4

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.

Share a link to this question
CC BY-SA 3.0
Improve this question
5
  • 1
    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, 2016 at 12:54
  • 4
    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, 2016 at 13:55
  • @JohnColeman, it's unfortunate that it can take this long to change notation.
    – Joonas Ilmavirta
    Jul 12, 2016 at 21:02
  • 1
    In my experience, AB is the length of AB¯.
    – John Molokach
    Jul 14, 2016 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, 2016 at 6:43

2 Answers 2

Reset to default
1

The situation you describe is common for mathematics. Take the notation ∑k=1∞ak which has two different meanings:

  • the sequence of partial sums, i.e. ∑k=1∞ak=(∑k=1nak)n∈N
  • the limit of the series, i.e. ∑k=1∞ak=limn→∞∑k=1nak

Another example is the symbol ⊂. 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 ⊂ I would use ⊆ or ⊊ since there is is no ambiguity how to interpret these symbols...

Share a link to this answer
CC BY-SA 3.0
Improve this answer
3
  • 3
    I don't believe I've ever seen ∑k=1∞ak 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, 2016 at 20:09
  • 1
    @DaveLRenfro: Take the sentence "∑k=1∞ak converges". Here the notation ∑k=1∞ak means the sequence of partial sums (since the sentence "the limit of ... converges" does not make any sense)
    – Stephan Kulla
    Jul 15, 2016 at 7:56
  • 1
    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, 2016 at 15:26
1

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)or‖A−B‖
for its length. In my experience this notation is used in many textbooks on Linear Algebra.

Share a link to this answer
CC BY-SA 3.0
Improve this answer
1

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.