# Is the absolute value of zero positive or negative?

If I had $|x|$, then we know, for pretty much any $x$, that the following is true:$$|x|\ge0$$$$|0|=0?$$Which, by the nature of how we usually apply the absolute value, the solution is positive and real.

But that would make $|0|$ positive?

And since it equals itself, then I have come to the solution that $0$ is positive.

Which has become a contradiction? Because $-0=0$, therefore what?

Is the absolute value of zero defined easily? And is it positive?

According to the comments, the absolute value of $x$ is not negative, so the absolute value of $0$ is not negative either?

• $|x|\ge 0$ for all $x\in\mathbb R$. Indeed $|0|=0$. – user236182 Dec 27 '15 at 17:38
• $| \cdot |$ is non-negative, but not positive – Alex Dec 27 '15 at 17:39
• $|x|\ge 0$ for all $x\in\mathbb R$ and $|0|=0$. There is no contradiction: we have $|0|=0\ge 0$. – user236182 Dec 27 '15 at 17:42
• To answer the question in the title: it's neither. – Wojowu Dec 27 '15 at 17:43
• I'm not sure why it hasn't showed up yet, but the usual definition is $$\lvert x \rvert = \begin{cases}x, &x \ge 0\\-x, &x < 0\end{cases}$$ – pjs36 Dec 27 '15 at 17:47

We have $$|x|\geqslant 0$$

Absolute value is not strictly positive but it's non-negative. Zero doesn't have any sign. For example we define $\operatorname{sgn}(0) = 0$ whereas all other numbers satisfy $\operatorname{sgn}(x)=\pm 1$.

• @SimpleArt Sign function – Wojowu Dec 27 '15 at 17:43
• So could you say that the absolute value is non-negative? – Simply Beautiful Art Dec 27 '15 at 17:47
• @SimpleArt yes, it is. You cannot say $|x|$ is always positive because for $x=0$ it's neither positive nor negative. – Kamil Jarosz Dec 27 '15 at 17:49
• So what does that mean for $0$? It's like saying 1 is on and 0 is off, but the solution is 1/2. – Simply Beautiful Art Dec 27 '15 at 17:50
• (sgn(x)) It's like saying a positive number is 1 and a negative number is -1, but 0 is 0, so what does that mean for 0? – Simply Beautiful Art Dec 27 '15 at 17:54

You mistake is the statement 'But that would make $|0|$ positive?'. No, it would not. Absolute value makes the expression non-negative, but not everywhere positive.

No, the absolute value of zero is zero. For real numbers, the law of trichotomy states that every real number is either positive, negative, or zero. $|0|=0$.

I'm not sure if this is necessarily the correct way to look at this, but could you not consider the absolute value function $|x|$ to be the distance function in $\mathbb R$, applied to $x$ and $0$, i.e. the distance of $x$ from $0$ (written $d(x,0)=d(0,x)$)? Then the distance from $0$ to $0$ is obviously $0$.

• That sort of brings me back to saying that the distance from $0$ to $0$ is positive, which it has appeared to obviously not be (completely). – Simply Beautiful Art Dec 27 '15 at 18:03

The word "positive" is ambiguous: it can mean "$\ge 0$" or "$>0$". You can distinguish these two cases by calling them "non-negative" and "strictly positive".

• That is helpful (+1) – Simply Beautiful Art Dec 27 '15 at 18:07
• And to add to the ambiguity, the usual meaning of "positive" alone is not the same in every country. In France, you usually say positive for $\geq0$ and strictly positive for $>0$. In anglo-saxon countries, I believe positive is $>0$ and $\geq0$ is non-negative. The best is to always be explicit. For $\geq$, it's also frequent to see "positive or null", at least in France. – Jean-Claude Arbaut Dec 27 '15 at 18:11