TITLE: Why do we define the modulus of a complex number as we do?
QUESTION [25 upvotes]: For a complex number $z = a+bi$, we say that its modulus is: $$|z|=\sqrt{a^2+b^2}$$
When we draw complex numbers in the Argand diagram, intuitively, this makes sense.
But if we used a different projection for the diagram (i.e. a different metric for distance) then it wouldn't necessarily. Of course, complex numbers can also be written as:
$$z = re^{i\theta} = r(\cos\theta +i\sin\theta)$$
so an equivalent question could be, if this is what we define, why we define that:
$$|e^{i\theta}| = |\cos\theta + i\sin\theta| = 1$$
for all values of $\theta$, rather than just $\theta = n\pi$.
The answer may simply be that it is convenient to work with this definition. But is there a deeper reason? Are there any problems for which it is convenient to define things differently? And what would be the consequences if we did things differently?

REPLY [31 votes]: Do you like the absolute value on $\mathbf R$?  Well, it turns out that the formula $|a+bi| = \sqrt{a^2+b^2}$ is the unique absolute value on $\mathbf C$ extending the absolute value on $\mathbf R$. That is not why it was originally defined, but it provides an excellent reason that this function plays such a prominent role in work on the complex numbers. You don't need anything like polar decomposition ($re^{i\theta}$) to prove that uniqueness. It follows from $\mathbf R$ being complete and $\mathbf C$ being finite-dimensional over $\mathbf R$.
An "absolute value" on a field $F$ is an $\mathbf R$-valued function $|x|$ for $x \in F$ such that (i) $|x| \geq 0$ with equality if and only if $x = 0$, (ii), $|xy| = |x||y|$ for all $x$ and $y$ in $F$, and (iii) $|x+y| \leq |x| + |y|$ for all $x$ and $y$ in $F$.  Using an absolute value on $F$ we get a metric on $F$ by $d(x,y) = |x-y|$ and that leads to associated topological and analytic ideas on $F$ if it has some nice properties for this metric (complete, locally compact, and so on).
You can apply this concept to a division ring in place of a field.  In that spirit, the absolute value on quaternions, given by $|a+bi+cj+dk| = \sqrt{a^2+b^2+c^2+d^2}$, is the unique absolute value on the quaternions that extends the absolute value on $\mathbf R$.