Quater-imaginary Base - Converting Into Quater-imaginary

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 1.0 = 0.9... in decimal notation, so 1/₅ has the two quater-imaginary representations 1.(0300)…_2i = 0.(0003)…_2i.

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 –1+4i is equal to –1 plus 4i, the quater-imaginary representation of –1+4i is the quater-imaginary representation of –1 (namely, 103) plus the quater-imaginary representation of 4i (namely, 20), which gives a final result of –1+4i = 123_2i.

To find the quater-imaginary representation of the imaginary component, it suffices to multiply that component by 2i, 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 2i). For example, the quater-imaginary representation of 6i is calculated by multiplying 6i • 2i = –12, which is expressed as 300_2i, and then shifting by one place to the right, yielding: 6i = 30_2i.

Finding the quater-imaginary representation of an arbitrary real number can be done manually by solving a system of simultaneous equations, as shown below.