The Coordinate Plane — Locating Points With Two Numbers at Once
One number line can only tell you how far along a single street you are. Cross two of them and you can pin down any exact spot on a whole map.
Intuition First
Think about how a city address works. "3rd Avenue" alone doesn't tell you where to stand — it's an entire street. "5th Street" alone is also an entire street. But "3rd Avenue & 5th Street" pins down one exact corner, because it's really two number lines laid on top of each other: one running east-west, one running north-south, crossing at some agreed-upon "home" corner.
The coordinate plane is exactly this idea, cleaned up. Take the number line from B1.1, and cross it with a second copy at a right angle, both sharing the same zero point. Now every location on the flat page needs exactly two numbers to describe it: how far to walk along the first line (left-right), and how far to walk along the second (up-down). Neither number alone locates anything — it takes both, together, in a fixed order, the same way a street address needs both the avenue and the street.
A Location You Can Plot by Hand
A treasure map says: starting from the fountain, walk 5 blocks east, then 3 blocks north. Where do you mark the X?
Notice the order matters: "5 east, 3 north" lands you somewhere completely different from "3 east, 5 north." That's why a coordinate is always written as an ordered pair — (5, 3) means x first, then y, no exceptions.
Try It — Locate the Point
Drag the x and y sliders to move the point around the grid. Watch its coordinate pair update as you move it.
Formal Definition
Every point P on the coordinate plane is described by an ordered pair of numbers, its coordinates:
x is the horizontal distance from the origin (right is positive, left is negative), and y is the vertical distance from the origin (up is positive, down is negative). Both are measured from the same fixed starting point: