quater-imaginary base

The quater-imaginary numeral system was first proposed by Donald Knuth in 1955, in a submission to a high-school science talent search. It is a positional system which uses the imaginary number 2i as base. By analogy with the quaternary numeral system, it is able to represent every complex number using only the digits 0, 1, 2, and 3, without a sign.

Powers of 2i

Note that i⁻¹ = −i. n>

math>(2i)^n
lign=right\|−8	align=right\|1/256
lign=right\|−7	align=right\|1/128 i
lign=right\|−6	align=right\|−1/64
lign=right\|−5	align=right\|−1/32 i
lign=right\|−4	align=right\|1/16
lign=right\|−3	align=right\|1/8 i
lign=right\|−2	align=right\|−1/4
lign=right\|−1	align=right\|−1/2 i
lign=right\|0	align=right\|1
lign=right\|1	align=right\|2i
lign=right\|2	align=right\|−4
lign=right\|3	align=right\|−8i
lign=right\|4	align=right\|16
lign=right\|5	align=right\|32i
lign=right\|6	align=right\|−64
lign=right\|7	align=right\|−128i
lign=right\|8	align=right\|256

Decimal to quater-imaginary

ase 10	Base 2i	Base 10	Base 2i	Base 10	Base 2i	Base 10	Base 2i
lign=right\|1	align=right\|1	align=right\|−1	align=right\|103	align=right\|1i	align=right\|10.2	align=right\|-1i	align=right\|0.2
lign=right\|2	align=right\|2	align=right\|−2	align=right\|102	align=right\|2i	align=right\|10.0	align=right\|-2i	align=right\|1030.0
lign=right\|3	align=right\|3	align=right\|−3	align=right\|101	align=right\|3i	align=right\|20.2	align=right\|-3i	align=right\|1030.2
lign=right\|4	align=right\|10300	align=right\|−4	align=right\|100	align=right\|4i	align=right\|20.0	align=right\|-4i	align=right\|1020.0
lign=right\|5	align=right\|10301	align=right\|−5	align=right\|203	align=right\|5i	align=right\|30.2	align=right\|-5i	align=right\|1020.2
lign=right\|6	align=right\|10302	align=right\|−6	align=right\|202	align=right\|6i	align=right\|30.0	align=right\|-6i	align=right\|1010.0
lign=right\|7	align=right\|10303	align=right\|−7	align=right\|201	align=right\|7i	align=right\|103000.2	align=right\|-7i	align=right\|1010.2
lign=right\|8	align=right\|10200	align=right\|−8	align=right\|200	align=right\|8i	align=right\|103000.0	align=right\|-8i	align=right\|1000.0
lign=right\|9	align=right\|10201	align=right\|−9	align=right\|303	align=right\|9i	align=right\|103010.2	align=right\|-9i	align=right\|1000.2
lign=right\|10	align=right\|10202	align=right\|−10	align=right\|302	align=right\|10i	align=right\|103010.0	align=right\|-10i	align=right\|2030.0
lign=right\|11	align=right\|10203	align=right\|−11	align=right\|301	align=right\|11i	align=right\|103020.2	align=right\|-11i	align=right\|2030.2
lign=right\|12	align=right\|10100	align=right\|−12	align=right\|300	align=right\|12i	align=right\|103020.0	align=right\|-12i	align=right\|2020.0
lign=right\|13	align=right\|10101	align=right\|−13	align=right\|1030003	align=right\|13i	align=right\|103030.2	align=right\|-13i	align=right\|2020.2
lign=right\|14	align=right\|10102	align=right\|−14	align=right\|1030002	align=right\|14i	align=right\|103030.0	align=right\|-14i	align=right\|2010.0
lign=right\|15	align=right\|10103	align=right\|−15	align=right\|1030001	align=right\|15i	align=right\|102000.2	align=right\|-15i	align=right\|2010.2
lign=right\|16	align=right\|10000	align=right\|−16	align=right\|1030000	align=right\|16i	align=right\|102000.0	align=right\|-16i	align=right\|2000.0

Examples

5 = 16 + (3\cdot-4) + 1 = 10301_{2i}

i = 2i + 2\left(-\frac{1}{2}i\right) = 10.2_{2i}

11210.31_{2i} = 1(16) + 1(-8i) + 2(-4) + 1(2i) + 3\left(-\frac{1}{2}i\right) + 1\left(-\frac{1}{4}\right) = 7 \frac{3}{4} - 7 \frac{1}{2}i

References

*D. Knuth. The Art of Computer Programming. Volume 2, 3rd Edition. Addison-Wesley. pp. 205, "Positional Number Systems"

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