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

4This is not exactly a duplicate but has the same answer as in http://math.stackexchange.com/questions/18464/ispositivethesameasnonnegative – Mitch Mar 13 '11 at 14:52

8The question is flawed. $\textbf{} \textbf{} \textbf{} $ – Bruno Joyal Oct 04 '13 at 03:03

3Why does it have to be either positive or negative? – mez Jul 05 '14 at 13:16

https://www.youtube.com/watch?v=8t1TC5OLdM – jimjim Oct 31 '14 at 01:49
7 Answers
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 semicontinuous 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.
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5To 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

2Some 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{} $
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3I agree and favour the conventional *positive means positive*, i.e. *strictly* positive. – Jack D'Aurizio Dec 11 '16 at 21:41
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11Wikipedia 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

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3So 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

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@Davos What is the difference between those two things? – Lightness Races in Orbit Jun 14 '19 at 16:53

@LightnessRacesinOrbit Specifically I mean it is not _objective_ truth, and by momentary consensus I mean that the content changes in subjective opinionated ways particularly for contentious topics. A lot of it is generated by opinionated bots, often in conflict with other bots. But yes, I get your philosophical question, and for that I have no answer. – Davos Jun 27 '19 at 05:21

2@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 in Orbit Jun 27 '19 at 09:27

@LightnessRacesinOrbit I used to think so, but in this posttruth world who knows. – Davos Jul 02 '19 at 01:13

$0$ 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.
From : http://mathforum.org/library/drmath/view/58735.html
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 "nonnegative" 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 nonnegative numbers is
that zero isn't in the first set, but it is in the second.
Similarly, the "nonpositive" numbers are the negative numbers
together with zero.
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If we are speaking about the element $0$ in an ordered field $F$, it is neither positive nor negative. By definition, an element $x\in F$ is called positive if $x>0$. Similarly, an element $x\in F$ is called negative if $x<0$. Since $0\not< 0$, we immediately see that $0$ cannot be positive nor negative. (See Definition 1.17 in Rudin's Principles of Mathematical Analysis.)
Outside of ordered fields, the question is trickier. In certain contexts, I would not say that it is wrong to view $0$ is positive, as long as you properly define the meaning of "positive" from the onset. To elaborate, I do not believe that the term "positive" is universally accepted as "strictly larger than $0$" because of the commonly used phrase "strictly positive number." This phrase is either redundant, or it illustrates the fact that the meaning of "positive" is context dependent.
There are examples throughout mathematics where the use of "positive" does not have a strict meaning. For example, a realvalued 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 "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 $0$. For these reasons, it seems to me like the language is not universally accepted. However, I should note that terms like "nonnegative function" or "nonnegative measure" seem to be a bit more awkward (and more verbose) than simply saying "positive function" or "positive measure". Could this be a reason why the word "positive" is not used in the strict sense in some contexts? Perhaps. Nevertheless, I think there are contexts where the terms "positive" and "nonnegative" are synonymous.
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2Of all the responses, I find this one the most... positive. I agree that it is best to warn the reader of what we mean by positive, increasing, convex, etc. When a text becomes unpleasant to read because it is full of words such as "nonnegative", "nondecreasing" or "nonstrictly convex", I believe it's time to revisit longheld opinions. – Oskar Limka May 05 '21 at 17:42
In neutral context, the number $0$ is distinct from each of positive and negative; it is the exactlymiddle value of the trichotomous nonprojective line of the real numbers, unique unto its own in regards to its sign, and as such should in a ‘default’ setting be treated precisely as such: neither $\text{negative}$ (arg=: 1⋅arg) nor $\text{positive}$ (arg=: +1⋅arg), but instead $\text{zero}$ (arg=: 0⋅arg).
However, since the numberline representable as $∞<x<∞$ for $ℝ∋x$ is continuous, its middlepoint $(0)$ at $x=0$ is contiguous with both $∞<x<0$ (when $x∈(∞,0)$, i.e. $x$ is strictlynegative) and $0<x<∞$ (when $x∈(0,∞)$, i.e. $x$ is strictlypositive), so it is reasonable to think of the value $0$ as belonging to one or both of the nondiscrete portions of the numberline (instead of a point separated from both). Since $1⋅0$ evaluates to the same as $1⋅0$, absent further qualification the value $0$ is in this regard equallyjustifiable as negative and positive, thus considering negative numbers as belonging to $(∞,0]$ and positives to $[0,∞)$, nondisjoint (the exhaustive ranges not mutually exclusive) from eachother —although moreprecisely (i.e., sans ambiguity) these intervals closed at zero are called "nonpositive" and "nonnegative" respectively, relegating the openinterval counterparts to respectively "negative" and "positive" (somewhat deferring the need to qualify these nonzero categories with "strictly") Pedantically, "nonpositive" and "nonnegative" could be stated nonambiguously as 'negative' or 'positive' respectively by prefixing with “nonstrictly..” or maybe “loosely” or “generically” (if choosing to describe this nonstrict negativityorpositivity as onexorth‘other with just words).
If be there a necessity to assign $\{0\}$ decidedly to exclusively either $(∞,0)$ (partitioning the numberline into $\text{negative:= }(∞,0]$ and $\text{positive:= }(0,∞)$) or $(0,∞)$ (partitioning the numberline into $\text{negative:= }(∞,0)$ and $\text{positive:= }[0,∞)$) but not_both (i.e. $∄(\mathrm{sgn}(x)=0)$), then the case for in the default calling $0$ 'positive' is stronger than for calling it 'negative', because nonnegative values are slightly less common than nonpositive values, at least insofar as defining principal domains accordingly (e.g. modulus, squareroot), since our physical world deals with movement and allocation of nonnegative quantities objects (which sometimes includes a value of 0 units). Conceiving of an additive complement hinges on the existence of an additive basis/default; the case of $0$ being a complement to itself. The nullset ($\{\}$ ≡ ∅) $≠ \{0\}$, even though $∅$ (the cardinality of {}) $=0$, since the object $0$ is an element (not null, though in many contexts 'empty').
Treating 0 as distinct from negatives orand positives proper (which is accurately is appropriate in general case absent context), the unity of all strictlynegative real numbers with all strictlypositive real numbers can be notated as "$(∞,0)∪(0,∞)$", which can be easily abbreviated succinctly as "$ℝ\backslash\{0\}$". In cases where $0$ is being used along with numbers smaller than it but non larger, such as a limit approaching from the left, the inclusion of a few symbol(s) makes clear the exact meaning; same when dealing with other larger numbers but none smaller than 0, though notationally outside a specialized domain requires less qualifiers for the simple fact that a constant consisting of only digits and no signage symbol (plus, minus, plusminus, or minusplus), the default understood sign is positive (+).
tl;dr:
${[\text{[definition]}]}_{\{\#_{\text{max}}=5\}:=⟨a,b,c,d,e,f⟩}(\text{“}0\text{”}) \overset{\text{sign}}{⊨} $
$ ⟨[\underset{a}{\text{zero}}],[\underset{b}{\text{pos. & neg.}]},[\underset{c}{\text{pos. notneg.}}],[\underset{\:\:\:\:\:\:\:\:\:\:\:\:\:\:d}{\text{[cntxt.dep.]}_{\text{sensible}}}],[\underset{e}{\text{neg. notpos.}}],[\underset{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:f}{\text{[cntxt.dep.]}_{\text{rando}}}]⟩$
where the defined meanings are listed in descending order ⟨a,..,f⟩ of general preference (corroborated by preceding paragraphs and other answers & comments). Note that def.$d$ is entailed in {defs.[$a,b,c,e$]} i.e. {[def.$a$]∪[def.$b$]∪[def.$c$]∪[def.$e$]} (but is listed separately since it deviates from a 'standard' *default*meaning, weakening need for explanation aligning with the standard but strengthening justification of explanation for deviation from nonstandard) whereas def.$f$ does not conform to any of the standard meanings (but is listed for sake of completeness).
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