# Notation of line segment and its length

I have sometimes seen a notation where $AB$ could mean either the line segment or its length. Why do the same notation can be mean both? Should one teach pupils to use for example notation $d(A,B)$ or $|AB|$ to mean the length of the line segment $AB$? I saw once in a text where $AB$ had two different meanings inside one solution.

• French notation: $[AB]$ means the segment (set of points), $AB$ is the length of $[AB]$ and there's the (much less used) notation $]AB[$. – user5402 Jul 12 '16 at 12:54
• The Greeks didn't distinguish between a line segment and the length of a line segment as clearly as we do. Hence geometric writing which follows Euclid's Elements tends to be similarly ambiguous. – John Coleman Jul 12 '16 at 13:55
• @JohnColeman, it's unfortunate that it can take this long to change notation. – Joonas Ilmavirta Jul 12 '16 at 21:02
• In my experience, $AB$ is the length of $\overline{AB}$. – John Molokach Jul 14 '16 at 2:11
• Good luck getting mathematicians to agree on a notation--that's what your question is about. This is a huge problem, imo, of mathematics--that it cannot agree on a formal language. – Jared Jul 16 '16 at 6:43

## 2 Answers

The situation you describe is common for mathematics. Take the notation $\sum_{k=1}^\infty a_k$ which has two different meanings:

• the sequence of partial sums, i.e. $\sum_{k=1}^\infty a_k= \left(\sum_{k=1}^n a_k\right)_{n\in\mathbb N}$
• the limit of the series, i.e. $\sum_{k=1}^\infty a_k= \lim_{n\to\infty} \sum_{k=1}^n a_k$

Another example is the symbol $\subset$. Some authors use it for the subset relation and some for the proper subset relation.

I cannot answer your question, why this happens. Since there is no overall style guide in the mathematics community for notations, it happens that different authors use the same notation for different mathematical concepts.

I would suggest the following: As an author or lecturer I would avoid notations with different meanings or notations which are used for different concepts in the literature (if this is possible). So instead of $\subset$ I would use $\subseteq$ or $\subsetneq$ since there is is no ambiguity how to interpret these symbols...

• I don't believe I've ever seen $\sum_{k=1}^\infty a_k$ identified as a sequence of partial sums, at least in a published paper or in a published book. However, the subset symbol is a good example and I do the same as you do when using it. – Dave L Renfro Jul 14 '16 at 20:09
• @DaveLRenfro: Take the sentence "$\sum_{k=1}^\infty a_k$ converges". Here the notation $\sum_{k=1}^\infty a_k$ means the sequence of partial sums (since the sentence "the limit of ... converges" does not make any sense) – Stephan Kulla Jul 15 '16 at 7:56
• O-K, I've seen this before, and I agree that it fits with what you were saying. That said, this is probably something more relevant in undergraduate or beginning graduate level textbook settings, where the reader is still learning standard notation and terminology, and so the writer should be a bit more careful. In more advanced situations, I would be more generous and consider the usage acceptable "abuse of notation" or acceptable "common usage". – Dave L Renfro Jul 18 '16 at 15:26

If you want to use a notation which clearly distinguishes between a line segment and its line, you can use the notation $$[A,B]$$ for the segment and $$d(A,B)\quad\text{or}\quad \|A-B\|$$ for its length. In my experience this notation is used in many textbooks on Linear Algebra.

• Or, more commonly, $| AB |$. – Joseph O'Rourke Jul 17 '16 at 22:50