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I always have problems with the use of colons when writing math papers. The following examples come from a published paper in a prestigious American journal.

Here is an example where the colon is not used:

... can be written as

"The equation is inserted here"

where...

And one where the colon is used:

... by accumulating the effects of earlier appreciation:

"The equation is inserted here"

A last one:

... is a function of the amount of search (...):

"The equation is inserted here"

where...

I am unable to draw a definitive conclusion from these examples. Is there any specific rule that applies?

  • I find periods more confusing here when the equation is supposed to end the sentence. Do you put the period at the end of the equation? That seems like it would get in the way. – Unrelated Sep 19 '17 at 18:07
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    @Unrelated I checked and there is no period at the end of the equations. I removed the one in the second example. – emeryville Sep 19 '17 at 18:52
  • Thanks for looking! And I didn't mean that as any criticism—I've always struggled with what to do when I am typing up a document with equations – Unrelated Sep 19 '17 at 19:00
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In your first example, the equation belongs grammatically to the same phrase as the preceding words. Imagine if it were not an equation but another word - then you could write e.g:

'The answer can be expressed as a formula'

i.e. a colon is not necessary because the equation can be seen as the grammatical complement of the phrase.

However, in the following two examples, the equation is not part of the phrase that precedes it but is juxtaposed as an illustration or demonstration of the preceding phrase, meaning that a colon is needed. It is the equivalent of e.g:

You know what to do: practice

<GrammarBook.com> (the source of the preceding example) also gives further tips on using colons, which may be applied to maths writing:

"A colon means "that is to say" or "here's what I mean.""

  • Grammatical colons signal a type of apposition: there are noun phrases on both sides. – AmI Sep 19 '17 at 19:20
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The part of a sentence that precedes a colon must be grammatically complete: it can be read as a complete sentence on its own.

With this logic, a colon cannot be used in the first example because '... can be written as' is not a complete sentence, regardless of how you fill in the ellipsis. The introductory clause of the second example, though, is grammatically complete and a colon therefore applies.

The APA is one style guide that presents this standard.

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