0
$\begingroup$

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?

$\endgroup$
  • 4
    $\begingroup$ $|x|\ge 0$ for all $x\in\mathbb R$. Indeed $|0|=0$. $\endgroup$ – user236182 Dec 27 '15 at 17:38
  • 5
    $\begingroup$ $| \cdot |$ is non-negative, but not positive $\endgroup$ – Alex Dec 27 '15 at 17:39
  • 3
    $\begingroup$ $|x|\ge 0$ for all $x\in\mathbb R$ and $|0|=0$. There is no contradiction: we have $|0|=0\ge 0$. $\endgroup$ – user236182 Dec 27 '15 at 17:42
  • 1
    $\begingroup$ To answer the question in the title: it's neither. $\endgroup$ – Wojowu Dec 27 '15 at 17:43
  • 1
    $\begingroup$ I'm not sure why it hasn't showed up yet, but the usual definition is $$\lvert x \rvert = \begin{cases}x, &x \ge 0\\-x, &x < 0\end{cases}$$ $\endgroup$ – pjs36 Dec 27 '15 at 17:47
1
$\begingroup$

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 $\operatorname{sgn}(x)=\pm 1$.

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

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.

$\endgroup$
1
$\begingroup$

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. $|0|=0$.

See https://en.wikipedia.org/wiki/Trichotomy_%28mathematics%29

$\endgroup$
1
$\begingroup$

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 $0$.

$\endgroup$
  • $\begingroup$ 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). $\endgroup$ – Simply Beautiful Art Dec 27 '15 at 18:03
1
$\begingroup$

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".

$\endgroup$
  • $\begingroup$ That is helpful (+1) $\endgroup$ – Simply Beautiful Art Dec 27 '15 at 18:07
  • $\begingroup$ 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. $\endgroup$ – Jean-Claude Arbaut Dec 27 '15 at 18:11

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.