# How to reasonably denote lines, line segments and rays?

I'm teaching geometry at high school for the first time soon and am struggling to find a reasonable notation for lines, line segments and rays defined by two points $A$, $B$ (and a direction). At the high school I'm teaching at, everyone just uses the notation they're familiar with already, and most of geometry is taught without a textbook.

I've come across rays starting at $A$ being denoted by $\overrightarrow{AB}$, which I'm not a fan of, because many people also use this for vector notation. Some people use $\overline{AB}$ for lines, some for line segments. Those people then use $AB$ for line segments and lines, respectively.

• French notation (following Bourbaki): $[AB]$ is the segment including endpoints. $(AB)$ is the line. $[AB)$ is the ray from $A$ to $B$ to infinity. $]AB)$ means $[AB)$ without the point $A$. $\overrightarrow{AB}$ is the line vector. Of course there's also $[AB[$, $]AB]$, $]AB[$ etc. This geometric notation is similar to the real intervals notation $[a,b]$ is the closed bounded interval, $]a,b[$ is the open interval etc. But in some countries, other ambiguous notations are used like $(a,b)$ for the open interval which look exactly like an ordered couple! – user5402 Aug 15 '16 at 12:52
• @whatever: Yes, over here, we almost exclusively use $(a,b)$ for open intervals. – Huy Aug 15 '16 at 12:54
• I've added the (secondary-education) tag because of the high school context. If you feel the question is more generally applicable, please feel free to rollback the edit. – J W Aug 15 '16 at 17:07
• @Huy In your comment you say "over here" but you haven't said what country you are in. Since notation is often region-specific that seems like pertinent information. – mweiss Aug 15 '16 at 17:35
• I use $(a,b)$ for both the open interval and the point with $x$-coordinate $a$ and $y$-coordinate $b$. I cannot remember the last time it caused confusion. It's almost always clear from context what is meant. – James S. Cook Aug 22 '16 at 3:11

The most important thing, in my opinion, is to adopt a notation that is consistent with whatever is most common in the broader culture. Geometry notation is highly context-specific; notation used at the secondary level tends to be different from that used at the undergraduate level, and there are country-to-country differences as well. If there is any kind of national or regional school-leavers exam (comparable to A-levels in the UK or to the SAT in the United States) you should use whatever notation is standard on that exam.

I don't know much about the Swiss educational system, but I find it frankly hard to believe that there is no standard notation in use in your context. The Swiss geometry curriculum was one of nine national curricula compared in:

Hoyles, C., Foxman, D. and Küchemann, D. (2002) A comparative study of geometry curricula. Qualifications and Curriculum Authority, London. ISBN 1858385091

...and while I don't have access to that book at the moment the mere fact that it exists suggests to me that there is a national geometry curriculum in Switzerland.

If you genuinely do teach in a vacuum, free of external encumbrances and with no cultural norms to align yourself with, then I guess you are free to choose whatever makes sense to you.

In the United States, standard notation at the secondary level is:

• $\overline{AB}$ denotes the line segment from $A$ to $B$
• $\overrightarrow{AB}$ denotes the ray with initial point $A$ and passing through $B$ (but note that a conventional Geometry class in the United States typically does not include vectors, so there is no conflict with that notation)
• $AB$ denotes the length of the segment $\overline{AB}$
• $\overleftrightarrow{AB}$ denotes the line through $A$ and $B$

At the advanced undergraduate or graduate level, different notations tend to be used; for example, I believe Greenberg's text uses $AB$ to denote a line, rather than the length of a segment, but otherwise uses the conventions above.

• I don't know what qualifies as a "curriculum", but with my interpretation, I can guarantee that there is no such thing as a national, not even cantonal standardized curriculum. As I mentioned in another comment, there are only (sort of) minimal expectations. Even within a high school, subjects that are covered can vary greatly. I honestly don't understand why that is hard to believe - the Swiss education works very differently than the US education, and that is just one part of it (not just in maths). – Huy Aug 15 '16 at 19:18
• Oh, the US system is completely decentralized, with no national curriculum at all -- but my understanding was that European educational systems tend to be more centralized. – mweiss Aug 15 '16 at 21:35
• This is the same notation that was taught to me in high school in the Philippines. – Joel Reyes Noche Aug 16 '16 at 0:59
• How do you simply denote the segment $AB$ without the point $B$? Also $\overleftrightarrow{AB}$ ($\overleftrightarrow{~}$ is sometimes used for tensors) is very similar to $\overrightarrow{AB}$ whereas in reality they are very different. – user5402 Aug 16 '16 at 11:23

I suggest not using a notation at all. "line $AB$", "segment $AB$", "ray $AB$" seems OK to me, and it forces you to write your proofs in text rather than using shorthand and over-formalization.