Editing Quater-imaginary base (section)

==Converting into quater-imaginary==
It is also possible to convert a decimal number to a number in the quater-imaginary system. Every [[complex number]] (every number of the form ''a''+''bi'') has a quater-imaginary representation. Most numbers have a unique quater-imaginary representation, but just as 1 has the two representations [[0.999...|1 = 0.{{overline|9}}]] in decimal notation, so, because of 0.{{overline|0001}}<sub>2''i''</sub> = {{sfrac|1|15}}, the number {{sfrac|1|5}} has the two quater-imaginary representations 0.{{overline|0003}}<sub>2''i''</sub> = 3·{{sfrac|1|15}} = {{sfrac|1|5}} = 1 + 3·{{sfrac|–4|15}} = 1.{{overline|0300}}<sub>2''i''</sub>.

To convert an arbitrary complex number to quater-imaginary, it is sufficient to split the number into its real and imaginary components, convert each of those separately, and then add the results by interleaving the digits. For example, since &minus;1+4''i'' is equal to &minus;1 plus 4''i'', the quater-imaginary representation of −1+4''i'' is the quater-imaginary representation of &minus;1 (namely, 103) plus the quater-imaginary representation of 4''i'' (namely, 20), which gives a final result of &minus;1+4''i'' = 123<sub>2''i''</sub>.

To find the quater-imaginary representation of the imaginary component, it suffices to multiply that component by 2''i'', which gives a real number; then find the quater-imaginary representation of that real number, and finally shift the representation by one place to the right (thus dividing by 2''i''). For example, the quater-imaginary representation of 6''i'' is calculated by multiplying 6''i'' × 2''i'' = &minus;12, which is expressed as 300<sub>2''i''</sub>, and then shifting by one place to the right, yielding: 6''i'' = 30<sub>2''i''</sub>.

Finding the quater-imaginary representation of an arbitrary real [[integer]] number can be done manually by solving a system of [[simultaneous equations]], as shown below, but there are faster methods for both real and imaginary integers, as shown in the [[Negative base#To negaquaternary|negative base]] article.

===Example: Real number===
As an example of an integer number we can try to find the quater-imaginary counterpart of the decimal number 7 (or 7<sub>10</sub> since the [[radix|base]] of the decimal system is 10). Since it is hard to predict exactly how long the digit string will be for a given decimal number, it is safe to assume a fairly large string. In this case, a string of six digits can be chosen. When an initial guess at the size of the string eventually turns out to be insufficient, a larger string can be used.

To find the representation, first write out the general formula, and group terms: 
:<math>
\begin{align}
7_{10}& = d_{0}+d_{1}\cdot b+d_{2}\cdot b^{2}+d_{3}\cdot b^{3}+d_{4}\cdot b^{4}+d_{5}\cdot b^{5} \\
& = d_{0}+2id_{1}-4d_{2}-8id_{3}+16d_{4}+32id_{5} \\
& = d_{0}-4d_{2}+16d_{4}+i(2d_{1}-8d_{3}+32d_{5}) \\
\end{align}
</math>

Since 7 is a real number, it is allowed to conclude that ''d<sub>1</sub>'', ''d<sub>3</sub>'' and ''d<sub>5</sub>'' should be zero. Now the value of the coefficients ''d<sub>0</sub>'', ''d<sub>2</sub>'' and  ''d<sub>4</sub>'', must be found. Because  d<sub>0</sub> − 4 d<sub>2</sub> + 16 d<sub>4</sub>  =  7 and because—by the nature of the quater-imaginary system—the coefficients can only be 0, 1, 2 or 3 the value of the coefficients can be found.  A possible configuration could be:  ''d<sub>0</sub>'' = 3,  ''d<sub>2</sub>'' = 3 and ''d<sub>4</sub>'' = 1. This configuration gives the resulting digit string for 7<sub>10</sub>.

:<math>7_{10} = 010303_{2i} = 10303_{2i}.</math>

===Example: Imaginary number===
Finding a quater-imaginary representation of a purely imaginary integer number {{math|∈ ''i'''''Z'''}} is analogous to the method described above for a real number. For example, to find the representation of 6''i'', it is possible to use the general formula. Then all coefficients of the real part have to be zero and the complex part should make 6. However, for 6''i'' it is easily seen by looking at the formula that if ''d<sub>1</sub>'' = 3 and all other coefficients are zero, we get the desired string for 6''i''. That is:

:<math>\begin{align}6i_{10} = 30_{2i}\end{align}</math>

===Another conversion method===
For real numbers the quater-imaginary representation is the same as negative quaternary (base −4). A complex number ''x''+''iy'' can be converted to quater-imaginary by converting ''x'' and ''y''/2 separately to negative quaternary. If both ''x'' and ''y'' are finite [[Binary number#fractions|binary fractions]] we can use the following algorithm using repeated [[Euclidean division]]:

For example: 35+23i=121003.2<sub>2i</sub>

                 35                                 23i/2i=11.5    11=12−0.5
             35÷(−4)=−8, remainder 3                12/(−4)=−3, remainder 0         (−0.5)×(−4)=2
             −8÷(−4)= 2, remainder 0                −3/(−4)= 1, remainder 1
              2÷(−4)= 0, remainder 2                 1/(−4)= 0, remainder 1
                2<small>0</small>0<small>0</small>3                    +              1<small>0</small>1<small>0</small>0<small>0</small>                         +  0.2 = 121003.2
                          32i+16×2−8i−4×0+2i×0+1×3−2×i/2=35+23i