The toy shop is minus one.

0:28And the question is: which of the four is lowest? Look at the shaft and it takes about a second. The cinema. But notice what you did to answer it, because you did not do any arithmetic. You looked. Start with two floors above the entrance. Plus three and plus four. Which of those is less? Plus three. And you have two different ways of knowing that, and both of them work here.

0:56The first way: three is a smaller number than four. The second way: plus three is lower down the shaft than plus four. They give the same answer, so you have never had any reason to ask which of the two you were actually using. Hold on to that. Now go below the entrance. Minus three and minus four. Which of those is less? Try the first way. Three is smaller than four, so minus three must be the smaller one.

1:29Now try the second way. Find them both on the shaft. Minus four sits one floor further down than minus three. So minus four is lower. And that is the opposite answer. Two methods that have agreed with each other your entire life have just disagreed. One of them is wrong, and you have to decide which. Here is the answer, and it is not the one most people expect.

1:58Less than was never about size. It was about position. When you first met those words, the picture was a line that started at zero and ran one way only. On that picture, further along and further from zero are the same direction. So bigger number and higher up meant the same thing. You could use either one and never once find out which you had picked. Extend the picture past zero and the two come apart. Less than stays with position, because position is what it always was.

2:35Two things follow immediately, and neither of them needs a rule. Every negative number is below zero. Not because of its sign, but because that is where it is drawn. And every positive number is above zero, for exactly the same reason. So any negative is less than any positive, no matter how large its digits happen to look. Minus twenty five is less than plus one. Twenty five is a far bigger number than one, and it makes no difference whatsoever.

3:11Zero sits between the two, belonging to neither of them. And it is no longer the smallest thing on the picture. Now strip the building down to what is really doing the work. A line, with evenly spaced marks. Eight of the marks are lettered, A at the bottom running up to H at the top. Only three of them carry numbers. A is minus twelve. D is minus one. E is plus one.

3:41There is one unlettered mark sitting between D and E, and it has to be zero. That is what fixes everything else. The rest you count. B is three marks up from A, which puts it at minus nine. C is six up from A, minus six. F is one above E, so plus two. G is five above E, plus six. And H is ten above E, plus eleven. No arithmetic beyond counting.

4:13Now twelve comparisons, and watch what they do to the picture. Minus two is less than plus five. Minus five is less than plus four. Minus five is less than minus three. Zero is greater than minus four. Zero is less than plus four. All five of those sit comfortably on the shaft as it is drawn. The sixth does not. Plus six against minus six. Minus six is one floor below the lowest floor there is, so the shaft has to grow by one before you can even ask.

4:51Then six more that no shaft could hold. Minus ten is greater than minus twelve. Plus seventeen is greater than minus ten. Zero is greater than minus twenty. Plus nine is greater than minus nine. Minus twenty five is less than minus seven. Plus fifteen is greater than minus seventeen. Which raises a question the building cannot answer. Is there a lowest floor? This one has a lowest floor, at minus five. But it has one because somebody built it that way.

5:30The mine went down to a hundred and seventy five metres below the surface, and there was nothing stopping it going further. And whatever number you name as the lowest, one less than it is a number as well. So there is no bottom, for exactly the same reason there is no top. The positives, the negatives and zero, all taken together, are called the integers. That is the whole collection, running away from zero in both directions and never stopping.

6:05Now the picture changes shape, and nothing whatsoever about it changes. Take the shaft and turn it a quarter turn, about zero. Watch the marks as it goes. Not one of them moves relative to any other. Every gap keeps its size, and every number keeps its neighbours. When it comes to rest it is lying flat, and it has a name. That is the number line. The turn has to go one particular way round, and this is it. The negatives finish on the left.

6:39Turn it the other way and they land on the right, and every picture you draw from then on reads backwards. Two things to say about the flat picture. The first is a convention. Above zero, the plus sign gets dropped. Plus three is written as three. Plus seven as seven. Nothing about the number changes. Only what is written down. And it is safe, because negatives never drop their sign. So a bare three could not possibly be anything else.

7:14The second thing is about direction. Lower has become further left. Every sentence you learned about the shaft still holds, with left in the place of down. Minus five is left of minus three, and that is exactly why it is less. Try it with numbers of your own. Any three positive ones and any three negative ones. Say minus fourteen, minus six, minus two, three, eight and fifteen. Mark them on the line, and then write them out in increasing order.

7:50Minus fourteen, minus six, minus two, three, eight, fifteen. Which is simply left to right along the line, exactly as they sit. You are not sorting them. You are reading them off. And notice that the negatives come out in what looks like the wrong order, largest digits first. They are not in the wrong order. They are where they are. One last problem, and this one is practical. Suppose you want to put minus one hundred and two hundred and fifty on the same picture.

8:25Marked at every unit, that is three hundred and fifty gaps. Squeeze all of those across the width of a page and they are far too close together to tell apart. So instead you draw a line, put an arrowhead on each end, and mark exactly one thing on it. Zero. That is an unmarked number line, and it is not an unfinished one. It promises you two things and no more. Where zero is, and that the order runs left to right. Which turns out to be everything you actually needed.