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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?

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    $\begingroup$ What is your definition of absolute value? That needs to be cleared up first. $\endgroup$ – alex.jordan Jun 30 '15 at 23:18
  • $\begingroup$ @alex.jordan I would say that it is the distance from that number to the origin when plotted on a...graph with an origin? With cardinal numbers it's simple on a number line, and with complex numbers you have the real part and the imaginary part. I don't know how such a number described in my question would be plotted, so that's up to the answerer's discretion. $\endgroup$ – Jason Chen Jun 30 '15 at 23:21
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    $\begingroup$ And so now, what is your definition of distance? $\endgroup$ – alex.jordan Jun 30 '15 at 23:23
  • $\begingroup$ @alex.jordan It would be the number of units two points are from each other. Cardinal distances would be measured in cardinal units (1, 2, 3) and imaginary distances would be measured in imaginary units (1i, 2i, 3i). If you had a "blah" distance it would be measured in "blah" units (1blah, 2blah, 3blah). Just take the coefficients of the units and use that in your calculations. $\endgroup$ – Jason Chen Jun 30 '15 at 23:26
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    $\begingroup$ Actually, "imaginary distances" are (typically) measured in real units, a measure known as the modulus $\endgroup$ – MichaelChirico Jun 30 '15 at 23:30
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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.

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