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I was absolutely certain that zero was both positive and negative. And zero was neither strictly positive nor strictly negative.

But today I made a few Google searches, and they all say the same thing: zero is neither positive nor negative.

I suppose that the definition of "positive" and "negative" depend on which country we're living in. In the U.S. "positive" and "negative" exclude zero. In France "positive" and "negative" include zero.

My question therefore is: does excluding or including zero from the definitions of "positive" and "negative" make any consequential difference in mathematics?

Omega Force
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    This is just a question about the definition of two terms; as such it cannot possibly make any consequential difference. A definition has no consequences on what facts are true or false, it simply has consequences for how various statements are spelled. – David C. Ullrich Apr 13 '16 at 13:39
  • I do not agree. Suppose, a claim is true for all integers $n>0$, but not for $n=0$. In this case, it matters whether we include $0$ or exclude $0$. – Peter Apr 13 '16 at 13:46
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    Then the claim is true for all $m\ge 0$ rewritten with $m=n-1$. – Dietrich Burde Apr 13 '16 at 13:50
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    No, @Peter, it just has consequences for how you write the result, not to the underlying mathematics. If we consider $0$ as both positive an negative, then you simply cannot replace "for all $x>0$" with "for all positive $x$." There are fields of mathematics where it is more convenient to use "positive" to include zero, just for brevity, since "non-negative" is a mouthful and doesn't immediately get processed as "positive." – Thomas Andrews Apr 13 '16 at 13:53
  • @DietrichBurde Assuming the question is about integers, of course. – Thomas Andrews Apr 13 '16 at 13:53
  • @Peter No. Say A says $0$ is positive and B does not. Then A and B disagree on whether the sentence ($$) "$P(n)$ holds for all positive $n$" is true. But A and B do not disagree regarding any mathematical facts! Their disagreement is just over what ($$) means. A says ($$) is false, and B agrees that _what A means by ($$)_ is false. B says ($$) is true, and A agrees that _what B means by ($$)_ is true. – David C. Ullrich Apr 13 '16 at 14:06
  • @Peter If we say a claim is true for all integers $n>0$, then it's true for all $n>0$, no matter whether we consider zero to be "non-negative" or "positif". At least I can't recall seeing such controversy about the $>$ symbol. – David K Apr 13 '16 at 15:01
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    The point is that the definition of "positive" determines the set of the positive integers. So, the truth of a claim in the form "for all positive integers ..." can depend on this definition. The OP did not mention the $>$-symbol, which is of course absolutely clear. Many claims have the form "for all positive integers ...", so the OP's question IS meaningful. – Peter Apr 13 '16 at 15:11
  • $0$ should be neither positive nor negative. It is somewhat odd to regard $0$ to be both positive and negative. Why do we have the names non-negative and non-positive ? They would be obsolete, if $0$ would be both negative and positive. – Peter Apr 13 '16 at 15:17
  • In particular, if I recall correctly, in some math books, mostly related to algorithms (linear programming and the like,) they write $\mathbb R^+$ for the non-negatives, and write $\mathbb R^{++}$ for the positives. This struck me as funny, and I've joked that $\mathbb R^{++}$ means "no really, I mean positive." However, it is natural to want to use shorter and simple words and notations for the concepts we need most often, even at the risk of confusion across disciplines. – Thomas Andrews Apr 13 '16 at 16:42
  • @Peter Of course the truth of a claim in the form "for all positive integers ..." can depend on this definition! That's obvious to everyone. That does not show that changing the definition would have mathematical consequences. – David C. Ullrich Apr 13 '16 at 21:08

5 Answers5

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Mathematical definitions are just entries in a dictionary, translating between one language and another.

There is certainly power in choosing names and formulating definitions -- I think of it as the "power of Adam". Good names and good definitions will get used a lot, poorer names and poorer definitions won't. There are even aesthetic issues that come into play in deciding between different terminology. For example, one of my personal aesthetic criteria is to avoid acronyms. Also, I know of mathematicians who dislike personal names being attached to mathematical objects, although that's a hard issue to fight against.

Nonetheless, for two different systems of mathematical terminology and definitions, there will be a dictionary that can be used to translate between them. Ideally there will even be a "compiler" that will do that translation automatically and efficiently, just as there are natural language translation devices that convert English to French and back (with admittedly comic outcomes sometimes...)

The translation between two different definitions of "positive and negative" in your question is a simple example of this. As long as the reader knows what "positive" means in the context of what they are reading -- and it is the author's responsibility to be clear on that point -- the reader should be able to make the translation automatically and efficiently into whatever language they are more confortable with.

Lee Mosher
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Depending on the context, sometimes we want $0$ to be excluded from the universe of discourse, and sometimes we want to have it included. Therefore people use notations like ${\mathbb R}_{>0}$ vs. ${\mathbb R}_{\geq0}$. These two sets are certainly different, and neglecting this fact can have detrimental consequences in mathematics, e.g., if you want to divide by $0$.

Now we like to have verbal descriptions of the the two properties $x\in{\mathbb R}_{>0}$, resp., $x\in{\mathbb R}_{\geq0}$. In English these descriptions are positive, resp., nonnegative, and similarly in German. It seems that in French they use strictement positif, resp. positif. But note that these "semantical differences" are of a purely linguistic character; and there is not the slightest mathematical truth bending involved.

Christian Blatter
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It makes no difference. It just means changing all statements of theorems and proofs. Where I (and most English-speakers) would say

$\text{positive}\qquad\qquad\text{nonnegative}$

Bourbaki (and many French speakers) would say

$\text{strictly positive}\qquad\text{positive}$

There are other, similar situations:

$\text{increasing}\qquad\qquad\text{nondecreasing}$

or

$\text{strictly increasing}\qquad\text{increasing}$

How about

$\subset \qquad \subseteq$

or

$\subsetneq \qquad \subset$

Maybe programming languages, which say

$=\qquad =\,=$

or

$:=\qquad =$

GEdgar
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    Comment: The OP, Omega Force, seems to be French. Surely French math teachers should learn to go over this difference with their students at some point! They can say: "Our system is of course better than the English system. But in order to read math papers in English, you need to know their inferior terminology." – GEdgar Apr 13 '16 at 14:45
  • The English/French definitions are also noted in http://math.stackexchange.com/questions/18464/is-positive-the-same-as-non-negative and later in http://math.stackexchange.com/questions/26705/is-zero-positive-or-negative?rq=1 – David K Apr 13 '16 at 14:58
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When multiplying value A by a positive value B, the sign of the result is identical to the sign of A:

  • If A is positive, then the result is positive
  • If A is negative, then the result is negative

When multiplying value A by a negative value B, the sign of the result is opposite to the sign of A:

  • If A is positive, then the result is negative
  • If A is negative, then the result is positive

Let's prove by contradiction that $0$ is not positive:

  • Assume that $0$ is positive
  • $-1$ is negative
  • Therefore $(-1)\cdot0$ is negative
  • But $(-1)\cdot0=0$, and $0$ is positive

Let's prove by contradiction that $0$ is not negative:

  • Assume that $0$ is negative
  • $-1$ is negative
  • Therefore $(-1)\cdot0$ is positive
  • But $(-1)\cdot0=0$, and $0$ is negative
barak manos
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  • So, you disagree that $0$ can be considered positive or/and negative, right ? – Peter Apr 13 '16 at 13:58
  • I think this is really missing the point. – David C. Ullrich Apr 13 '16 at 14:09
  • @Peter: Right... – barak manos Apr 13 '16 at 14:09
  • @DavidC.Ullrich: Why? – barak manos Apr 13 '16 at 14:09
  • You start with certain assumptions about positive and negative, and show from those assumptions that $0$ is not positive. Fine. But this is irrelevant to the question of whether changing the definitions to include $0$ as a positive number would have mathematical consequences! If we included $0$ as a positive number that would not change any of the mathematical facts - it would just mean that the English sentences expressing those facts would have to be written differently. – David C. Ullrich Apr 13 '16 at 14:16
  • @DavidC.Ullrich: So you don't think that if we change the definition of zero, then the definition of multiplication (as stated above) would immediately change as well (which is a profound implication on Mathematics)??? – barak manos Apr 13 '16 at 14:22
  • To put it another way: The OP says that in France $0$ is considered to be both positive and negative. Let's assume that's so. Does it follow that French mathematics is different from US mathematics? Of course not. Mathematical facts are true in the US if and only if they're true in France - the way those facts are written is different, but the French calling $0$ positive has no consequences for the actual mathematics. – David C. Ullrich Apr 13 '16 at 14:22
  • @DavidC.Ullrich: Please read the (fresh) comment above yours. – barak manos Apr 13 '16 at 14:23
  • Of course the definition of multiplication would change as well! Many many statements would look different. That would not have a profound implication for mathematics, it would have no implications for mathematics whatever. It would have implications for what words you need to write to express a given mathematical fact. – David C. Ullrich Apr 13 '16 at 14:24
  • Do you really think that some mathematical facts are true in France but false in the US? – David C. Ullrich Apr 13 '16 at 14:25
  • @DavidC.Ullrich: No, I think what OP has stated about France is wrong. I was trying to explain that in the answer above, by implying that if it had been true, then that would have made our definition of multiplication false!!! – barak manos Apr 13 '16 at 14:26
  • "I think what OP has stated about France is wrong". That's irrelevant. I said "Let's assume so". Suppose for the sake of argument that what he says about France is true. Now: Does it follow that some mathematical facts are true in France but false in the US? – David C. Ullrich Apr 13 '16 at 14:28
  • @DavidC.Ullrich: If what OP states about $0$ being negative and/or positive in France is true, then the definition of multiplication in France would have to be different in some manner (different than what I have described above - perhaps the description is wrong and there should be a special case for $0$, but it would still be a different definition in each country, as a consequence of the different definitions for $0$). – barak manos Apr 13 '16 at 14:30
  • Yes, I've agreed several times that obviously a lot of things would have to be changed. That is, the way a lot of things are written would have to be changed. That doesn't answer the question: Are there mathematical facts that are true in France but false in the US? – David C. Ullrich Apr 13 '16 at 14:35
  • @DavidC.Ullrich: That's not what the question says. It says 'does excluding or including zero from the definitions of "positive" and "negative" make any consequential difference in mathematics?'. And I tried to answer that it would make a consequential difference - it would alter our definition of multiplication. – barak manos Apr 13 '16 at 14:38
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    And you're wrong about that. It would not change the meaning of the definition of multiplication. It would change how that definition must be worded - that's obvious. You are in fact missing the point. I started this when there were two answers, the one with more votes being the wrong one - seemed like a Good Thing to try to set the record straight. That's changed - now that you're substantially outnumbered we can drop this. – David C. Ullrich Apr 13 '16 at 14:41
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Injecting a little chauvinism just for fun:

Positive and negative, maybe the French are simply wrong. Consider $\Bbb N$, the set of natural numbers. English-speaking mathematicians do disagree on whether or not $0\in\Bbb N$. And that disagreement has no mathematical consequences whatever.

David C. Ullrich
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