# Is there a number whose absolute value is negative? I've recently started to think about this, and

Is there a number whose absolute value is negative?
I've recently started to think about this, and I'm sure a couple of you out there have, too.
In Algebra, we learned that $|x|\ge 0$ , no matter what number you plug in for x. For example:
$|-5|=5\ge 0$
We also learned that ${x}^{2}\ge 0$. For example:
$\left(-5{\right)}^{2}=25\ge 0$
The exception for the ${x}^{2}$ rule is imaginary numbers (which we learn later on in Algebra II). Imaginary numbers are unique, in that their square is a negative number. For example:
$4{i}^{2}=-4$
These imaginary numbers can be used when finding the "missing" roots of a polynomial equation.
My question to you is this: Is there any number whose absolute value is negative, and how could it be used?
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Jordin Church
If such a number were allowed to exist, it could not be a part of ${\mathbb{R}}^{n}$, with $n\in \mathbb{N}$, because the absolute value of any such number is $\sqrt{{x}_{1}^{2}+{x}_{2}^{2}+\dots +{x}_{n}^{2}}\ge 0$, since ${x}_{i}\in \mathbb{R}$. But could it be part of ${\mathbb{R}}^{a}$, with $a\in {\mathbb{Q}}_{+}^{\star }\setminus \mathbb{N}$? Unfortunately, such factional-order sets have yet to be studied. Or perhaps part of something else altogether ? We don't know.
In my opinion, this is the real question... because, if someone were to find a “practical” use for such a quantity (inside mathematics itself, at the very least), then people would allow it to exist, and study it, and research it, just like they did with the imaginary unit i.