# Math terminology: What are rules regarding hyphens? (Nonzero vs. non-zero)

This question is geared toward clarifying terminology in writing math.

Which terms are correct and why?

1. A set $E$ is non-empty.
2. A set $E$ is nonempty.

1. The number $x$ is non-negative.
2. The number $x$ is nonnegative.

1. The number $y$ is non-positive.
2. The number $y$ is nonpositive.

1. The number $z$ is non-zero.
2. The number $z$ is nonzero.

As a personal preference, I like the way nonzero looks but I prefer to use non-empty so I don't know what the consistency with hyphens is when prefixing "non". I also like the way nonnegative reads but I feel like non-positive looks better than without a hyphen. I don't have any rhyme or reason why I have these preferences and that's why I'm curious to learn if there is a correct version of each term listed above.

• This would be more suitable on english.stackexchange.com as the word-formation rules probably don't have much to do with actual math. – Fizz Feb 7 '15 at 18:26
• I'd be surprised if the technicality of the terms is relevant to answer this question. Have you tried asking at English Language S.E. or English Language Learners S.E.? – Git Gud Feb 7 '15 at 18:26
• @RespawnedFluff I considered that SE but I figured that mathematicians would have more familiarity seeing these terms and have a better sense of which one is more common or be able to give reasons why they prefer/avoid hyphens. – Xoque55 Feb 7 '15 at 18:29
• Both are generally accepted. I much prefer and always use the hyphenated versions. – Brian M. Scott Feb 7 '15 at 20:38
• I use hyphens before capital letters, so "non-Abelian" instead of "nonAbelian". – bof May 19 '16 at 11:08

Since many (or a nonnegligible amount) of authors just do not care, I guess it's up to personal preference. However, there is a semantic code, like pointed out here. Thereafter, the prefix non is not hyphenated. And I really think this should be consistent. I had nonzero, nonempty and nonholonomic occuring in my thesis and it was "corrected" by different supervisors into different versions, mostly where non-holonomic was hyphenated and the rest not. I really think this is bad practice, since consistency is lost by this "to highlight" mentality. If on tries to highlight something, why not explain it and treat it consistently in semantics.

To underline my point consider using $$X$$ for vectors in one direction. For some other vector, to hightlight that it's not in this direction you use $$\alpha$$. But why not $$Y$$ with an explanation?! Therefore, semantics is consistent and the highlights are where they belong, in the "comments".

What got me to enforce this was that every title I cited contained the nonhyphenated versions of all those words. Thus, (from a far too small sample, but of good works) I conclude that this is common practice in this field (between robotics and automation).

Oxford Dictionary also advises to use hyphens for prefixes, which lead to vowel collisions, like pre-advised, although nothing further is stated there.

• What does "nonnegatable" mean? – bof May 19 '16 at 11:07
• I was stuck between languages and just wrote some mixture, what I meant was nonnegligible. – mike May 19 '16 at 13:49

Donald Knuth says we should drop the hyphens.

Here is the link to his post in which he says so:

https://cs.stanford.edu/~knuth/email.html

In mathematics, it is usual to formulate above words without hyphens. Some People want to clarify difficult words by separating These with Hyphens; so words with Hyphen arise.

• If I search in Google Books "non-negative" vs. "nonnegative" they're about tied, and the top hits are all math books for both searches. – Fizz Feb 7 '15 at 18:28

It seems that positive is better word usage than "nonnegative" or "non-negative"? Also should it be not equal to zero as opposed to "nonzero" or 'non-zero"?

• Positive and nonnegative are not the same though, as 'nonnegative' implies zero or positive, while 'positive' only implies positive. – T. Eskin Oct 13 '16 at 12:13
• I did not know this. I have always taken zero to be a positive whole even number. – Antonio Hernandez Maquivar Oct 13 '16 at 12:15
• @AntonioHernandezMaquivar This is just a convention. In French positive also includes zero, and I suppose the same holds for Spanish? – Roberto Rastapopoulos Feb 18 '19 at 18:23
• @RobertoRastapopoulos I was taught that in school, yes. Is there a sound mathematical reason why $0$ would not be a positive number ? – Antonio Hernandez Maquivar Jun 3 '19 at 12:36