Complex Distance
Why
The identification of $\C $ with a plane leads $\C $ to naturally inherit $\R ^2$'s notion of distance.
Definition
The absolute value or modulus of $z = (x, y) \in \C $ is the distance of $z$ to the origin. If $z \in \C $, then the modulus of $z$ is
\[ \sqrt{x^2 + y^2}. \]
In other words, the modulus of $z$ is the distance (in $\R ^2$ of $z = (x,y)$ from the origin $(0,0)$.
Notation
We denote the modulus of $z$ by $\Cmod{z}$.
Properties
For all $z, w \in \C $,
\[ \Cabs{z + w} \leq \Cabs{z} + \Cabs{w}. \]
Also, for all $z \in \C $, we have $\Cabs{\Re (z)} \leq \Cabs{z}$ and $\Cabs{\Im (z)} \leq \Cabs{z}$, and for all $z, w \in \C $,1\[ \Cabs{\Cabs{z} - \Cabs{w}} \leq \Cabs{z - w}. \]
- This follows from the triangle inequality. Future editions will include an account. ↩︎