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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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
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.
The best advice about this sort of stuff:
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.