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

Question

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?

Answer 1

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

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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

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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

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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

Answer 2

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.

Answer 3

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

$:=\qquad =$

Answer 4

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.

Answer 5

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.