Cartesian Coordinates

A movie ticket reads "Row 5, Seat 3" — no need to ask anyone, you walk straight to your seat: find row 5, then count to seat 3. Over three hundred years ago the mathematician Descartes carried this seat-reporting trick into mathematics: draw two perpendicular number lines, and every point on the plane owns a one-of-a-kind address.

Two number lines, one map

Each of the two lines has a job:

The horizontal one is the x-axis, positive to the right;
The vertical one is the y-axis, positive upward;
Where they cross sits the origin $O$ , with the address $(0, 0)$ .

A point's address is written as an ordered pair $(x, y)$ : horizontal first, vertical second. $(3, 5)$ and $(5, 3)$ are two different points — the order is the rule, and that is exactly what "ordered" means.

How to walk to (2, 3)

Reading coordinates is like walking, with the chant horizontal first, then vertical: start at the origin, step 2 to the right along the x-axis, then 3 up — and you land on $(2, 3)$ . To read off a point's coordinates, see which number it lines up with on the x-axis and which on the y-axis, then report them in the same order.

What if you swap the order

Writing $(2, 3)$ as $(3, 2)$ means walking 3 right and 2 up — you land somewhere else entirely. The most common way to lose marks on coordinates is reporting the pair backwards.

The sign code of the four quadrants

The two axes slice the plane into four pieces, numbered counterclockwise from the top right: quadrants I, II, III, IV. No need to memorize the map — just watch the signs:

Quadrant I $(+, +)$ : x and y both positive;
Quadrant II $(-, +)$ : x negative, y positive — upper left;
Quadrant III $(-, -)$ : both negative — lower left;
Quadrant IV $(+, -)$ : x positive, y negative — lower right.

The two axes themselves belong to no quadrant — a point on an axis has at least one coordinate equal to $0$ , like $(4, 0)$ .

Coordinate Board

InteractiveCoordinate Board

x3y-2

(3, -2) → Quadrant IV

Walk x steps horizontally, then y steps vertically — the ordered pair (x, y) pins down exactly one point.

Drag the x and y sliders and watch the point roam the plane while the sign panel announces which quadrant it lives in. Give yourself three missions: move the point into quadrant II; find a point "living on the x-axis"; then make x and y share a sign — and see which two regions the point can land in.

What addresses are good for

Maps and navigation: longitude and latitude are a pair of coordinates — "30° N, 120° E" pins a spot on Earth itself;
Game worlds: many games write a character's position as $(x, y)$ , and 3D worlds carry an extra $z$ ;
Cinemas and classrooms: row and seat are coordinates in their plainest form.

Later, in Equation of a Straight Line, you will see that a line is simply the neat queue of all points $(x, y)$ satisfying $y = mx + b$ — and every $(x, y)$ there is a coordinate from today. The inputs and outputs of the Variables & the Function Machine slot into the same pair of brackets.

Check yourself

Quick quiz

1. Which quadrant does the point (-3, 4) live in?
2. From the origin you walk 4 steps right and 2 steps down. Where do you land?
3. A point lies exactly on the x-axis. What is special about its coordinates?