Points, lines, line segments, rays — notation and differences: Foundations

Before you can read $A B$ or $\overline{A B}$ , you must own the small pile of ideas hiding underneath them. This page unpacks that pile — one idea per row, each built from the one before it — so that nothing on the parent note is a mystery.

0 · The very first thing: a dot that has no size

The picture. Look at figure s01. The dot is drawn fat enough to see, but that fatness is a lie for your eyes only — the true point is the exact centre, infinitely small.

Why the topic needs it. Every other object (line, ray, segment) is made of points. A point is the atom; you cannot describe "start" or "end" without first having something that a start or end could be. If a point had size, "the endpoint" would be a fuzzy blob and we could never measure exactly.

1 · Two numbers that pin a point down: coordinates

The picture. In figure s02 the point $A = (1, 2)$ is reached by walking $1$ step right, then $2$ steps up. The two arrows are the two numbers, made visible.

This is the doorway to Coordinate Geometry. The parent note quietly used $A = (1, 2)$ and $B = (4, 6)$ — those are coordinate pairs.

$x$ ::: how far right of the origin (negative = left)
$y$ ::: how far up from the origin (negative = down)

2 · The idea of "straight": a path that never bends

Why the topic needs it. "Line", "ray", and "segment" are all straight. The three words differ only in how much of that one straight path you keep. So "straight" is the shared skeleton; the notation just clips it at zero, one, or two ends. (That "exactly one straight path between two points" fact is one of the Euclidean Postulates.)

3 · Infinity — the honest meaning of the arrows

The picture. Figure s03 shows the three objects stacked, so you can see the difference an arrowhead makes.

4 · The arrows and bars themselves (the notation decoded)

Now every ingredient is in place, we can read the parent's symbols. Each symbol is a cap placed over two point-names.

Why the design is clever. Count the arrowheads = count the infinite ends. Zero arrows → zero infinite ends → a finite segment. One arrow → one infinite end → a ray. Two arrows → two infinite ends → a full line. The notation is the picture.

$\overline{A B}$ has how many endpoints? ::: two ( $A$ and $B$ )
$A B$ has how many endpoints? ::: one (the start, $A$ )
$A B$ has how many endpoints? ::: zero

5 · The vertical bars $∣ \cdot ∣$ — "how long"

This connects to the Distance Formula, which the parent used as $(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$ — see Distance Formula for the full build.

Why can a line not have a length? ::: it never ends, so the number would be infinite (not a real length)

6 · The direction arrow $v$ and the parameter $t$

The parent wrote a ray as ${P + t v ∣ t \geq 0}$ . Two new symbols hide there.

The picture. Figure s04 shows one $v$ -step and the dial $t$ selecting positions.

Why each range. At $t = 0$ you sit on $A$ . At $t = 1$ you land exactly on $B$ (since $A + 1 \cdot (B - A) = B$ ). Values between fill the segment. Letting $t$ pass $1$ shoots past $B$ (ray). Letting $t$ go negative backs up behind $A$ (needed for a full line). Every case is covered by choosing the dial's limits.

$t = 0$ puts you at which point? ::: $A$ (the start)
$t = 1$ puts you at which point? ::: $B$
What does negative $t$ do? ::: backs up behind $A$ , opposite to $v$ (only a line allows it)

7 · How it all stacks

Read it upward: a point splits into "where it is" (coordinates) and "the straight path through it". Coordinates grow into a direction vector; the straight path grows an infinity notion. Those feed the cap notation and the step dial $t$ , and together they are the parent topic.

Equipment checklist

Test yourself — you are ready for the parent note only when every line below is easy.

I can say what a point is without mentioning size ::: a location only — no length, width or thickness
I can plot $(3, 4)$ from the origin ::: 3 steps right, 4 steps up
I can state why "line" always means infinite ::: it has no endpoints; the arrowheads promise it never ends
I can count endpoints from the cap ::: bar = 2, one arrow = 1, two arrows = 0
I know what $∣ \overline{A B} ∣$ asks for ::: the length (a single positive number) of the segment
I can compute $v = B - A$ for $A = (1, 2), B = (4, 6)$ ::: $(3, 4)$
I know what turning the dial $t$ does ::: slides the point $A + t v$ along the straight path
I can name the object for each range of $t$ ::: $0 \leq t \leq 1$ segment, $t \geq 0$ ray, $t \in R$ line
I know what $R$ means ::: all real numbers — the whole number line