Editing Quater-imaginary base (section)

==Z-order curve==
The representation
:<math>z = \sum_{k\ge n} z_k \cdot (2i)^{-k}</math>
of an arbitrary complex number <math>z\in \Complex</math> with <math>z_k\in \{0,1,2,3\}</math> gives rise to an [[Injective function|injective]] mapping
:<math>\textstyle 
\begin{array}{llcl}
\varphi \colon & \Complex & \to & \R \\
 & \sum_{k \ge n} z_k \cdot (2i)^{-k} & \mapsto & \sum_{k \ge n} z_k \cdot r^{-k}\\
\end{array}
</math>
with some suitable <math>r \in \Z</math>. Here <math>r = 4</math> cannot be taken as base because of
:<math>\textstyle \sum_{k>0} 3\cdot (2i)^{-k} = \tfrac{-3-6i}5 \; \; \; \; \ne \; \; \; \; 1 = \sum_{k>0} 3 \cdot 4^{-k}.</math>
The [[Image of a function|image]] <math>\varphi(\Complex) \subset \R</math> is a [[Cantor set]] which allows to linearly order <math>\Complex</math> similar to a [[Z-order curve]]. Since the image is disconnected, <math>\varphi</math> is not [[Continuous function|continuous]].