The coordinate system

Picture a treasure map: walk 4 steps east and 3 steps north, and there lies the treasure. Or a city map where you look up a square like C7. Both use the same idea: two pieces of information are enough to pinpoint any spot on a surface. In maths, this idea is called the coordinate system.

Two axes, one origin

A coordinate system consists of two number lines crossing at a right angle: the horizontal one is the x-axis, the vertical one the y-axis. Their crossing point is the origin, where both are 0. Every point gets a pair of numbers (x, y): first how far right or left, then how far up or down. So the point (4, 3) sits 4 to the right and 3 up. Careful: (4, 3) and (3, 4) are different points, the order matters!

The four quadrants

The two axes divide the plane into four regions called quadrants. They are counted counterclockwise starting at the top right: in the first quadrant at the top right, x and y are both positive; in the second at the top left, x is negative and y positive; in the third at the bottom left, both are negative; in the fourth at the bottom right, x is positive and y negative. So the point (−2, 5) sits top left, in the second quadrant: 2 to the left, 5 up. The signs alone tell you which region a point is in.

Reading and plotting points

To plot a point, always start at the origin: first walk along the x-axis, then move up or down parallel to the y-axis. Reading a point works in reverse: from the point go straight up or down to the x-axis, and the value there is the x-coordinate. From the point go straight across to the y-axis, and the value there is the y-coordinate. With a little practice you'll read points as quickly as squares on a city map.