# Is zero positive or negative?

Follow up to this question. Is $0$ a positive number?

It really depends on context. In common use in English language, zero is unsigned, that is, it is neither positive nor negative.

In typical French mathematical usage, zero is both positive and negative. Or rather, in mathematical French "$x$ est positif" (literally "$x$ is positive") allows the case $x = 0$, while "$x$ est positif strictement" (literally "$x$ is strictly positive") does not.

Sometimes for computational purposes, it may be necessary to consider signed zeros, that is, treating $+0$ and $-0$ as two different numbers. One may think of this a capturing the different divergent behaviour of $1/x$ as $x\to 0$ from the left and from the right.

If you are interested in mathematical analysis, and especially semi-continuous functions, then it sometimes makes more sense to consider intervals that are closed on one end and open on the other. Then depending on which situation are in it may be more natural to group 0 with the positive or negative numbers.

There are certainly much more subtleties, but unless you clarify why exactly you are asking and in what context you are thinking about this, it is impossible to give an answer most suited to your applications.

• To a computer programmer a significant context might be the IEEE 754 standard for floating point arithmetic, which distinguishes a +0 from a -0 representation. Of course mathematically there is only one Zero, but if your context is floating point representations, you could have it either way! – hardmath Mar 13 '11 at 15:05
• Some people (like me) might be looking for specific domains where people are concerned about the positivity of zero. A reasonable place to start is to search "strictly positive" or "strictly negative" in your favorite academic search engine. Those terms are used when there must be no ambiguity in whether or not a set of positive or negative numbers includes zero. I'd like criticisms of that search technique or other search suggestions if anyone has any! – kdbanman Mar 17 '17 at 16:11

No. $\textbf{} \textbf{} \textbf{}$

• I agree and favour the conventional positive means positive, i.e. strictly positive. – Jack D'Aurizio Dec 11 '16 at 21:41
• Wikipedia is not the Holy Bible. Some people (like me) regard all Wikipedia content as false until proven correct. – Lisa Oct 30 '14 at 22:28
• @Lisa then it's like the Holy Bible, for some (myself included) – Marc.2377 May 11 '16 at 6:09
• So what IS the Holy Bible / The Great Standardization Document of All Definitions for Mathematics? Because people are often fighting over different definitions of mathematical entities, 0 being one of such examples (French always start a flamewar when someone says 0 is not positive, because for French, 0 is positive and negative at the same time :P ). Same goes with definitions of angles, or square roots (only positive? positive and negative?) Being able to refer to some standard reference source with all the definitions agreed upon by the majority of mathematicians would be great. – BarbaraKwarc Jul 20 '16 at 11:07
• Wikipedia is momentary consensus not truth – Davos Jun 13 at 1:13
• @Davos: What is "objective truth"? Not asking for a definition; I mean, does it exist? I don't have an answer either :P – Lightness Races with Monica Jun 27 at 9:27

$0$ is the result of the addition of an element ($x$) in a set with its negation ($-x$). Hence, it is not necessary to conceive $0$ as having a negative element since it would produce itself. Therefore, by Occam's razor (i.e., the simplicity clause) it is not necessary for $0$ to have a negative element. However, by definition, the given set must have a negative element for all the positive elements. Therefore, it makes no sense to conceive it as a positive number.

Hence, $0$ is neither positive nor negative. That is intuitive since $0$ is null, defines nullity which is the absence of some abstract object.

However, if one does not agree with the simplicity clause, he can admit it as being both a positive and a negative number.

Therefore, as many things it is a matter of definition.

Actually, zero is neither a negative or a positive number.  The
whole idea of positive and negative is defined in terms of zero.
Negative numbers are numbers that are smaller than zero, and
positive numbers are numbers that are bigger than zero.  Since
zero isn't bigger or smaller than itself (just like you're not
older than yourself, or taller than yourself), zero is neither
positive nor negative.

People sometimes talk about the "non-negative" numbers, and what
that means is all the numbers that aren't negative, in other words
all the positive numbers and zero.  So the only difference between
the set of positive numbers and the set of non-negative numbers is
that zero isn't in the first set, but it is in the second.
Similarly, the "non-positive" numbers are the negative numbers
together with zero.


I think it is a matter of convention. I would not say that it is "wrong" to say $$0$$ is positive, as long as you properly define the meaning of "positive" from the onset.

Part of the reason I do not believe that "positive" has a universally accepted definition of meaning "strictly larger than $$0$$", is because of the (common) usage of the phrase "strictly positive number". Perhaps it is very redundant, or it illustrates the fact that "positive" does not have a universally accepted meaning.

Furthermore, there are examples throughout mathematics where the use of "positive" does not have a strict meaning. For example, a real-valued function $$f:X\rightarrow \mathbb{R}$$ is typically called a "positive function" if $$f(X)\subseteq[0,\infty)$$. In measure theory, one typically calls a "measure" $$\mu: \mathcal{M} \rightarrow [0,\infty]$$ on a measurable space $$(X, \mathcal{M})$$ a "positive measure" (to distinguish from so-called "complex measures" $$\mu:\mathcal{M}\rightarrow\mathbb{C}$$). In the first case, we still call $$f$$ positive even if there are $$x\in X$$ such that $$f(x) =0$$. In the second case, we still call $$\mu$$ positive even if there are $$E\in\mathcal{M}$$ such that $$\mu(E)=0$$.

There are many other examples in mathematics where "positive" is used to include $$0$$. So for these reasons, either the language is not universally accepted or is misleading. (I.e., misleading in the sense that does the word "positive" in the strict sense only apply to taking about the number $$0$$ itself, and not functions, measures, etc.?) I also admit that using language like "non-negative function" or "non-negative measure" is a bit more awkward than saying "positive function" or "positive measure". Nevertheless, I think it is too quick to jump to the conclusion that it is incorrect to view "positive" and "non-negative" as synonyms. I believe it is context dependent.

As a final remark, I do think using the word "positive" in the strict sense is more common. But, nevertheless, it does not seem incorrect to use "positive" in the non-strict sense as well and (dare I say) perhaps has merit in some contexts.