# 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 $]AB[$. Jul 12, 2016 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. Jul 12, 2016 at 13:55
• @JohnColeman, it's unfortunate that it can take this long to change notation. Jul 12, 2016 at 21:02
• In my experience, $AB$ is the length of $\overline{AB}$. Jul 14, 2016 at 2:11

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. Jul 14, 2016 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) Jul 15, 2016 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". Jul 18, 2016 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, $| AB |$. Jul 17, 2016 at 22:50

Definitions are not universal. You need to specify your definitions for each new context, and that goes for line segments and lengths as well. Today, I came across the notation in a textbook that

[...] in $$\triangle ABC$$, $$a=|BC|$$,

that is $$a$$ is to be considered a number (indicating length). I must confess that I so much more often see the length extracted from a segment in the two-letter form, i.e. $$|BC|$$ than in the one-letter form, i.e. $$|a|$$. But would that distinction be consistent?

I tend to see segments both in $$BC$$ and in $$a$$, but acknowledge that I subscribe to the notation of e.g. the Pythagorean theorem $$c^2=a^2+b^2$$ which indicates that small-letter notation does indeed signify lengths, that is numbers.