The Cartesian Plane: Axes, Coordinates and Quadrants

Why two num­bers?#

On a num­ber line, one num­ber is enough to say where a point is. On a flat sur­face like a page or a floor, one num­ber is not enough. You need two. In this les­son we set up two num­ber lines at right angles, called the axes. With them, any posi­tion on a flat sur­face becomes an ordered pair of num­bers. Then we will see how the signs of those two num­bers tell you where the point is.

Two lines at right angles#

Draw a hor­i­zon­tal num­ber line and call it the $x$-axis. Through its zero, draw a ver­ti­cal num­ber line and call it the $y$-axis. Their cross­ing point is the ori­gin, writ­ten $O$, with coor­di­nates $(0, 0)$.

Along the $x$-axis, dis­tances to the right of $O$ are pos­i­tive and to the left neg­a­tive. Along the $y$-axis, up is pos­i­tive and down is neg­a­tive. We use the same unit on both axes.

The flat sur­face hold­ing both axes is called the Carte­sian plane. You will also hear it called the coor­di­nate plane or the $xy$-plane. It is named after Descartes, who in 1637 set out the idea that two num­bers can fix a point. But the think­ing in grids is much older. The streets of the Sindhu-Saras­vatī cities were laid out north to south and east to west at reg­u­lar spac­ing. And it was Brah­magup­ta's work on zero and neg­a­tive num­bers in the sev­enth cen­tury that makes the neg­a­tive halves of the axes pos­si­ble at all.

Coor­di­nates of a point#

Take any point $P$ in the plane. Drop a per­pen­dic­u­lar from $P$ to the $x$-axis; the num­ber where it lands is the $x$-coor­di­nate of $P$. Drop a per­pen­dic­u­lar to the $y$-axis; that num­ber is the $y$-coor­di­nate. Together they are writ­ten $(x, y)$, and $x$ always comes first.

Another way to say the same thing: $x$ is the signed dis­tance of $P$ from the $y$-axis, and $y$ is the signed dis­tance of $P$ from the $x$-axis. Mind the order. $(2, 5)$ and $(5, 2)$ are dif­fer­ent points, and they are the same only when the two num­bers are equal.

Points that sit on an axis always have one coor­di­nate equal to zero.

• On the $x$-axis: $(x, 0)$. For exam­ple $(6, 0)$ is $6$ units right of $O$; $(-2.5, 0)$ is $2.5$ units left.
• On the $y$-axis: $(0, y)$. For exam­ple $(0, 3)$ is $3$ units above $O$; $(0, -7)$ is $7$ units below.

The four quad­rants#

The axes cut the plane into four regions, called quad­rants. We num­ber them anti­clock­wise, start­ing from the top right.

One small point that often appears in tests: a point on an axis belongs to no quad­rant at all.

Read­ing a floor plan#

Here is a place where coor­di­nates are really use­ful. They make the plan of a room exact. Put the ori­gin at one cor­ner of the room, run the $x$-axis along one wall and the $y$-axis along the wall at right angles to it, and let $1$ unit $= 1$ foot.

Notice that a plan like this shows only the floor. Some­thing at a height, such as a win­dow in a wall, has no place on it. Every point on the floor is fixed by two num­bers, and a height would need a third.

Prac­tice#

1. Write the quad­rant or axis for each point: $(5, -1)$, $(-2, -2)$, $(0, 8)$, $(-9, 4)$, $(3, 0)$, $(1, 1)$.
2. What are the coor­di­nates of the ori­gin? What is the $y$-coor­di­nate of every point on the $x$-axis?
3. Name a point in Quad­rant III whose $x$-coor­di­nate is $-5$, and a point on the $y$-axis $4$ units below the ori­gin.
4. $P(a, b)$ lies in Quad­rant II. In which quad­rant does $(b, a)$ lie? What about $(-a, -b)$?
5. A rec­tan­gu­lar table has three feet at $(6, 8)$, $(10, 8)$ and $(10, 5)$. Where is the fourth foot? What are the table's length and width?
6. A room is $16$ ft by $12$ ft with a cor­ner at the ori­gin and sides along the axes. A $5$ ft by $3$ ft desk is placed exactly in the cen­tre with its longer side par­al­lel to the $x$-axis. Give the coor­di­nates of its four cor­ners.

Answers#