Should You Write “Nonzero” or “Non-Zero” in Mathematics?

In mathematics, nonzero and non-zero mean the same thing: not equal to zero. There is no mathematical rule that makes one spelling correct and the other incorrect. The choice is an editorial one, so the best spelling is usually the one required by your publisher, instructor, or department.

“Nonzero” is a common spelling in mathematical texts, especially in American usage. “Non-zero” is also established and appears in mathematical writing, including in styles that hyphenate some prefix forms. Usage varies, so treat either form as acceptable unless a particular style guide says otherwise.

What “nonzero” means

“Nonzero” is an adjective formed from the prefix non- and zero. It describes a number or quantity that is not zero. For a real number $x$, the definition is simply

$$x \ne 0.$$

The word itself does not specify the number system. If $x$ is known to be real, “a nonzero real number” is clear. If the surrounding context could involve different systems, name the relevant one: for example, “a nonzero complex number” or “a nonzero element of the ring $R$.” The spelling does not affect that mathematical qualification.

How to choose a spelling

Use this order of preference:

1. Follow an explicit style rule. A journal, publisher, course, or thesis guide may require one form. That instruction takes precedence over a general preference.
2. Otherwise, choose one form and be consistent. If there is no house style, “nonzero” is a straightforward default for mathematical prose. “Non-zero” is equally defensible when it matches the spelling convention you are following.
3. Do not treat the choice as a change in meaning. A proof does not become more or less correct because it uses a hyphen.

Consistency means more than choosing the form once in an isolated sentence. Avoid switching between “nonzero” and “non-zero” in the same article, paper, or set of notes without a reason. If you are editing an existing document, consistency may be more important than changing every instance to your personal preference.

Worked example

Suppose you want to state that the equation $x^2=1$ has nonzero real solutions. The solutions are $x=1$ and $x=-1$, and both differ from zero. You could write:

The equation $x^2=1$ has two nonzero real solutions, $1$ and $-1$.

If your style guide uses a hyphen, write “two non-zero real solutions” instead. The mathematical statement is identical. Specifying “real” also makes the domain explicit; in this example, the solutions happen to be real, but the word nonzero alone does not tell the reader which number system is under consideration.

A useful check is to replace the adjective with its definition. “A nonzero real number $a$” should mean “a real number $a$ such that $a\ne0$.” If that replacement does not capture your intended claim, the issue is probably the mathematical wording, not the hyphen.

Related terms and constructions

The same editorial question arises with other words formed using non-, such as nonempty, nonnegative, nontrivial, and noncommutative. Mathematical publications may close these up or use hyphens depending on house style. Do not assume that the spelling of “nonzero” determines the spelling of every other term; consult the relevant guide or follow an established convention in your field.

When a prefix makes a phrase difficult to parse, rewriting is often better than adding a hyphen arbitrarily. For example, instead of building a long chain of hyphenated modifiers, state the relationship directly: “a game whose payoffs do not sum to zero.” Clarity matters more than enforcing a mechanical rule in a construction that is hard to read.

Common mistakes

• Calling one form mathematically wrong. The hyphen is a spelling convention, not part of the definition.
• Switching spellings without a reason. Pick a form for the document and apply it consistently.
• Assuming “nonzero” identifies a domain. It means “not zero,” but you may still need to say whether the object is real, complex, an integer, or an element of some algebraic structure.
• Confusing a style choice with notation. The condition $x\ne0$ states the mathematical restriction directly; “nonzero” expresses it in prose.

If you are writing for a specific journal or institution, search its author instructions for the preferred form. If no preference is stated, use “nonzero” or “non-zero” consistently and spend your attention on making the domain and claim precise.