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Quater-imaginary base

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The quater-imaginary numeral system (with radix and digits from the set ) allows the representation, in a unique way, of any complex number, no negative sign or imaginary being required.

It is like the negabinary (base –2) or negaquartal (base –4) numeral systems, but using the powers of . Those powers, from down to , are

Since we have the even powers of to represent the real part and the odd powers of to represent the imaginary part, this numeral system thus requires the digits 0, 1, 2, and 3.

(used with digit of negaquartal (base –4) representation of real part of )
(used with digit of negaquartal (base –4) representation of real part of )
(used with digit of negaquartal (base –4) representation of real part of )
(used with digit of negaquartal (base –4) representation of real part of )
(used with digit of negaquartal (base –4) representation of real part of )
(used with digit of negaquartal (base –4) representation of real part of )
(used with digit of negaquartal (base –4) representation of real part of )

For example, the quater-imaginary representation of 201 is

s s s s s s s s s
1 0 1 0 1 0 2 0 1

This particular example demonstrates that the base representation of real numbers is the same as that in negaquartal except for the "riffling" in of 0s corresponding to the odd-indexed powers of . To obtain the base representation of a complex number , do as above for the real part, then again for the real part of then shift right to divide it back by , giving the digits corresponding to the odd-indexed powers of . Like negabinary and negaquartal, quater-imaginary dispenses with the need for a sign bit, but additionally, eliminates the need to represent the real and imaginary parts of a complex number separately. Thus, rather than writing , we can simply write 102300.2, where

-12 =    300 (real part of , 30 interleaved with 0s)  
15i = 102000.2 (real part of , 1202 interleaved with 0s, then shift right to divide it back by )
      --------
      102300.2

Conversion tables

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


Base to base 10
Base Base 10
0 0
1 1
2 2
3 3
10 0+2i
11 1+2i
12 2+2i
13 3+2i
20 0+4i
21 1+4i
22 2+4i
23 3+4i
30 0+6i
31 1+6i
32 2+6i
33 3+6i
Base Base 10
100 −4
101 −3
102 −2
103 −1
110 −4+2i
111 −3+2i
112 −2+2i
113 −1+2i
120 −4+4i
121 −3+4i
122 −2+4i
123 −1+4i
130 −4+6i
131 −3+6i
132 −2+6i
133 −1+6i
Base Base 10
200 −8
201 −7
202 −6
203 −5
210 −8+2i
211 −7+2i
212 −6+2i
213 −5+2i
220 −8+4i
221 −7+4i
222 −6+4i
223 −5+4i
230 −8+6i
231 −7+6i
232 −6+6i
233 −5+6i
Base Base 10
300 −12
301 −11
302 −10
303 −9
310 −12+2i
311 −11+2i
312 −10+2i
313 −9+2i
320 −12+4i
321 −11+4i
322 −10+4i
323 −9+4i
330 −12+6i
331 −11+6i
332 −10+6i
333 −9+6i

Arithmetic operations

For the examples in the discussion below, we will use two pairs of integers: 12 and 8 (10100 and 10200, respectively), and (10303.2 and 102300, respectively).

Addition

Addition amounts to base −4 addition of real parts (digits corresponding to even powers of ) and base −4 addition of imaginary parts (digits corresponding to odd powers of ). Remember that "carrying" in base −4 (in fact for all negative bases) is subtracting, and this is applied separately for even indexed digits (real part) and for odd indexed digits (imaginary part).

In some cases, addition is as straightforward as in decimal, with no carrying needed at all.

  10100
+ 10200
  -----
= 20300

That is 12 + 8 = 20.

But for our second pair of operands, it is necessary to "carry." (Here 3 + 3 gives 2 carry 1, i.e. subtract 1 two positions to the left.)

   10303.2
+ 102300.0
  --------
= 102203.2

That is .

Subtraction

Subtraction amounts to base −4 subtraction of real parts (digits corresponding to even powers of ) and base −4 subtraction of imaginary parts (digits corresponding to odd powers of ). Remember that "borrowing" in base −4 (in fact for all negative bases) is adding, and this is applied separately for even indexed digits (real part) and for odd indexed digits (imaginary part).

In some cases, subtraction is as straightforward as in decimal, with no borrowing needed at all.

  20300
- 10100
  -----
= 10200

That is 20 – 12 = 8.

For our second pair of operands, it is necessary to "borrow." (Here 2 − 3 borrow 1, i.e. add 1 two positions to the left, gives 2.)

  102203.2
-  10303.2
  --------
= 102300.0

That is .

Multiplication

(...)

Division

Division within n-imaginary numeral systems is a challenge!

Algebraic operations

Exponentiation

(...)

Root extraction

Root extraction within n-imaginary numeral systems is a challenge!

Sequences

A212494 Base representation of nonnegative integers.

{0, 1, 2, 3, 10300, 10301, 10302, 10303, 10200, 10201, 10202, 10203, 10100, 10101, 10102, 10103, 10000, 10001, 10002, 10003, 20300, 20301, 20302, 20303, 20200, 20201, 20202, 20203, ...}

A007608 Nonnegative integers in base −4. (digits of base representation corresponding to even powers of , i.e. real part)

{0, 1, 2, 3, 130, 131, 132, 133, 120, 121, 122, 123, 110, 111, 112, 113, 100, 101, 102, 103, 230, 231, 232, 233, 220, 221, 222, 223, 210, 211, 212, 213, 200, 201, 202, 203, ...}

A177505 Base representation of nonnegative integers reinterpreted in base 4.

{0, 1, 2, 3, 304, 305, 306, 307, 288, 289, 290, 291, 272, 273, 274, 275, 256, 257, 258, 259, 560, 561, 562, 563, 544, 545, 546, 547, 528, 529, 530, 531, 512, 513, 514, 515, ...}

A212542 Base representation of negative integers.

{103, 102, 101, 100, 203, 202, 201, 200, 303, 302, 301, 300, 1030003, 1030002, 1030001, 1030000, 1030103, 1030102, 1030101, 1030100, 1030203, 1030202, 1030201, 1030200, ...}

A212526 Negative integers in base −4. (digits of base representation corresponding to even powers of , i.e. real part)

{13, 12, 11, 10, 23, 22, 21, 20, 33, 32, 31, 30, 1303, 1302, 1301, 1300, 1313, 1312, 1311, 1310, 1323, 1322, 1321, 1320, 1333, 1332, 1331, 1330, 1203, 1202, 1201, 1200, ...}

See also

  • n-imaginary numeral systems (with radix , and digits from the set )
  • Bi-imaginary numeral system (with radix and digits from the set )
  • Negabinary (base –2) numeral system
  • Negaquartal (base –4) numeral system

References

  • Donald Knuth, The Art of Computer Programming. Volume 2, 2nd Edition. Reading, Massachusetts: Addison-Wesley (1981): 189.

External links