We built the number plane using two copies of the number line and we explored how it can be used to describe the location of shapes and points in a 2D space.
Now we can extend this coordinate system using directed numbers, which will allow us to describe the location of points in any direction from the origin.
The applet below demonstrates the main features of this extended number plane.
Drag the point P to explore the other quadrants.
Both axes now have positive and negative coordinates.
Moving around anticlockwise we cover the other three quadrants, which have the following features:
2nd quadrant: x-coordinates are negative, y-coordinates are positive
3rd quadrant: both coordinates are negative
4th quadrant: x-coordinates are positive, y-coordinates are negative
Points that lie on an axis, like (-5,\,0) or (0,\,4), are not in any quadrant.
The advantage of using directed numbers on the number plane is that we no longer have boundaries for the coordinates. If an object begins at some point on the plane, we can move it any which way we like, as far as we like, and still be able to describe its location with respect to the origin.
What are the coordinates of the point shown in the number plane?
In which quadrant does the point (3,\,-2) lie?
What is the distance between A\left(6,\,8\right) and B\left(-3,\,8\right)?
Each quadrant of the coordinate plane has distinct characteristics: