Is zero positive or negative?

Question

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

Answer 1

$0$ is neither positive nor negative

Answer 2

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.

Answer 3

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

Answer 4

$0$ 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.

Answer 5

In neutral context, the number $0$ is distinct from each of positive and negative; it is the exactly-middle value of the trichotomous non-projective 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 number-line 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 strictly-negative) and $0<x<∞$ (when $x∈(0,∞)$, i.e. $x$ is strictly-positive), so it is reasonable to think of the value $0$ as belonging to one or both of the non-discrete 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 equally-justifiable as negative and positive, thus considering negative numbers as belonging to $(-∞,0]$ and positives to $[0,∞)$, non-disjoint (the exhaustive ranges not mutually exclusive) from eachother —although more-precisely (i.e., sans ambiguity) these intervals closed at zero are called "non-positive" and "nonnegative" respectively, relegating the open-interval counterparts to respectively "negative" and "positive" (somewhat deferring the need to qualify these non-zero categories with "strictly") Pedantically, "non-positive" and "non-negative" could be stated nonambiguously as 'negative' or 'positive' respectively by prefixing with “non-strictly..” or maybe “loosely” or “generically” (if choosing to describe this non-strict negativity-or-positivity as one-xor-th‘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 non-negative values are slightly less common than non-positive values, at least insofar as defining principal domains accordingly (e.g. modulus, squareroot), since our physical world deals with movement and allocation of non-negative 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 strictly-negative real numbers with all strictly-positive 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, plus-minus, or minus-plus), 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. not-neg.}}],[\underset{\:\:\:\:\:\:\:\:\:\:\:\:\:\:d}{\text{[cntxt.-dep.]}_{\text{sensible}}}],[\underset{e}{\text{neg. not-pos.}}],[\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 non-standard) whereas def.$f$ does not conform to any of the standard meanings (but is listed for sake of completeness).