# Why do we define the modulus of a complex number as we do?

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?

• Definitions are almost always made because something is used repeatedly and is worth defining. This is one such example. – CyclotomicField Apr 11 at 2:34
• It seems quite natural to define the magnitude (or "norm") of a complex number $z$ to be the distance from $z$ to the origin, and Euclidean distance is probably the most intuitive way of measuring distance. One nice thing about this definition of the norm of $z$ is that norms are multiplicative: $\| z_1 z_2 \| = \| z_1 \| \| z_2 \|$. – littleO Apr 11 at 2:39