Can straight lines be denoted by capital letters?

Question

I've recently noticed an answer where a line was represented by a capital letter (not anymore, it's been edited).

Can straight lines be represented by capital letters given that the letters are initially defined?

Are there any cases when denoting an object by some unusual symbol is considered wrong?

Could I define two points as, e.g., $\beta$ and $\zeta$, and the segment that they create as $\zeta\beta$? Could that be legitimately marked wrong on a serious exam?

And can $AB$ (where $A$ and $B$ are points) be considered a straight line instead of just a segment? A straight line is defined by two points, so that notation sounds correct to me. Is it?

Update:

I've seen this written on an exam: "Two parallel lines $a$ and $b$ are intersected by the line $AB$." The point $A$ was on the line $a$ and the point $B$ was on the line $b$ (on an added diagram). Is this wrong?

Answer 1

In a high school geometry course, it is common to always denote lines by pairs of capital letters which represent points, i.e. $AB$ or $\overline{AB}$ (often these two are distinguished).

In practice, this notation is extremely unstandardized and the exact form you use is widely considered irrelevant. In a classroom setting, of course, you get points based largely on how your grader feels about the correctness of your work, and so if E asks for a specific notation, you would do best to give it to em.

To answer your questions directly:

• Yes, a line can be reasonably denoted by a single letter, capital or otherwise.
• No, if you define $\pi=2$ then it is not wrong to write $\pi+\pi=\pi^\pi$. That said, as the previous example shows, it's not necessarily a good idea. You should only override strong intuition if you have to, and even then a reader could reasonably want some carefully explained motivation.
• Yes, if $\zeta$ and $\beta$ are points, then you can talk about the line $\zeta\beta$ (but see below).
• Yes, doing so could very legitimately be marked incorrect on an exam.
• Yes, $AB$ could be a line instead of a segment (but see below)
• It is almost true that a line is defined by two points (see below), assuming Euclidean geometry.
• No, the exam snippet in your edit is not "wrong" in any meaningful sense. This doesn't mean it wouldn't be reasonable for it to be marked incorrect— for example, if the grader had mentioned that answers which used lower case letters to denote lines would be marked incorrect, or if there was some other strong indication in that direction. [For what it's worth, I rather like the notation it uses.]

"below":

In Euclidean geometry, the observation that a straight line is defined by two points is almost true, but the only thing truly defined by two points is a pair of points. What is true is that a line is defined by two points and the assumption that the object you are considering is a line. This is what the fuss about $AB$ or $\overline{AB}$ or $m(AB)$ or arrows or half-arrows is really about; the notation is trying to convey not only the two points, but also what kind of object is being considered. In practice, you can just say "The line $AB$", where $A$ and $B$ were previously understood to be points, and you are perfectly understandable.