Real Line

Why

We are constantly thinking of the real numbers as the points of a line.¹

Discussion

We commonly associate elements of the real numbers (see Real Numbers) with points on a line (see Geometry).

Given a line, there exists a set of its (infinite) points.

Let $P$ be the set of points for a line. There exists a one-to-one correspondence mapping elements of $P$ onto elements of $\R $.

For this reason, we sometimes call elements of the real numbers points. We call the point associated with 0 the origin.

Visualization

To visualize the correspondence we draw a line. We then associate a point of the line with the $0 \in \R $. We can label it so. We then pick a unit length. We associate the points a unit length away from zero with $1 \in \R $ (on the right) and $-1 \in \R $ (on the left). We do the same for two and $2$ and $-2$, $3$ and $-3$, and then we say that we could continue the process indefinitely. We can visualize the image below

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