I didn't understand why the absolute value of $(a+bi)$ is equal to $\sqrt{a^2+b^2}$ but not $\sqrt{(a+bi)^2}$ like $|x|=\sqrt{x^2}$
if $|x|=\sqrt{x^2}$ is right and if we give $x = a+bi$ it should be $\sqrt{(a+bi)^2}$
and why this proof is wrong:
$z=a+bi$
$|z|^2=z^2$
$|z|^2=(a+bi)^2$
$|z|=\sqrt{(a+bi)^2}$
Note: I don't want "Cartesian representation of a complex number" I don't see it as proof.
So I wrongly generalized the equation of the absolute value of a real number:
my generalization:
$|x|=\sqrt{x^2}$ (only right for real numbers)
right definition: (by my understanding so far)
$|a_1x_1+a_2x_2…a_nx_n| = \sqrt{\Sigma_n{(a_kx_k)^2}}$
Best Answer
For a complex number $z$, we don't speak about the "absolute value of $z$" but we speak about the modulus of $z$ denoted by $|z|$. If the algebraic form of $z$ is $a + ib$, then by definition you set $|z| = \sqrt{a^2 + b^2}$. Note that this definition extends the absolute value on $\mathbb R$.
Also, if $z = i$, how do you define $\sqrt{i^2}$ ? More generally, how do you define the square root of a complexe number ? How can you distinguish the two roots ?