Placing a rational number on the line, and distance as |a − b|

0:00A line, and nothing on it yet. To put numbers on this line you have to make exactly one arbitrary decision, and after that you are not allowed to make any more. Choose a point. Call it nought. Nothing about the line made that point special. You chose it. And now the freedom is over, because every other label will be forced by that choice and by one more rule.

0:28Move the origin somewhere else and every tick moves with it, but the distance between neighbouring ticks does not change. That is the shape of the whole topic: one decision, and then consequences. The second rule is that the steps are all the same size. Lay down seven of them going rightwards and label them one to seven. That sounds like bookkeeping, and it is not. Because the steps are equal, the label on a tick is the number of steps from the origin. The label and the measurement are the same thing.

1:08Let the step drift by one part in a hundred and the seventh tick is no longer at seven, and the error grows the further out you go. Equal spacing is what turns a label into a measurement instead of a name. Now the question this whole idea exists to answer. Where is p over q? Not roughly. Exactly. The rule has two instructions in it and a direction. The lower number cuts: it says how finely to slice the unit. The upper number walks: it says how many of those slices to take. And the sign steers.

1:49Say it in that order every single time, because the commonest error here is swapping the two. Across five hundred and eighty eight fractions, walking as the rule says lands on the right point every time. Swap the cutting and the walking and you get the right answer twenty four times, and only when the two numbers happen to be equal. Three quarters. Watch the rule run. The lower number is four, so cut the unit from nought to one into four equal parts. The upper number is three, so take three of them. One. Two. Three. Stop.

2:31That is the point, and notice what did not happen. Nobody estimated. Nobody looked for where three quarters usually sits on a picture. Two numbers, two instructions, and the position came out. Had you swapped them, cutting into three and walking four, you would have landed on four thirds, which is past one and is not the number you were placing. Nine quarters. A fraction bigger than one. The reason people believe a fraction like this cannot go on the line is that they are still looking inside the first unit. It is not in the first unit. So find the right unit first.

3:15Nine quarters is two and a quarter, so it lives between two and three. Now the construction runs exactly as before, except inside that interval: cut from two to three into four, and walk one. Same rule, different stretch of line. The mixed number was never a change of subject. It is how you find out which stretch to cut. Minus seven quarters. Same construction, opposite direction. Cut into quarters and walk seven of them leftwards from nought. One motion, not a walk to the right followed by a flip at the end.

3:56If you would rather start from a whole number, minus seven quarters is minus one and three quarters, so stand at minus one and take three quarter steps left. Both routes land on the same point. Take those same three quarter steps to the right instead and you arrive at minus one quarter, missing by one and a half. The direction is not decoration. Here is what a line looks like once you stop treating whole numbers and fractions as different kinds of thing.

4:31Twenty labels. Seven above the axis and thirteen below. Whole numbers, and fractions cut into quarters, fifths, sevenths and eighths, all on one line, each sitting on its own single point. Why split them above and below? Because the two closest labels anywhere on this line are one eighth apart, and at that spacing two labels written side by side would collide. Split them across the axis and the closest two on the same side are two fifths apart, which is a bit over three times the room. The split is not decoration either.

5:14Now a problem the rule as stated does not solve. Place two thirds, minus five quarters, and one and a half on one line. The rule handles one fraction at a time, and thirds and quarters do not land on the same ticks. So you need a cutting that serves all three at once. Try them in turn. Halves, thirds, quarters, sixths: none of them works for all three until you reach twelfths.

5:43In twelfths the three numbers read eight, minus fifteen, and eighteen. One cutting, three positions, and now they can be compared by eye. Decimals join the same way. Nought point five three two is one hundred and thirty three over two hundred and fifty, which sits exactly on a tick once the unit is cut into two hundred and fifty. Once every one of these numbers has a definite point, you can start measuring between points. Begin with the measurement from the origin.

6:20Five thirds is five thirds away from nought. Minus five thirds is also five thirds away from nought: the same distance, on the other side. And nought is no distance at all from itself. That quantity has a name and a notation. It is the absolute value, written between two upright bars, and the definition that matters is this one: it is the distance from nought. There is a rival description that most people carry instead. Absolute value means delete the minus sign.

6:59That gives the right answer for every case you are likely to meet, which is exactly why it survives. It is still the wrong idea, and here is how you can tell. Ask why an absolute value is never negative. Under the deleting rule there is no reason. It is simply what deleting happens to do. Under the distance definition the answer arrives immediately: a length cannot be less than nothing. And one number is its own absolute value and its own opposite at the same moment, which is nought, the only point that is no distance from itself.

7:43Now the distance between two points when neither of them is the origin. Take three and minus four. Subtract one from the other and you get seven. Subtract the other way round and you get minus seven. But the gap between them is seven either way. You can count the units along the line and check. So the distance is the difference with the sign taken off it, which is the absolute value again, now doing the job it was built for: reporting the length of a segment however you approached it.

8:20And this is why the bars are not tidying up after the arithmetic. Over two hundred and twenty five ordered pairs of points, the bare difference comes out negative one hundred and five times. It agrees with the distance on one hundred and twenty of them. That is often enough that people stop distinguishing the two, and then meet a negative distance the first time the order flips on them.

8:48On all two hundred and ten pairs of two different points, the difference depends on which one you subtract first. The distance never does. A gap has one length. The absolute value is what makes the formula behave the way the thing it measures already behaves.