I am confused whether to use singular or plural for "degree" used in directions/bearings. Is it grammatically correct to write "1 degree East of South", and "0.23 degrees West of South"?
Answer
Generally with any units we use the singular with 1 and the plural with anything else.
Some cases where there is disagreement are:
-1. Some take this as a case of one and some as a case of not-one and pluralise accordingly. (see Is -1 followed by a singular or plural noun?)
1.0 or other cases of 1 with greater precision than integer. Again, some take this as a case of one but generally it's taken as a case of not-one (see Is 1.0 singular or plural?)
Adjectival use. This doesn't so much happen with directions but does with other uses of degrees. An angle of 30 degrees is a 30-degree angle. This is an extension of how attributive uses of nouns are singular. (Even in always-plural cases; a press for trousers is a trouser-press, a sharpener for scissor-sharpener, most book shops have more than one book). (This seems to be why the EU advised at one point that the plural for Euro in English should be Euros despite the general advice being to follow the normal pluralisation rules of the language in question; someone got thrown by such expressions as "ten-pound note").
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