Distance and Length

Dependency: Ruler axiom. A derived definition: how far apart two points are is reduced to a real number supplied by the ruler axiom.

Definition

Let $A$ and $B$ be points in the plane. The Ruler axiom axiom gives every line a coordinate scale, assigning a real number to each point on it.

When $A$ and $B$ lie on the same line, the absolute difference of their coordinates is independent of the chosen ruler. This number is the distance from $A$ to $B$, denoted by $|AB|$ (or simply $AB$), and is also the length of segment $AB$.

This turns the visual idea of “how far” into a real number that can be added and compared. It is the point where geometry becomes measurable rather than merely visible.

Remarks

• Nonnegative, symmetric, and zero only for the same point. $|AB|\ge 0$; $|AB|=0$ if and only if $A=B$; and $|AB|=|BA|$. These follow directly from the Ruler axiom axiom.
• This definition concerns point-to-point distance. Distance from a point to a line, or between parallel lines, is derived later after results such as the shortest-perpendicular theorem make “nearest” precise.
• The triangle inequality is not part of the definition. The fact that $|AC|\le|AB|+|BC|$, with equality exactly when $B$ lies on segment $AC$, must be proved as a theorem.