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Needs:
Real Order
Lists
Geometry
Needed by:
Area Measure
Circular Coordinates
Complex Plane
Plane Distance
Plane Vectors
Real Space
Rectangles
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Real Plane

Why

We are constantly thinking of the elements of $\R ^2$ as points of a plane.1

Discussion

We commonly associate elements of $\R ^2$ with points on a plane. (see Geometry).

Given a plane, there exists a set of its (infinite) lines.
Let $L$ be the set of lines of a plane. Then $\cup L$ is the set of points of the plane. There exists a one-to-one correspondence mapping elements of $\cup L$ onto elements of $\R ^2$.
For this reason, we sometimes call elements of $\R ^2$ points. We call the point associated with $(0, 0)$ the origin. We call the element of $\R ^2$ which corresponds to a point the coordinates of the point.

Visualization

To visualize the correspondence we draw two perpendicular lines. We then associate a point of the line with $(0, 0) \in \R ^2$. We can label it so. We then pick a unit length. And proceed as usual.2

Given that we have identified a plane with $\R ^2$ in this way, we call $(x, y) \in \R ^2$ the coordinates of the point it corresponds to. Many authors refer to this identification as a Cartesian coordinate system (or Rectangular coordinate system, $x$-$y$ coordinate system).


  1. Future editions will modify this sheet. ↩︎
  2. Future editions will expand this. ↩︎
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