The Cartesian Plane

If a single number line can name every point on a line, then two perpendicular number lines should be able to name every point on a plane , and they can. This idea, due to René Descartes in the seventeenth century, is the foundation of coordinate geometry. The plane equipped with these two number lines is called the Cartesian plane, and every point on it is labelled by a pair $(x, y)$.

Definitions

The Cartesian plane (or coordinate plane) is the ordinary geometric plane on which we have drawn two number lines:

• A horizontal number line, the $x$-axis, with positive direction to the right.
• A vertical number line, the $y$-axis, with positive direction upward.

The two axes are perpendicular to each other. They meet at a single point called the origin, denoted $O$. The origin has coordinates $(0, 0)$.

Every point $P$ in the plane corresponds to a unique pair of numbers $(x, y)$, called the coordinates of $P$. The first number $x$ is the abscissa; the second number $y$ is the ordinate.

Concept and setup

Drawing the plane. Start with a clean sheet of grid paper. Draw a horizontal line across the middle and mark it as the $x$-axis. Draw a vertical line through the middle and mark it as the $y$-axis. Mark the point where they meet as $O = (0, 0)$. Now choose a unit: usually one grid square equals one unit. Mark $1, 2, 3, \ldots$ to the right of $O$ along the $x$-axis and $-1, -2, -3, \ldots$ to the left. Similarly mark $1, 2, 3, \ldots$ above $O$ on the $y$-axis and $-1, -2, -3, \ldots$ below.

Why two axes? A single number line has a fundamental limit: it can only name points on the line itself. To name a point that is not on a particular line, you need a second coordinate that tells you how far from that line the point is. The two axes give us two independent directions; their pair gives any point.

The "coordinate" idea. For any point $P$ in the plane, drop a perpendicular from $P$ to the $x$-axis. The foot of this perpendicular has a value on the $x$-axis: that is the abscissa of $P$. Now drop a perpendicular from $P$ to the $y$-axis. The foot has a value on the $y$-axis: that is the ordinate of $P$. Together, abscissa and ordinate form the ordered pair $(x, y)$.

Order matters. $(3, 5)$ is not the same point as $(5, 3)$. The first coordinate always means "$x$", the second always means "$y$". This is why the pair is called ordered.

Special positions. Points on the $x$-axis itself have $y$-coordinate zero: they are of the form $(x, 0)$. Points on the $y$-axis have $x$-coordinate zero: they are of the form $(0, y)$. The origin $(0, 0)$ is on both axes.

Distances. For a point $P = (x, y)$, the perpendicular distance from $P$ to the $x$-axis is $|y|$ (the absolute value of the ordinate). The perpendicular distance from $P$ to the $y$-axis is $|x|$. We need absolute values because distances are never negative; the coordinates themselves can be of either sign.

Why the axes are perpendicular. Mathematically, we could use any two non-parallel directions. But perpendicular directions are special: they describe motion in two independent ways (horizontal and vertical), and they let us use the Pythagoras theorem to compute distances. We will see this in later chapters.

The plane has no boundary. The grid stretches infinitely far in all four directions. A point like $(100, -50)$ is just as legitimate as $(1, 2)$ , we simply might not draw it on a small page.

Worked examples

Example 1. What are the coordinates of the origin?

The origin is the meeting point of the two axes. Its coordinates are $(0, 0)$.

Example 2. Where do points of the form $(0, y)$ lie?

They have abscissa $0$, so they sit on the $y$-axis. Their height above or below the origin is $|y|$.

Example 3. Find the perpendicular distance from $(3, -4)$ to the $x$-axis.

The perpendicular distance to the $x$-axis is $|y| = |-4| = 4$.

Example 4. A point lies on the $y$-axis at a distance of $5$ units below the origin. Find its coordinates.

It is on the $y$-axis, so $x = 0$. It is $5$ below the origin, so $y = -5$. The point is $(0, -5)$.

Example 5. Are the points $(2, 3)$ and $(3, 2)$ the same? Justify.

No. The first has abscissa $2$ and ordinate $3$; the second has abscissa $3$ and ordinate $2$. These are different positions in the plane.

Try it yourself

1. Name the point at the intersection of the $x$- and $y$-axes.
2. What are the coordinates of a point on the $x$-axis at distance $4$ to the right of the origin?
3. What are the coordinates of a point on the $y$-axis $7$ units below the origin?
4. Find the distance from $(6, -2)$ to the $x$-axis.
5. Find the distance from $(-3, 5)$ to the $y$-axis.
6. Are $(0, 4)$ and $(4, 0)$ the same point? Justify.
7. The point $(a, 0)$ for $a > 0$ lies on which axis?
8. The point $(0, -7)$ lies on which axis?
9. Sketch the axes and mark the points $A(2, 0)$, $B(0, 3)$, $C(-1, 0)$, $D(0, -2)$.
10. Give the coordinates of any three different points on the $x$-axis.

Pitfalls / Insight

• The convention is $(x, y)$, never $(y, x)$. A point named $(2, 5)$ is $2$ to the right and $5$ up , never $2$ up and $5$ to the right.
• "On the $x$-axis" means the whole horizontal line, not just the right half. Both $(3, 0)$ and $(-3, 0)$ are on the $x$-axis.
• Distances are non-negative. When asked for "distance from the $x$-axis", use $|y|$.

Insight. The Cartesian plane turns geometry into algebra. Every shape , line, circle, parabola, triangle , becomes an equation. Every algebraic question , solve, simplify, compare , becomes a question about points or curves. The two halves of mathematics meet here.