Two axes, an origin, and why the order of the pair matters

0:00Here is a number line. Every point on it has exactly one address, and one number is enough to say which point you mean. Now put that line along the bottom wall of a room. It can tell you that something stands eight feet from the corner, along that wall. But the room is not a wall. Lift your finger off the floor and move it into the middle of the room, and the line has nothing left to say.

0:28One number buys you one direction of freedom. The room has two. So we are one number short, and no amount of care with the first one will make up for it. The repair is not clever. Take a second number line, identical to the first, and stand it up at a right angle to it. Now every point in the room can be asked two questions instead of one. How far across, and how far up.

0:58The two lines cross at exactly one point, and that point is where both answers are zero. We give it a name: the origin, written with the letter O. Notice what just happened. The origin is not a feature of the room. Somebody chose it, by choosing where to lay the two lines, and once it is chosen everything else follows. Two crossed lines are not yet a frame. They need two more agreements before a single number means anything.

1:33First, equal steps, marked the same size on both. Second, a direction on each line that counts as positive: to the right on the horizontal one, upward on the vertical one. The reversed directions get a minus sign, and that is all a minus sign is doing here. It is not a smaller number. It is the same distance, counted the other way. And here is a thing worth being clear about. The first number is measured from the vertical line, not from the edge of your paper. Slide the whole picture across the page and not one coordinate changes.

2:17Let us mark some points. Here is O, at zero and zero. Here is B, four and a half steps to the right and none up. Here is E, two point nine steps to the left. Here is H, four steps up. And here is G, four and a half steps down. Now look at what these five have in common, because it is not an accident. Every one of them is sitting on one of the two lines.

2:45That is a deliberately narrow start. These are the easy cases, and we are going to use them to get the rules exactly right before we let a point wander off into open ground. Two of those numbers should bother you slightly. Four and a half. And two point nine. There is a strong instinct that coordinates are counts of whole steps. It does not. Zoom in between the minus three mark and the minus two mark, and cut that single step into ten. E sits one tenth of the way along, from minus three heading towards minus two.

3:26You can say the same thing from the other end: nine tenths of the way back from minus two towards minus three. Both readings are correct, and they add to one, which is the whole step. The line carries every distance, not only the ones we bothered to tick. Because those five points sit on the lines, they fall into two families, and each family has a signature you can read straight off the pair.

3:56Anything of the form something, zero is on the horizontal line. To the right of the origin when the first number is positive, to the left when it is negative. Anything of the form zero, something is on the vertical line. Above the origin when the second number is positive, below it when it is negative. So here is a question with a very tidy answer. Which points belong to both families at once? A point in both must have its first number zero and its second number zero.

4:34There is exactly one such point, and it is the origin. Which is another way of saying the two lines meet once, and we already knew that, and now the arithmetic says it too. Now the part that matters most, and that people get wrong for years afterwards. A pair of coordinates has two slots. The first slot holds the answer to how far across. The second holds the answer to how far up. They are answers to different questions.

5:08And here is the trap. Nothing about the numeral three records which question it answered. Written down on its own, a three is just a three. The order of the slots is carrying that information, and it is the only thing carrying it. Take the order away and you have destroyed half of what the pair was telling you. Watch. To plot three, four, go three across and then four up. To plot four, three, go four across and then three up. Same two numerals. Two different places in the room.

5:46So exchanging the two numbers usually moves the point. When does it not? Draw the line that runs through the origin at forty five degrees, the line where the across answer and the up answer are equal. Exchanging a point's two numbers reflects it in that line. That is not a rule we are imposing. Drop a perpendicular from the point to the line, continue the same distance past it, and where you land is the point with its numbers exchanged. Every time.

6:21And a reflection leaves alone exactly the points that were already on the mirror. So the swap changes nothing precisely when the two numbers were equal to begin with, and it changes something in every other case. Here is a concrete pair. Minus five, three sits up and to the left. Three, minus five sits down and to the right. Same two numerals, opposite corners of the picture. A small note on writing, because you will meet both forms.

6:56You can write B equals four and a half, zero. Or you can write B, then the pair, with no equals sign at all. They say the same thing. The second is the one to use when you are actually marking points, because you are labelling a dot on a drawing rather than making a statement about a quantity. Now take the frame back to the room we could not describe. Lay the origin on the near left corner. Run the horizontal line along the bottom wall and the vertical one up the left wall.

7:33Instantly the corners name themselves. Zero, zero. Twelve, zero. Twelve, ten. Zero, ten. And the floor area falls straight out of those, one hundred and twenty square feet, without measuring anything again. The doorway in the bottom wall runs from eight, zero to eleven and a half, zero. Both ends sit on the horizontal line, so the width is a subtraction: three and a half feet. The bathroom doorway runs from zero, one and a half to zero, four. Both ends on the vertical line. Two and a half feet.

8:13One foot narrower. Forty two inches against thirty. That is a difference you would feel carrying anything through it, and we got it from four numbers and no ruler. One last thing, and it is the honest limit of what we have built so far. Every point we have marked has been sitting on a line. Look at how much of the picture that leaves untouched. Tick the horizontal line from minus seven to seven and the vertical from minus five to five, and the window we have drawn holds one hundred and sixty five whole number points.

8:52Twenty five of them are on a line. That is fifteen along one, eleven along the other, and one shared, because the origin got counted twice. Which leaves one hundred and forty points sitting in open ground, off both lines, and we have not marked a single one of them. The machinery already reaches them. Two questions, two answers, and every one of those points has a pair waiting for it.

9:23So what did the second line actually buy us. Not a picture. A way of asking two independent questions about the same point, and a rule for writing the two answers down so that neither can be mistaken for the other. The origin is a choice. The unit is a choice. The positive direction is a choice. And the order of the two slots is a choice. Everything after that is forced.