a line

Picture a taut string stretched between two faraway posts, or the crease where you fold a sheet of paper flat: perfectly straight, and continuing — in the geometer's mind — without end in both directions. A line is this idea made exact: it is straight, it has length but no width, and it goes on forever.

Like a point, a line in Euclidean geometry is an undefined term, pinned down by axioms rather than a definition. The key axioms are that through any two distinct points there passes exactly one line, and that a line contains infinitely many points and extends without bound. We name a line by any two of its points with a double-headed arrow, line AB, or by a single lower-case letter such as l. A line is one-dimensional: you need only one number to say where you are along it once you fix a starting point and a unit, which is exactly what the ruler postulate makes precise.

Be careful to keep three relatives apart. A line has no endpoints and runs forever; a ray (such as ray AB) has one endpoint and goes on in just one direction; a line segment (AB) has two endpoints and a finite length. When people say 'line' loosely in everyday speech they often mean a segment, but in proofs the distinction matters: 'the line AB' and 'the segment AB' are not the same object.