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https://blog.naver.com/ghangth/221236461513

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복소수 진법 : 네이버 블로그

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복소수 진법 위키백과 (영문)

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복소수 진법 위키백과 (영문)

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https://en.m.wikipedia.org/wiki/Quater-imaginary_base

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The quater-imaginary numeral system is a numeral system, first proposed by Donald Knuth in 1960. Unlike standard numeral systems, which use an integer (such as 10 in decimal, or 2 in binary) as their bases, it uses the imaginary number (such that ) as its base. It is able to ( almost) uniquely represent every complex number using only the digits 0, 1, 2, and 3.[1] Numbers less than zero, which are ordinarily represented with a minus sign, are representable as digit strings in quater-imaginary; for example, the number −1 is represented as "103" in quater-imaginary notation.

A radix point in the decimal system is the usual . (dot) which marks the separation between the integer part and the fractional part of the number. In the quater-imaginary system a radix point can also be used. For a digit string the radix point marks the separation between non-negative and negative powers of b. Using the radix point the general formula becomes:

or

If the quater-imaginary representation of the complex unit i has to be found, the formula without radix point will not suffice. Therefore, the above formula should be used. Hence:

for certain coefficients dk. Then because the real part has to be zero: d ₄ = d ₂ = d ₀ = d−2 = 0. For the imaginary part, if d ₅ = d ₃ = d−3 = 0 and when d ₁ = 1 and d−1 = 2 the digit string can be found. Using the above coefficients in the digit string the result is:

It is possible to add and subtract numbers in the quater-imaginary system. In doing this, there are two basic rules that have to be kept in mind:

1. Whenever a number exceeds 3, subtract 4 and "carry" −1 two places to the left.

2. Whenever a number drops below 0, add 4 and "carry" +1 two places to the left.

Or for short: "If you add four, carry +1. If you subtract four, carry −1". This is the opposite of normal long addition, in which a "carry" in the current column requires adding 1 to the next column to the left, and a "borrow" requires subtracting. In quater-imaginary arithmetic, a "carry" subtracts from the next-but-one column, and a "borrow" adds.

Below are two examples of adding in the quater-imaginary system:

   1 − 2i                1031
   1 − 2i                1031
   ------ +      <=>     ---- +
   2 − 4i                1022


   3 − 4i                1023
   1 − 8i                1001
   ------ +      <=>    ----- +
   4 −12i               12320

In the first example we start by adding the two 1s in the first column (the "ones' column"), giving 2. Then we add the two 3s in the second column (the "2 i s column"), giving 6; 6 is greater than 3, so we subtract 4 (giving 2 as the result in the second column) and carry −1 into the fourth column. Adding the 0s in the third column gives 0; and finally adding the two 1s and the carried −1 in the fourth column gives 1.

Subtraction is analogous to addition in that it uses the same two rules described above. Below is an example:

         − 2 − 8i                       1102
           1 − 6i                       1011
           -------           <=>        -----
         − 3 − 2i                       1131

In this example we have to subtract from . The rightmost digit is 2−1 = 1. The second digit from the right would become −1, so add 4 to give 3 and then carry +1 two places to the left. The third digit from the right is 1−0 = 1. Then the leftmost digit is 1−1 plus 1 from the carry, giving 1. This gives a final answer of .

For long multiplication in the quater-imaginary system, the two rules stated above are used as well. When multiplying numbers, multiply the first string by each digit in the second string consecutively and add the resulting strings. With every multiplication, a digit in the second string is multiplied with the first string. The multiplication starts with the rightmost digit in the second string and then moves leftward by one digit, multiplying each digit with the first string. Then the resulting partial products are added where each is shifted to the left by one digit. An example:

              11201
              20121  ×
        ---------------
              11201      ←--- 1 × 11201
             12002       ←--- 2 × 11201
            11201        ←--- 1 × 11201
           00000         ←--- 0 × 11201
          12002      +   ←--- 2 × 11201
        ---------------
          120231321

Below is a table of some decimal and complex numbers and their quater-imaginary counterparts.

Below are some other examples of conversions from decimal numbers to quater-imaginary numbers.

The representation

of an arbitrary complex number with gives rise to an injective mapping

with some suitable . Here cannot be taken as base because of

The image is a Cantor set which allows to linearly order similar to a Z-order curve. Since the image is disconnected, is not continuous.

In a positional system with base ,

and because , the entire series of powers can be separated into two different series, so that it simplifies to for even-numbered digits (digits that simplify to the value of the digit times a power of -4), and for those digits that still have an imaginary factor. Adding these two series together then gives the total value of the number.

Because of the separation of these two series, the real and imaginary parts of complex numbers are readily expressed in base −4 as and respectively.

k

(2i)k

−4

1/16

−2

−1/4

0

1

2

−4

4

16

6

−64

8

256

To convert a digit string from the quater-imaginary system to the decimal system, the standard formula for positional number systems can be used. This says that a digit string in base b can be converted to a decimal number using the formula

Additionally, for a given string in the form , the formula below can be used for a given string length in base

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 = 0.9 in decimal notation, so, because of 0.0001 2i = ⁠1 / 15 ⁠, the number ⁠1 / 5 ⁠ has the two quater-imaginary representations 0.0003 2i = 3·⁠1 / 15 ⁠ = ⁠1 / 5 ⁠ = 1 + 3·⁠-4 / 15 ⁠ = 1.0300 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+4 i is equal to −1 plus 4 i, the quater-imaginary representation of −1+4 i is the quater-imaginary representation of −1 (namely, 103) plus the quater-imaginary representation of 4 i (namely, 20), which gives a final result of −1+4 i = 123 2i.

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 = −12, which is expressed as 300 2i, and then shifting by one place to the right, yielding: 6 i = 30 2i.

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 article.

As an example of an integer number we can try to find the quater-imaginary counterpart of the decimal number 7 (or 7 ₁₀ since the 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:

Since 7 is a real number, it is allowed to conclude that d₁, d₃ and d₅ should be zero. Now the value of the coefficients d₀, d₂ and d₄, must be found. Because d ₀ − 4 d ₂ + 16 d ₄ = 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: d₀ = 3, d₂ = 3 and d₄ = 1. This configuration gives the resulting digit string for 7 ₁₀.

Finding a quater-imaginary representation of a purely imaginary integer number ∈ 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 d₁ = 3 and all other coefficients are zero, we get the desired string for 6 i. That is:

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 fractions we can use the following algorithm using repeated Euclidean division:

For example: 35+23i=121003.2 2i

                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
               20003 + 101000 + 0.2 = 121003.2 32i+16×2−8i−4×0+2i×0+1×3−2×i/2=35+23i

# data[].attributes.attachment_link.description
The quater-imaginary numeral system is a numeral system, first proposed by Donald Knuth in 1960. Unlike standard numeral systems, which use an integer (such as 10 in decimal, or 2 in binary) as their bases, it uses the imaginary number (such that ) as its base. It is able to (almost) uniquely represent every complex number using only the digits 0, 1, 2, and 3.[1] Numbers less than zero, which are ordinarily represented with a minus sign, are representable as digit strings in quater-imaginary; for example, the number −1 is represented as “103” in quater-imaginary notation.

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https://dl.acm.org/doi/pdf/10.1145/367177.367233

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복잡한 진법 개괄 (영문)

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복잡한 진법 개괄 (영문)

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https://en.wikipedia.org/wiki/Complex-base_system

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In arithmetic, a complex-base system is a positional numeral system whose radix is an imaginary (proposed by Donald Knuth in 1955[1][2]) or complex number (proposed by S. Khmelnik in 1964[3] and Walter F. Penney in 1965[4][5][6]).

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고1에서 배우는 복소수

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고1에서 배우는 복소수

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https://m.blog.naver.com/holywater_p/222618895609

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https://dlxdlx.tistory.com/5

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복소수 진법 개괄 관련

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복소수 진법 개괄 관련

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복소수 진법에서의 기수법

- 복소평면에서 원점과의 거리

- (3+2i) 진법이라면 √(3^2+2^2)= √(13) 근사로 0123 사용

- 복소평면에 나타내보기

복소평면

x+yi 를 (x,y)의 형태로 평면 그래프 상에 나타냄.

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프랙탈 관련 중요 자료 (영문)

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프랙탈 관련 중요 자료 (영문)

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https://www.math.uwaterloo.ca/~wgilbert/Research/MathIntel.pdf

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https://en.m.wikipedia.org/wiki/Negative_base

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Complex base 복소수 진법

Quater Imaginary base 복소수 진법 개선 체계

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https://youtu.be/DkPu7GyVTas?feature=shared

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Complex-base numeral systems - allowing for fractal covering of complex plane