The modulus of a complex number being negative?

Question

Why can't the modulus of a complex number considered negative? I mean it is obvious but has it ever been considered to invent any new maths in order to analyze such cases similarly to what created the imaginary unit $i$? Edit: why solving the equation $|z|^2= \alpha^2 + \beta^2 = k\; with\; k<0 $ using the tools from complex numbers does not give any appropriate result?

Answer 1

The modulus of a complex number, $\displaystyle z=a+b\cdot i$, where $a,b\in \mathbb R$, is defined as $$|z|=\sqrt{a^2+b^2}.$$ By convention, the range of $\sqrt x$ is $[0,\infty)$, i.e. it is always non-negative. It is also always real because the sum of squares of two real numbers is always non-negative.

$|z|$ can also be visualised as the distance of a point $(a,b)$ representing the complex number $\displaystyle z=a+b\cdot i$ , from the origin in the Argand plane. Distance is always positive (zero in case of origin).

Answer 2

Since the modulus is the distance from the point $(0,0)$ to the $(a,b)$, that represents the complex number $a+bi$, so it does not make sense to consider it negative, since it's a magnitude of its length. The idea behind create the complex unity $i$ is for the real polynomials with no solutions in real line, not to study the negative modulus, concept that cannot be negative, at least in euclidean space