There are numbers which are square roots of negative numbers and they're are those which are logarithms of negative numbers(both are complex numbers).
Are there any numbers whose moduli are negative? I mean, $|x|<0$
If yes, what are they?
If no, why?
On
The modulus, by definition, is a positive number as it represents a concept similar to the "magnitude" in Physics. So the fact that the value of a modulus is always positive does not depend on the complex number you are considering at all but is in fact an attribute of the modulus itself.
In $\mathbb C$, given a complex number $x = a + ib$ (with $a$, $b ∈ \mathbb R$) the modulus is defined as
$$\sqrt{x² + y²} $$
Since $y$ itself real, the number under the square root is always positive, hence the modulus is always real and positive.
From a more general and abstract point of view, the modulus is a special case of the norm in $\mathbb C$. By definition, for any vector space, the norm satisfies the below conditions:
On
The modulus is meant to represent the magnitude of a number, regardless its direction (also for the complex). This is precisely why it is defined to be a non-negative number.
You are free to define another "modulus", for example by keeping the ordinary modulus and doubling the argument:
$$[z]:=|z|e^{i2\angle z}=ze^{i\angle z}=\frac{z^2}{|z|}.$$
With this definition,
$$[1]=[-1]=1,\\ [i]=[-i]=-1,\\ [1\pm i]=[-1\pm i]=\pm i\sqrt2,\\\cdots$$
This is perfectly artificial.
In the complex numbers, no. $|z|$ is defined to be $\sqrt{z*\overline z}$ which is non negative as $z\overline z = \Re(z)^2 + \Im(z)^2$ is always non-negative real.
I suspect you are asking as we extended the reals to the complex (depending upon your philsophy) by allowing square roots of negatives which as a consequence allowed for logarithms of negative numbers, you are asking if we can create another number system that will allow $|z| < 0$.
It's hard for me to imagine how or why we would do so. There are no numbers that "want to exist but can't" because if they existed there modulus would have to be negative. And as modulus was defined to be non-neg real, it's hard to imagine in what sense we'd define another meaning for it that satisfy any condition associated with it. Primarily the condition that the modulus represents the quantitative measure of the absolute non-negative size of something.
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Okay, more.
In the reals $|x|$ is defined as $|x| = x$ if $x \ge 0$ and $|x| = -x$ if $x < 0$.
In extending to the complex numbers we could have kept that definition as close as possible by saying $|a+ib| = a+ib$ if $a \ge 0$ and $|a+ib| = -a-ib$ if $a < 0$. Admittedly we wouldn't ever have $|z| < 0$ but we would have $|z| \not \ge 0$. We could define, god knows why we'd want to but we could, $|a+ib| = a+b$ and the we could have $|z| < 0$.
So we don't we? Why instead to we replace the simple $|x| = \pm x$ with the scary looking $|a+bi| = \sqrt{(a+bi)(a-bi)} = \sqrt{a^2 + b^2}$?
Well, BECAUSE $|a+bi|= \sqrt{a^2 + b^2}$ IS greater than or equal to $0$. That $|z| \ge 0$ is a requirement of the definition of anything that we wish to call a "modulus".
Y2H in his/her answer lists some of the requirements for what something we call a "modulus" must obey. Why must we obey it to call it a modulus? Because if we didn't there wouldn't be any meaning to the word "modulus".
(Why is an elephant large, gray and wrinkly? Because if it were small, white and smooth it would be an aspirin.)
The very first requirement is ... that the "modulus" is real, and non-negative.
The modulus is essentially the "size" of a number. And size is positive value (or zero if and only if the number is zero). That's just.... axiomatic.
There are other conditions. By definition:
In an algebra, If $a,b \in F$, a field, $|a|$ is called the modulus, we must have:
i)$|a| \in \mathbb R; |a| \ge 0$.
ii) $|a| = 0$ if and only if $a$ is the multiplicative identity of $F$.
iii) $|ab| = |a||b|$
iv) $|a+b| \le |a| + |b|$
In vector spaces, $|x|$ is a norm and we must have (by definition) where $V,W$ are vectors in a vector space and $a$ is an element of a field:
i) $|V| \ge 0; |V| \in \mathbb R$
ii) $|V| = 0 \iff V = 0$
iii) $|aV| = |a||V|$ where $|a|$ is a modulus of $a$.
iv) $|V + W| \le |V| + |W|$.