The four quadrants, and reading a point's signs off its position

0:00So far every point we have marked has been sitting on one of the two lines. That was on purpose: it kept one of the two numbers at zero. Now let us let go of that. Take a point somewhere out in the open, off both lines, and ask what its pair looks like. The answer turns out to be completely determined by two things, and neither of them is anything you have to remember.

0:29Here are the two lines, and here is the only thing they do to the plane. They cut it. The vertical one splits everything into left and right. The horizontal one splits everything into below and above. Two cuts, made independently. Ask any point in the open two questions. Is it to the right of the vertical line, yes or no. Is it above the horizontal line, yes or no.

0:59Two questions, two answers each, so four possible pairs of answers. Not three, not five. Four. And that is why the plane comes out in four pieces. The four pieces have numbers, and the order they run in is worth watching. Start where both answers are yes, up and to the right, and call that the first region. Then turn anticlockwise. Left and up is the second. Left and down is the third. Right and down is the fourth.

1:33It does not begin at the bottom left the way you would number rows in a table. It begins where both numbers are positive, and it turns the way an angle turns. Now here is the part people try to memorise, and there is nothing there to memorise. The first number is positive exactly when the point is to the right. The second is positive exactly when the point is above.

2:03That is not two rules. It is the same two questions we already asked, written down as signs instead of as yes and no. So plus plus, minus plus, minus minus, plus minus, going round. If you can see where the point is, you have already read its signs. If you cannot see it, no amount of recitation will help. There is a second way to say the same thing, and it fixes a mix-up that costs people marks for years.

2:36Drop a perpendicular from your point to each line. The length of the drop to the horizontal line is the second number. The length of the drop to the vertical line is the first. Read that again, because it is crossed over. The first number is measured from the vertical line. The second is measured from the horizontal one. The name of a coordinate tells you which line you read it against, not which line you measure it from. Get those two the wrong way round and every plot you draw will be a reflection of the one you meant.

3:16Let us put two points down. One at minus five, three. One at three, minus five. Same two numerals. Opposite orders. And look where they land: one up and to the left, one down and to the right. Diagonally opposite regions. Every question you can ask them comes out reversed. Is it right of the vertical line? No for the first, yes for the second. Above the horizontal? Yes, then no.

3:46Both answers flip, so the point moves to the region diagonally across. That is what exchanging the two slots does, every single time. One thing the four regions do not include: the two lines themselves. Take a point and slide it towards the vertical line. Its first number shrinks. Right up until the moment it lands, it is in a region. Then it lands, the first number becomes zero, and zero is neither positive nor negative. The point is not in the region it came from and not in the one it is heading for.

4:27The lines are the boundaries. A point sitting on one is in no region at all, and that is not a technicality. It is why every statement about regions is about points with neither number zero. Now we are going to use all four regions for something real, and the first move is one people skip. Before you mark a single point, decide how far your lines have to reach. Across: from minus seven to thirteen. Up and down: from minus fifteen to twelve.

5:04That is twenty units wide and twenty seven tall, and at a centimetre to the unit it is a sheet twenty by twenty seven centimetres. Why those numbers? Because of what has to fit. There is one spare unit at the left, one at the right, two at the top, and at the bottom there is nothing spare at all. The drawing reaches minus fifteen exactly. A range is not a decoration you add afterwards. Get it wrong and you find out halfway through, with half a plan drawn and nowhere to put the rest.

5:42Here is what has to fit. A whole home, with the origin sitting in one corner of one room. The bedroom runs from zero to twelve across and zero to ten up. Both numbers positive throughout. It is entirely in the first region. The bathroom is on the other side of that corner. Six feet across and nine feet up, and because it is to the left, every one of its first numbers is negative. Second region.

6:14And the dining room is below. Eighteen feet long, fifteen feet deep, hanging under the horizontal line and straddling the third region and the fourth. This is the answer to why negatives exist. Not for symmetry, and not to make the picture look balanced. Somebody put the origin in the corner of a bedroom, and the house carried on in the other directions. Once the whole plan is in coordinates, things you would have measured become things you can work out.

6:49A small table has three feet marked: eight nine, eleven nine, and eleven seven. Where is the fourth? You do not need to look. Two feet share a first number of eleven, two share a second number of nine, so the missing one takes the leftovers: eight and seven. The table is three by two, six square feet. Now the shower, which is a better puzzle. Its four corners are minus six six, minus three six, minus two nine, and minus six nine.

7:24Two of its sides are horizontal, one three feet long and one four, three feet apart. One side is vertical. And one is slanted, which is why this is not a rectangle. It is a trapezium, and its area is ten and a half square feet, out of a bathroom floor of fifty four. Here is a question that sounds physical and turns out to be arithmetic. A door is hinged on the vertical line at zero, one and a half, and it swings inwards. The leaf is two and a half feet long. The wardrobe stands with its near edge at three across, between zero and two up.

8:08Will the door hit it? Swing the door and the tip traces part of a circle, always exactly two and a half feet from the hinge. So the question is how far the wardrobe is from the hinge. The nearest corner of that wardrobe is exactly three feet away. Two and a half is less than three. The door clears it, by half a foot, and it never gets past two and a half across.

8:35Which also tells you the thing the drawing alone would not. Make that door three feet wide and it just touches. Any wider and it sweeps in. So a wider door means moving the wardrobe, or hanging the door on its other edge. Four regions, and not one of them had to be learnt. There are four because there are two independent questions, and two answers each. The signs in each region are those two answers, written down in order.

9:08The lines themselves belong to none of them, because zero is on neither side. And the negatives are not an exercise. They are what happens when somebody chooses a corner to count from, and the world continues past it in every direction.