Why there is no value for $x$ if $|x| = -1$?

Question

According to the definition of absolute value negative values are forbidden.

But what if I tried to solve a equation and the final result came like this: $|x|=-1$ One can say there is no value for $x$, or this result is forbidden.

That reminds me that same thinking in the past when mathematical scientists did not accept the square root of $-1$ saying that it is forbidden.

Now the question is :"is it possible for the community of math to accept this term like they accept imaginary number.

For example, they may give it a sign like $j$ and call it unreal absolute number then a complex number can be expanded like this:

$x = 5 +3i+2j$ , where $j$ is unreal absolute number $|x|=-1$

An other example, if $|x| = -5$, then $x=5j$

The above examples are just simple thinking of how complex number may expanded

You may ask me what is the use of this strange new term? or what are the benefits of that?

I am sure this question has been raised before in the past when mathematical scientists decided to accept $\sqrt{-1}$ as imaginary number. After that they knew the importance of imaginary number.

Answer 1

The beauty of math is that you can define everything. The question is: what properties you want this "j" to satisfy? For example, I guess that you want the absolute value $|\cdot|$ to satisfy the triangle inequality. Note that $$ 0=|0|=|j+(-j)|\leq|j|+|-j|=-1-1=-2 $$ a contradiction.

Answer 2

Absolute value is a special case of the definition of norm. We need to impose some properties on it so that the domain of the function (or the space which the norm is defined on) can have special structures. The properties of a certain function result from a specific purpose.