Two Axes, Four Quadrants: The Cartesian Plane

In earlier classes you met the number line — a single horizontal line with 0 in the middle, positive numbers stretching to the right, and negative numbers stretching to the left. The number line is one-dimensional: each point on it is described by exactly one number.

The 2-D Cartesian coordinate system is what you get when you take two number lines and lay them at right angles to each other, crossing at zero on both. By convention:

• The horizontal number line is called the x-axis.
• The vertical number line is called the y-axis.
• The point where the two axes meet is called the origin, written as O. Its coordinates are $(0, 0)$.

Distances are measured along each axis in equal units — every step of one unit is the same length on the x-axis and on the y-axis. By convention:

• Moving right of O on the x-axis or upward of O on the y-axis is counted as positive.
• Moving left of O on the x-axis or downward of O on the y-axis is counted as negative.

That single set of choices — two perpendicular axes, an origin at zero, equal units, positive to the right and up — gives you the Cartesian plane. Every point in the flat plane now has a precise address. There is no spot, however small, that this system cannot describe.

Look at the diagram above. To pin down any point in the plane, you need two numbers — and the order in which you give them matters.

The first number is always the x-coordinate — the perpendicular distance from the y-axis (i.e., how far left or right of O the point is). The second number is the y-coordinate — the perpendicular distance from the x-axis (how far up or down from O). The point is then written as $P = (x, y)$.

Let's read off the four labelled points from the diagram:

• B = (4.5, 0) — B is on the x-axis, 4.5 units to the right of O. So x = 4.5; y = 0.
• G = (0, −4.5) — G is on the y-axis, 4.5 units below O. So x = 0; y = −4.5.
• H = (0, 4) — H is on the y-axis, 4 units above O. So x = 0; y = 4.
• E = (−2.9, 0) — E is on the x-axis, 2.9 units to the left of O. So x = −2.9; y = 0.

A few useful patterns fall out immediately:

• A point of the form $(x, 0)$ — with $y = 0$ — lies exactly on the x-axis. If $x > 0$, it's to the right of O; if $x < 0$, it's to the left.
• A point of the form $(0, y)$ — with $x = 0$ — lies exactly on the y-axis. If $y > 0$, it's above O; if $y < 0$, it's below.
• The single point where both coordinates are zero — $(0, 0)$ — is the origin itself.

Shorthand. While writing coordinates, it is often convenient to drop the equals sign. Instead of writing $P = (x, y)$, mathematicians simply write $P(x, y)$. So $H = (0, 4)$ and $H(0, 4)$ mean exactly the same thing. The shorter form is what you will see most often when points are marked on a graph.