Is the absolute value of zero positive or negative?

Question

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?

Answer 1

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

Answer 2

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.

Answer 3

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

Answer 4

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 $0$.

Answer 5

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".