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 $|x|\geq0$, no matter what number you plug in for $x$. For example: $$|-5|=5\geq0$$

We also learned that $x^2\geq0$. For example: $$(-5)^2=25\geq0$$ 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: $$4i^2=-4$$ 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?

Solution 1:

Is there any number whose absolute value is negative ?

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 $\sqrt{x_1^2+x_2^2+\ldots+x_n^2}\ge0$, 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.

How could it be used ?

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.