The plane as pairs of numbers

A point is not just a mark on a page — it is exactly one pair of numbers.

Every lesson so far has drawn points wherever they happened to land. From here, every point gets an address.

A labelled x and y axis, crossing at a fixed origin. One draggable point, P. A live reading counts how many steps right or left, and up or down, P sits from the origin.

A labelled x and y axis crossing at a fixed origin, with one draggable point P. A live reading gives how many steps right or left, and up or down, P sits from the origin, and states that pair of numbers as P's coordinates. This needs JavaScript switched on.

Drag P anywhere. Its position is always exactly this one pair of numbers.

Never two different pairs for the same spot, and never one pair for two different spots — the match between point and pair runs both ways.

The crossing point is the origin. The pair of numbers that names a point’s distance from it, across then up, are its coordinates — written $(x, y)$ , in that order, always.

Two independent counts, not one

Every point on the plane sits some number of steps right or left of the origin, and some number of steps up or down from it. Those two counts are independent of each other: knowing how far across a point is says nothing about how far up it is.

(x, y) \quad \text{where } x = \text{steps right (negative if left)}, \quad y = \text{steps up (negative if down)}

Here $x$ is not one fixed unknown to solve for, the way it is in an equation like $2x + 1 = 7$ ; it varies across every point on the plane, not just one line. It’s also G0.1’s “distance from a fixed point” idea, applied in two directions at once instead of one.

A single number is not enough to say where something is once “where” has more than one direction to it — a point on a line needs only one number, but a point on a page needs two, and a point in a room would need three. This is the idea every later lesson in this unit builds on directly: the distance between two points, the midpoint of a segment, and eventually a whole graph, are all just this one pairing, used more than once.

One to watch out for.

This is also where the two halves of this course — geometry and algebra — properly become one subject. A triangle’s corner is a shape you can drag; from here, it is also a pair of numbers you can calculate with.

What are the coordinates of a point 5 steps right and 2 steps down from the origin?

$(5, -2)$ — right is positive $x$ , down is negative $y$ .

A point has coordinates $(-3, 4)$ . Describe its position from the origin in words.

3 steps left, 4 steps up.

Two points are $(0, 0)$ and $(-6, 0)$ . What do both have in common, and what is different?

Both sit on the x-axis, since neither has moved up or down from the origin — their $y$ -coordinate is 0 in both cases. They differ only in $x$ : one is the origin itself, the other is 6 steps left of it.

Do the exercise

This lesson counts as done once you get its exercise right. Nothing to tick off by hand.