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At first, the number zero looked like it was positive to me because positive numbers can be written with or without a plus sign to the left of them, but it's false. I was surprised when I heard that zero is neither positive nor negative, but it's still a number and it's still even. At least I know it's in between the positive and the negative numbers, so that must be why. Also, because zero is neither negative nor positive, it's also known as neutral. Am I on the right track? Also, is there such thing as ±0, since it's neutral? That's why I put the plus/minus sign there. Is this how zero is neither negative nor positive? I can see happy faces in your answers!

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    $\begingroup$ Look: the law of trichotomy $\endgroup$ – ThePortakal Oct 29 '14 at 21:03
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    $\begingroup$ Now, there are some ways in which $0$ is more like positive numbers than negative ways. For example, the square root of a positive number is another positive number. The square root of a negative number is an imaginary number (have you learned about those yet? it can be a very difficult concept to comprehend). But the square root of $0$ is $0$ itself, and the only other number that is its own square root is $+1$. $\endgroup$ – Mr. Brooks Oct 29 '14 at 22:29
  • $\begingroup$ Not to get carried away with closing for being "off-topic" or anything, but here's a very similar question: math.stackexchange.com/questions/26705/… $\endgroup$ – Lisa Oct 30 '14 at 22:02
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    $\begingroup$ If zero would be positive then $(-1) \cdot 0$ would be negative... $\endgroup$ – N. S. Jun 7 '15 at 17:14
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I've never heard it called "neutral," but if you must absolutely have an adjective for it, I suppose that's as good as any.

Think about your bank account. If the balance is negative, then that means you owe the bank money. If the balance is positive, then that means you have money with which to pay for goods or services. But if your bank account is exactly $\$0.00$, you don't owe the bank money but your don't have any money to spend. It's a good thing that you don't owe, but a zeroed out balance is neither positive nor negative.

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Would it make sense to adjust the definition of positive or negative so that one of them includes $0$? The following pretty theorems

  • The product of a positve and a negative number is negative
  • The product of two negative numbers is positive
  • The product of two positive numbers is positive

would require ugly adjustments, e.g.

  • The product of a positive number and a negative number is negative, except when the positive number happens to be zero, in which case the result is zero, hence positive

The emergence of ugly theorems is often a hint that the definitions are bad (as in: not very useful - recall that definitions cannot be "wrong")

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    $\begingroup$ Worth noting: In French, 0 is both “positif” and “negatif.” To express English “positive” in French, say “strictment positif” or “positif et non nul.” fr.wikipedia.org/wiki/Nombre_positif $\endgroup$ – Steve Kass Oct 29 '14 at 21:30
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    $\begingroup$ You gotta take Wikipedia with a huge grain of salt. $\endgroup$ – Mr. Brooks Oct 29 '14 at 22:20
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So much ground to cover, I'm going to try to address every part of your question, though not quite in the order you presented it.

How is it true that zero is neither a positive number nor a negative number?

Do you know about additive inverses? Define $f(x)$ to be the number such that $x + f(x) = 0$. Turns out that $f(x) = -1 \times x = -x$. The additive inverse of a positive number is a negative number. For example, the additive inverse of 8 is $-8$. The additive inverse of a negative number is a positive number. For example, the additive inverse of $-\frac{3}{2}$ is $\frac{3}{2}$. We can say that $x \neq -x$. Except if $x = 0$, in which case $-1 \times 0 = 0$. This means that 0 is its own additive inverse. The point is that no positive number is its own inverse, nor does any negative number have this property either.

Hagen von Eitzen has already brought up the multiplicative perspective, but allow me to expand on his point. Consider the equation $ax = b$. Suppose I tell you exactly what $a$ and $b$ are. Can you determine what $x$ is?

  • $a = -5$, $b = \frac{22}{7}$
  • $a = \sqrt{43}$, $b = -3698$

Now let's make the game a little more difficult. I'll still tell you what $b$ is, but instead of telling you what $a$ is, I will give you hints.

  • $a$ in an integer and $a < 0$, $b = 197$
  • $a$ might be an integer and $a \geq 0$, $b = 0$

In the former, there are two possibilities for $x$. But in the latter, there are infinitely many possibilities. If $a \neq 0$, then $x$ must be 0. But if $a = 0$, then $x$ can be anything, including 0.

At first, the number zero looked like it was positive to me because positive numbers can be written with or without a plus sign to the left of them ... Also, is there such thing as $\pm0$

Think of 0 as the point of origin, and think of positive as meaning to the right and negative to the left (or vice-versa, if you like). $-4$ means you go 4 to the left, $+7$ means 7 to the right. So $-0$ means you go 0 to the left, that is, you stay at the point of origin, and similarly for $+0$.

However, this reminds me about something I read somewhere about two's complement, which is used by almost all computers today. Without two's complement, 0 might have more than one representation as 0s and 1s inside a computer, including a bona fide $-0$ that is different than $+0$. But I guess that's a digression into computer programming.

but it's still a number and it's still even.

Yes, it's still an integer, and it's still even. If $m$ and $n$ are both even, then $m - n$ is also even, correct? What if $m = n$? Then $m - n = 0$. For example, $12 - 10 = 2$ and $12 - 14 = -24$, so what is $12 - 12$?

Also, because zero is neither negative nor positive, it's also known as neutral.

Sure, you can call it neutral and people will understand what you mean, but this usage is hardly common.

I can see happy faces in your answers!

Not from me. I've had kind of a confused face on. I've been going back and forth beteween thinking this stuff should be obvious and thinking this stuff is taken for granted but shouldn't be.

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This is just a matter of definition.

If you define positive numbers as all the real numbers $x$ such that $x=|x|$, then $0$ is a positive number. If you define positive numbers as all the real numbers $x$ such that $x=|x|$ except $0$, then $0$ is not a positive number.

Added: Maybe I have to admit, sometimes it is not only a matter of definition, but also a matter personal preference.

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  • $\begingroup$ Well, absolute value is zero or positive unless it has a negative sign in front of it to make it zero or negative. $\endgroup$ – Mathster Oct 29 '14 at 20:55
  • $\begingroup$ @Mathster yeah, but first of all, before using terms as "positive" or "negative", we should give them definitions, right? $\endgroup$ – Petite Etincelle Oct 29 '14 at 20:58
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    $\begingroup$ And for things other than numbers, "positive" may or may not include $0$. For example, a "positive operator" or a "positive linear functional" or a "positive measure" could be $0$. $\endgroup$ – Robert Israel Oct 29 '14 at 22:24
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    $\begingroup$ You both got off-track. I was hoping to see someone explain: What do we gain from defining 0 as a positive number? $\endgroup$ – user155234 Oct 30 '14 at 23:03
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    $\begingroup$ There are some things to be gained by defining 0 to be positive, but they are rather minor. For example, you can then say that the norm of a number in an imaginary quadratic field is always positive. But it would much easier to say that that norm is never negative. $\endgroup$ – Robert Soupe Oct 31 '14 at 2:14

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