Editing Quater-imaginary base (section)

==Decomposing the quater-imaginary==
In a [[Positional notation|positional system]] with base <math>b</math>, 

<math>\ldots d_3d_2d_1d_0.d_{-1}d_{-2}d_{-3}\ldots</math> 

represents<math>\dots + d_3\cdot b^3+d_2\cdot b^2+d_1\cdot b+d_0+d_{-1}\cdot b^{-1}+d_{-2}\cdot b^{-2}+d_{-3}\cdot b^{-3}\dots</math>

In this numeral system, <math>b = 2i</math>,

and because <math>(2i)^2=-4</math>, the entire series of powers can be separated into two different series, so that it simplifies to <math>\begin{align}
&{} [\dots+d_4\cdot(-4)^2 +d_2\cdot(-4)^1+d_0+d_{-2}\cdot(-4)^{-1}+\dots ]
\end{align}</math> for even-numbered digits (digits that simplify to the value of the digit times a power of -4), and <math>\begin{align}
2i\cdot[\dots+d_3\cdot(-4)^1+d_1+d_{-1}\cdot (-4)^{-1}+d_{-3}\cdot (-4)^{-2} + \dots]
\end{align}</math> for those digits that still have an imaginary factor. Adding these two series together then gives the total value of the number.

Because of the separation of these two series, the real and imaginary parts of complex numbers are readily expressed in base −4 as <math>\ldots d_4d_2d_0.d_{-2}\ldots</math> and <math>2\cdot(\ldots d_5d_3d_1.d_{-1}d_{-3}\ldots)</math> respectively.