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In much of the scientific literature, the words 1D, one-dimensional, 1-dimensional, and 1-D occur frequently. Which of these is the best practice? Are there general principles for deciding which is the best format of these types

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The general principle must surely be to copy other writers dealing with similar subject matters at similar levels of detail. I've no doubt that would vary according to the specific context and target audience, so I don't see how ELU can offer constructive "general" advice. –  FumbleFingers Oct 25 '12 at 21:47
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3 Answers

Is there any prospect that you will need to refer to an n-dimensional vector space? If so, there is no other sensible way to write that (nD or n-D is open to all sorts of misinterpretation) and your other references need to be consistent (as in 1-dimensional and 3-dimensional).

In words, "one-dimensional" might refer to the performance of a poor actor or an aspiring politician.

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-1 "One-dimensional" is indeed used in a technical sense. –  Kris Oct 26 '12 at 6:33
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The best advice about this sort of stuff:

(A) Read the style manual prescribed by whoever is going to publish your work. Follow the rules.
(B) When the publisher doesn't prescribe a format for those words, imitate what it publishes in its articles that do use those words.
(C) If you're not writing for publication but for your colleagues, imitate their practice. It's usually, but not always, standard in the field.
(D) If you want to use the short form, spell it out the first time (unless the publisher's style manual says that it's unnecessary) -- one-dimensional {(1D) / (1-D)} -- and then give the abbreviation you will use throughout the article.

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I was taught to write out numbers in formal writing, so I would tend towards "one-dimensional". I don't have a good source with me to look this up, but the first search result to grammarbook.com says to

Spell out single-digit whole numbers. Use numerals for numbers greater than nine.

Examples:

I want five copies.

I want 10 copies.

This rule is actually contrary to what I learned, though, as I was told to write out all numbers.

However, it is widely accepted to forgo this rule in scientific, mathematical, and similar writings and simply use numberals instead, since they tend to contain calculations, measurements, formulas, etc. As FumbleFingers recommended, you should observe what other similar resources use.

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Actually, I was taught to write out all numbers up to twelve and use numerals for numbers greater than that. –  Anke Apr 9 '13 at 12:46
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