Does excluding or including zero from the definitions of "positive" and "negative" make any consequential difference in mathematics?

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?

• 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. Apr 13, 2016 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 $0$. Apr 13, 2016 at 13:46
• Then the claim is true for all $m\ge 0$ rewritten with $m=n-1$. Apr 13, 2016 at 13:50
• 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." Apr 13, 2016 at 13:53
• 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. Apr 13, 2016 at 15:11

5 Answers

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.

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.

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

$:=\qquad =$

• 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." Apr 13, 2016 at 14:45
• The English/French definitions are also noted in math.stackexchange.com/questions/18464/… and later in math.stackexchange.com/questions/26705/… Apr 13, 2016 at 14:58

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
• So, you disagree that $0$ can be considered positive or/and negative, right ? Apr 13, 2016 at 13:58
• I think this is really missing the point. Apr 13, 2016 at 14:09
• @Peter: Right... Apr 13, 2016 at 14:09
• @DavidC.Ullrich: Why? Apr 13, 2016 at 14:09
• 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. Apr 13, 2016 at 14:41

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.