Quater-imaginary base

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Positional systems with various bases:
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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 non-standard positional numeral 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.

Contents 1 Powers of 2i 2 Decimal to quater-imaginary 3 Examples 4 References

Powers of 2i

Note that i⁻¹ = −i.

<math>n</math>	<math>(2i)^n</math>
−8	1/256
−7	1/128 i
−6	−1/64
−5	−1/32 i
−4	1/16
−3	1/8 i
−2	−1/4
−1	−1/2 i
0	1
1	2i
2	−4
3	−8i
4	16
5	32i
6	−64
7	−128i
8	256

Decimal to quater-imaginary

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

Examples

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

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

<math>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</math>

References

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