11.01 The number plane

The grid reference system commonly used in street directories and topographic maps makes use of ordered pairs of numbers, or sometimes letter-number pairs, to describe different regions within an area.

In mathematics, the Cartesian plane (or number plane) is used to describe the location of actual points, not regions, in a two-dimensional space. These points are labelled by an ordered pair of numbers of the form $\left(x,y\right)$(x,y), called coordinates.

We can think of the Cartesian plane as being built from two separate number lines. The first number line we lay down horizontally on the plane. This line is called the $x$x-axis, and for every point in the plane we can talk about its $x$x-coordinate, which is its horizontal position along this axis.

The second number line is placed in a vertical direction, perpendicular to the first number line. This vertical line is called the $y$y-axis, and every point in the plane has a $y$y-coordinate, which is its vertical distance along this axis.

The point at which the $x$x-axis and the $y$y-axis intercept is called the origin. The coordinates of the origin are $\left(0,0\right)$(0,0).

Use the applet below to see how drawing a Cartesian plane over an area can be used to describe the location of different objects.

The coordinates of a point are given with respect to the origin. In the image above, we can see that the cat is $6$6 units to the right of the origin, and $2$2 units above the origin. So we can say the cat has the coordinates $\left(6,2\right)$(6,2). The $x$x-coordinate is $6$6 and the $y$y-coordinate is $2$2.

Notice that it would be incorrect to say the cat has the coordinates $\left(2,6\right)$(2,6). These coordinates refer to the point $2$2 units to the right of the origin, and $6$6 units above the origin. The order of the numbers is important!

Worked example

Consider the point $A$A plotted on the Cartesian plane.

 

a. What are the coordinates of $A$A?
Starting at the origin, we can move $5$5 units across to the right along the $x$x-axis until point $A$A is directly above us. This means that the $x$x-coordinate of point $A$A is $5$5. Next, we will need to move $4$4 units up in the vertical direction to get to $A$A, so the $y$y-coordinate of point $A$A is $4$4. We write the coordinates as $\left(5,4\right)$(5,4).

 

b. What are the coordinates of the point that is $4$4 units to the right of $A$A?
This new point is directly to the right of $A$A, so it will have the same $y$y-coordinate. If we start at $A$A, we can increase the $x$x-coordinate by $4$4 units, which gets us to the point $\left(5+4,4\right)=\left(9,4\right)$(5+4,4)=(9,4).

 

c. What are the coordinates of the point that is $3$3 units to the left and $2$2 units above $A$A?
Starting at point $A$A with coordinates $\left(5,4\right)$(5,4), we subtract $3$3 units from the $x$x-coordinate (because we are moving to the left), and add $2$2 units to the $y$y-coordinate (because we are moving upward). This gives $\left(5-3,4+2\right)=\left(2,6\right)$(5−3,4+2)=(2,6).

 

d. If point $B$B has the coordinates $\left(5,7\right)$(5,7), what is the distance between $A$A and $B$B?
Notice that point $B$B has the same $x$x-coordinate as point $A$A. This means that the distance between the two points is given by the difference in the two $y$y-coordinates. So the distance is $7-4=3$7−4=3 units.

 

 

Practice questions

Question 1

Question 2

Question 3