Abscissa and ordinate, and what a coordinate pair actually records

0:00Here is a line, and a point somewhere on it. One number locates that point completely - say how far along, and you are done. Now lift the point off the line and into the plane. Say five. Which point? Five could be five across, five up, five along some diagonal. One number names a whole line's worth of places, and you needed one place. So a plane needs two numbers. That is the entire reason a coordinate pair has two entries, and it is worth saying out loud before anything else.

0:37And once every point carries two numbers, something larger becomes possible. A question about a picture turns into arithmetic, and arithmetic turns back into a picture. Sample a first-degree equation at nine values and the nine points lie in one straight line. Sample a squared one and they do not - they bend. The differences of the differences give it away: nothing but zero for the line, nothing but four for the squared one. Traffic runs both ways.

1:15Two numbers, then. But two numbers taken how? Lay down two rulers at right angles. One horizontal, one vertical, crossing at a place we will call the origin. Now take a point and ask two questions of it. How far is it from the vertical ruler? And how far is it from the horizontal one? Read those two answers off, in that order, and you have the pair. Notice which is which, because this is where nearly everything goes wrong. The FIRST number is measured across to the VERTICAL line. The second is measured across to the horizontal one.

1:57Each reading is taken against the ruler it is not named after. Call that the crossed reading. It has to be crossed. If both numbers were measured from the same ruler they would be the same number twice, and you would be back to knowing one thing. The two entries have proper names. The first one is the abscissa. The second is the ordinate. They are also called the x-coordinate and the y-coordinate, and those names are easier to say.

2:30But the plain names hide the crossing, and the crossing is the point, so here is a way to keep it straight. The x-coordinate is not the distance from the x-axis. It is the distance from the OTHER one. So keep it straight this way: the first number is the one that a vertical slide leaves alone. Slide a point sideways and its first number changes; slide it up and down and its first number does not.

3:00Take the point minus five, minus three. Drop a perpendicular to the vertical line and the leftover, measured sideways, is minus five. That is the abscissa. Drop a perpendicular to the horizontal line and the leftover, measured up and down, is minus three. That is the ordinate. Two readings, two directions, neither one derived from the other. That last phrase is worth testing rather than trusting. Take a grid of a hundred and thirty-two points - twelve first entries against eleven second ones.

3:41Ask how many of them share their first number with some other point in the grid. All hundred and thirty-two of them do. Ask how many share their second number. All hundred and thirty-two again. So neither reading, on its own, tells you which point you have. Say the abscissa is four and you have named eleven different points - a whole vertical line's worth. Say the ordinate is five and you have named twelve, a whole horizontal line's worth.

4:13Say both and you have named exactly one. Across the whole grid, no two different points agree in both entries. Two readings, and it takes both. Some points are special, and it is worth being precise about why. Put a point on the horizontal ruler itself. How far is it from that ruler? Zero. So its second reading is zero, and the pair looks like something, zero. Put a point on the vertical ruler and the same argument runs the other way: its first reading is zero, and the pair is zero, something.

4:53The origin is where both readings vanish. Zero, zero. Now the trap. It is tempting to say that a point on the horizontal axis has no second coordinate. It has one. It is the number zero, and that is not the same as not having one. The difference shows up the moment a formula asks you to substitute. There is a slot, and something goes into it. What goes in is zero.

5:24One more thing about the pair before we use it: it is ORDERED. Three, five and five, three are not two ways of writing the same thing. They are two different points. Three, five sits three across and five up. Five, three sits five across and three up. Look at them and they are plainly in different places. On the grid of a hundred and thirty-two, swapping the two entries moves a hundred and twenty-three of the points somewhere else.

5:55It leaves nine of them alone, and those nine are exactly the ones whose two entries were already the same number - the diagonal. So a pair is not a set of two numbers. The comma is doing work. First entry, then second entry, and the order is part of the name. Which sounds fussy until you watch what happens when a drawing depends on it. Here is a list of pairs. Plot each one, and join them up in the order they are given.

6:28Four eight. Three nine. Three eight. One six. One five. Three three. Six three. Eight five. Eight six. Six eight. Six nine. Five eight. And back to the start. That is an outline, twelve points and twelve strokes. Then three small triangles: one near three and a half, seven; one near five and a half, seven; and one near four and a half, four. And four straight strokes running out to the edges, two from the left of the middle triangle and two from the right.

7:04Twenty-five distinct points in all, twenty-one of them lettered, joined by twenty-five strokes. Nothing about that list looks like a picture. Plot it, and it is a face. Two spikes for ears, two small triangles for eyes, one for a nose, and four whiskers. The nice thing about the face is that you do not have to take my word for it, because the drawing has a property you can check by arithmetic.

7:34Take every point and replace its first entry by nine minus that entry. Leave the second entry alone. That is a reflection in the vertical line at four and a half. Every one of the twenty-five points lands on another point of the drawing. Four eight goes to five eight. Three nine goes to six nine. One five goes to eight five. And here is the stronger statement: every one of the twenty-five STROKES lands on a stroke that was already drawn. Three of them land exactly on themselves.

8:13That is worth separating out. Reflect instead in the line at four - nine minus becomes eight minus - and eleven of the twenty-five points still land on the drawing. But not one of the twenty-five strokes does. Points landing back is cheap. Strokes landing back is the symmetry. Now break one pair. Write the third point, three eight, backwards as eight three, and the outline's symmetry drops from twelve points to ten. One comma, and the face is crooked.

8:49A word about the word distance, which we have been using loosely. A distance cannot be negative. A coordinate can, and does - on that grid of a hundred and thirty-two points, fifty-one carry a negative reading. Take minus five, minus three again. Its abscissa is minus five. Its distance from the vertical line is five. Those are different numbers. The minus sign is not measuring anything extra. It is recording a direction: which side of the ruler the point is on.

9:25So a coordinate is a signed displacement, not a distance. It carries a size and a direction, and the sign is the direction. How do you get from one to the other? Square it. Minus five squared is twenty-five, and so is five squared. Squaring throws the direction away and keeps the size. Which is exactly why every distance rule you are about to meet squares a coordinate difference before it does anything else with it.

9:58The two rulers cut the plane into four regions, and the signs tell you which one you are in. Both readings positive: the first quadrant, up and to the right. First negative, second positive: the second. Both negative: the third. First positive, second negative: the fourth. So minus five, minus three is in the third quadrant, and you can say that without plotting it. But four regions is not the whole answer, and the grid says so. Sort all hundred and thirty-two points by where they sit.

10:37Sixty-four in the first quadrant, twenty-four in the second, six in the third, sixteen in the fourth. That accounts for a hundred and ten of them. The other twenty-two are on the rulers themselves - eleven on the horizontal one, ten on the vertical, and one at the origin. A quadrant needs both readings to be non-zero. There are seven places a point can be, not four, and the axes are not a rounding error.

11:10Now put the readings to work. Two points on the horizontal ruler: one at four, zero, and one at six, zero. How far apart? Six take away four. Two. Two points on the vertical ruler: zero, three and zero, eight. Eight take away three. Five. That feels obvious, and it is - but it is worth naming why it works, because the reason is not the one people give. It is not that the points are on an axis. It is that they AGREE in their other reading. Both of the first pair have second reading zero, so the whole separation lies along one direction and a single subtraction catches it.

11:56Along one ruler, the ruler IS a number line, and subtracting on a number line is what you already know how to do. So: two, and five. Both of them read straight off the coordinates with nothing but a subtraction. And now the two separations on that same figure that will not come out that way. From four, zero across to zero, three. And from six, zero across to zero, eight. Neither pair agrees in either reading.

12:29Try subtracting anyway. Take the first entries: four. Take the second entries: three. Add them: seven. The true separation is five. None of the three is right. That is not bad luck. Run all three of those moves over every pair of points in the hundred-and-thirty-two grid - eight thousand six hundred and forty-six pairs of them. On the one thousand three hundred and eighty-six pairs that agree in a reading, one of the subtractions lands. On the other seven thousand two hundred and sixty, not one of the three ever does.

13:10Subtracting coordinates gives you a separation exactly when the two points share a reading, and never otherwise. That is the boundary. Four and three, and the answer five. Six and eight, and the answer ten. Those two answers need a rule that subtraction cannot supply, and building that rule is where we go next.