Placing a large number on the number line by feel

0:00Chapter Three keeps teaching the same lesson from new angles, and here it comes again, on a number line. You have used number lines before, with small numbers. One, two, three, four, five, evenly spaced. Now the numbers are big, and something changes. Nobody is going to label every mark. So a line arrives with a handful of labels on it, and every other position is something you have to work out.

0:30And you can work it out. That is the whole point of this section. Two labelled marks tell you what one step is worth. And once you know the step, the entire line is decided. So this is not about eyeballing, or having a feel for numbers. It is about reading the step off the labels, and then counting. Here is the line your book prints. It has ten marks on it.

0:58A thousand. Two thousand. Three thousand. Four, five, six, seven, eight, nine thousand. And ten thousand at the far right. One mark for every thousand, all the way along. Which means every gap between neighbouring marks is worth a thousand. That is a lot of room. A thousand whole numbers live inside each gap, and not one of them gets a mark of its own. So placing a number is really two questions. Which gap does it fall into? And then, where inside that gap?

1:34Almost everybody answers the first question and forgets the second one. Your book places two numbers for you, with little curved pointers. Two thousand one hundred and eighty. And two thousand seven hundred and fifty-four. And look where they both go. Between the two thousand mark and the three thousand mark. The same gap. Both of them. Which is the first useful thing this exercise does to you. Two numbers that do not look especially alike land inside one interval.

2:11So the gap cannot be the answer on its own. We have to say where inside it. Let's blow that gap up so it fills the screen. The left end is two thousand. The right end is three thousand. The gap is a thousand wide. Now, two thousand one hundred and eighty. How far in is it? It is a hundred and eighty past the left end. Out of a thousand.

2:38A hundred and eighty out of a thousand is under a fifth of the way across. So it sits just clear of the two thousand mark. Nowhere near the middle. And that is the mistake to avoid. People drop a number in the middle of its gap because the middle is easy to find. Ask how far through instead. Now the other one. Two thousand seven hundred and fifty-four. Seven hundred and fifty-four past the left end, out of a thousand.

3:10That is just over three-quarters of the way across. Much closer to the three thousand mark than to the two thousand mark. And notice what we just did. We never measured anything. We read the last three digits and treated them as how far through the gap to go. That works because the gap is exactly a thousand wide, so the last three digits are the distance from the left-hand mark.

3:38It is worth saying that out loud, because it is doing all of the work. Then your book gives you nine more numbers and asks you to place all of them. Fifteen hundred. Three thousand six hundred. Nine thousand nine hundred and fifty. Nine thousand five hundred and ninety. One thousand and fifty. Three thousand and fifty. Five thousand and thirty. Five thousand three hundred. Eight thousand four hundred. And they are not random. They arrive in pairs that share a gap.

4:16One thousand and fifty, and fifteen hundred, both live between one thousand and two thousand. But one is five per cent in, and the other is halfway. Three thousand and fifty and three thousand six hundred share the next gap up. Five per cent in, and sixty per cent in. Five thousand and thirty and five thousand three hundred share a gap too, and this pair is the tightest of the lot. Three per cent in, and thirty per cent in.

4:49Same gap, wildly different places inside it. Which is exactly what this exercise wants you to feel. Now the pair your book really wants you to look at. Nine thousand five hundred and ninety, and nine thousand nine hundred and fifty. The same four digits. Nine, nine, five and zero. Just written in a different order. So people assume the two numbers are near each other. They are not. Both of them land in the last gap, between nine thousand and ten thousand. So go straight to the second question. Where inside?

5:30Nine five nine zero is five hundred and ninety in. That is just past halfway. Nine nine five zero is nine hundred and fifty in. That is almost at the far end. Three hundred and sixty apart, which is more than a third of the whole gap. On the line you could never confuse them. Sharing digits buys you nothing. Only distance from zero shows on a line, and the order of the digits is what decides it.

6:03Now your book turns the exercise round, and this is the better half of the section. Instead of a fully labelled line, you get four lines with almost nothing on them. Ten marks each, and only two or three labels. Everything else you have to work out. Start with the first one. Two marks are labelled: two thousand and ten, and two thousand and twenty. And they are two marks apart.

6:32So going from one to the other costs two steps, and covers ten. Ten divided by two is five. Each step is worth five. And now the line is finished. Every other mark follows. Two thousand and five, two thousand and fifteen, two thousand and twenty-five, and on to the end. One subtraction and one division, and a blank line becomes a full one. Second line. The two labels are nine thousand nine hundred and ninety-six, and nine thousand nine hundred and ninety-seven, right next to each other.

7:12So the step is one. The simplest possible line. But watch what happens as you walk along it. Nine thousand nine hundred and ninety-eight. Nine thousand nine hundred and ninety-nine. And then ten thousand. The number of digits changes. Four digits become five. The spacing does not change at all. It is still one step, exactly as wide as every other step on the line. That is worth pausing on, because plenty of people quietly believe the line does something dramatic at ten thousand.

7:50It does not. Only our way of writing the number changes. Third line, and this one hands you something extra. Two labels next to each other: fifteen thousand and seventy-seven, and fifteen thousand and seventy-eight. So the step is one again. But there is a third label further along. Fifteen thousand and eighty-three, five marks past the seventy-eight. Now, you did not need that label. You already had the step.

8:22So use it as a check. Fifteen thousand and seventy-eight, plus five steps of one, is fifteen thousand and eighty-three. It matches. Which means the step you found was right. Whenever a problem hands you more information than you need, that extra is a free check — and you should always spend it. Fourth line. Two labels, next to each other: eighty-six thousand seven hundred and five, and eighty-seven thousand seven hundred and five.

8:56Subtract. The step is a thousand. And now put this line beside the second one, the one that stepped by one. They look identical. Ten marks, evenly spaced, the same drawing. One of them advances by one each time. The other advances by a thousand. Nothing about the picture tells you which. Only the labels do. And that is the honest summary of this whole section. A line is not a fact about numbers. It is a fact about what somebody wrote on it.

9:32Last instruction. On each line, circle the smallest number and draw a box around the largest. And now it is trivial, because we have labelled everything. On a number line, smallest just means furthest left, and largest just means furthest right. You do not compare anything. You look. That is the real gift of putting numbers on a line. Ordering stops being an arithmetic job and becomes a visual one.

10:04Nine five nine zero against nine nine five zero took real attention as digits. As positions, one is simply left of the other. Next time we count instead of placing: how many numbers have a given number of digits, and what digit sums are quietly doing.