Imaginary base numbers: Difference between revisions
Line 1,082: | Line 1,082: | ||
15i -> 102000.2 -> 15i -15i -> 2010.2 -> -15i |
15i -> 102000.2 -> 15i -15i -> 2010.2 -> -15i |
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16i -> 102000.0 -> 16i -16i -> 2000.0 -> -16i |
16i -> 102000.0 -> 16i -16i -> 2000.0 -> -16i |
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</pre> |
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=={{header|Haskell}}== |
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<lang Haskell>import Data.Maybe |
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import Data.List |
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import Data.Complex |
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import Data.Char |
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base = 0 :+ 2 |
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quotRemPositive :: (Integral a) => a -> a -> (a, a) |
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quotRemPositive a b= |
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let (q, r) = quotRem a b in |
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if (r < 0) then (1 + q, (floor $ realPart (- base^^2)) + r) else (q, r) |
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digitToIntQI :: Char -> Int |
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digitToIntQI c = if (isDigit c) then (digitToInt c) else ((ord c) - (ord 'a') + 10) |
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shiftRight :: [Char] -> [Char] |
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shiftRight n = let (l, h) = (last n, init n) in |
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if (l == '0') then h else h ++ "." ++ [l] |
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intToDigitQI :: Int -> Char |
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intToDigitQI i = if (elem i [0..9]) then (intToDigit i) else (chr (i - 10 + (ord 'a'))) |
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fromQItoComplex :: (RealFloat a) => [Char] -> Complex a -> Complex a |
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fromQItoComplex num b = |
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let dot = fromMaybe (length num) (elemIndex '.' num) in |
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fst $ foldl |
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(\(acc, indx) x -> |
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(acc + (fromIntegral $ digitToIntQI x)*(b^^(dot - indx)), indx + 1)) |
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(0, 1) |
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(delete '.' num) |
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euclidEr :: Int -> Int -> [Int] -> [Int] |
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euclidEr a b l = |
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if (a == 0) then l else let (q, r) = quotRemPositive a b in euclidEr q b (0:r:l) |
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fromIntToQI :: Int -> [Int] |
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fromIntToQI 0 = [0] |
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fromIntToQI x = tail (euclidEr x (floor $ realPart (base^^2)) []) |
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getCuid :: Integral a => Complex a -> a |
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getCuid c = (imagPart c)*(floor $ imagPart (-base)) |
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qizip :: Complex Int -> [Int] |
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qizip c = let (r, i) = (fromIntToQI (realPart c) ++ [0], fromIntToQI (getCuid c)) in |
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let m = min (length r) (length i) in |
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(take ((length r) - m) r) ++ (take ((length i) - m) i) |
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++ reverse (zipWith (+) (take m (reverse r)) (take m (reverse i))) |
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fromComplexToQI :: Complex Int -> [Char] |
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fromComplexToQI c = shiftRight (map intToDigitQI (qizip c)) |
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main = print (fromComplexToQI (35 :+ 23)) >> |
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print (fromQItoComplex "10.2" base) |
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</lang> |
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{{out}} |
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<pre> |
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"121003.2" |
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0.0 :+ 1.0 |
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</pre> |
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With base = 8i (you may choose any base): |
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<pre> |
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"3z.8" |
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0.0 :+ 7.75 |
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</pre> |
</pre> |
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Revision as of 05:29, 7 April 2019
![Task](http://static.miraheze.org/rosettacodewiki/thumb/b/ba/Rcode-button-task-crushed.png/64px-Rcode-button-task-crushed.png)
You are encouraged to solve this task according to the task description, using any language you may know.
Imaginary base numbers are a non-standard positional numeral system which uses an imaginary number as its radix. The most common is quater-imaginary with radix 2i.
The quater-imaginary numeral system was first proposed by Donald Knuth in 1955 as a submission for a high school science talent search. [Ref.]
Other imaginary bases are possible too but are not as widely discussed and aren't specifically named.
Task: Write a set of procedures (functions, subroutines, however they are referred to in your language) to convert base 10 numbers to an imaginary base and back.
At a minimum, support quater-imaginary (base 2i).
For extra kudos, support positive or negative bases 2i through 6i (or higher).
As a stretch goal, support converting non-integer numbers ( E.G. 227.65625+10.859375i ) to an imaginary base.
See Wikipedia: Quater-imaginary_base for more details.
For reference, here are some some decimal and complex numbers converted to quater-imaginary.
|
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|
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C++
<lang cpp>#include <algorithm>
- include <complex>
- include <iomanip>
- include <iostream>
std::complex<double> inv(const std::complex<double>& c) {
double denom = c.real() * c.real() + c.imag() * c.imag(); return std::complex<double>(c.real() / denom, -c.imag() / denom);
}
class QuaterImaginary { public:
QuaterImaginary(const std::string& s) : b2i(s) { static std::string base("0123.");
if (b2i.empty() || std::any_of(s.cbegin(), s.cend(), [](char c) { return base.find(c) == std::string::npos; }) || std::count(s.cbegin(), s.cend(), '.') > 1) { throw std::runtime_error("Invalid base 2i number"); } }
QuaterImaginary& operator=(const QuaterImaginary& q) { b2i = q.b2i; return *this; }
std::complex<double> toComplex() const { int pointPos = b2i.find('.'); int posLen = (pointPos != std::string::npos) ? pointPos : b2i.length(); std::complex<double> sum(0.0, 0.0); std::complex<double> prod(1.0, 0.0); for (int j = 0; j < posLen; j++) { double k = (b2i[posLen - 1 - j] - '0'); if (k > 0.0) { sum += prod * k; } prod *= twoI; } if (pointPos != -1) { prod = invTwoI; for (size_t j = posLen + 1; j < b2i.length(); j++) { double k = (b2i[j] - '0'); if (k > 0.0) { sum += prod * k; } prod *= invTwoI; } }
return sum; }
friend std::ostream& operator<<(std::ostream&, const QuaterImaginary&);
private:
const std::complex<double> twoI{ 0.0, 2.0 }; const std::complex<double> invTwoI = inv(twoI);
std::string b2i;
};
std::ostream& operator<<(std::ostream& os, const QuaterImaginary& q) {
return os << q.b2i;
}
// only works properly if 'real' and 'imag' are both integral QuaterImaginary toQuaterImaginary(const std::complex<double>& c) {
if (c.real() == 0.0 && c.imag() == 0.0) return QuaterImaginary("0");
int re = (int)c.real(); int im = (int)c.imag(); int fi = -1; std::stringstream ss; while (re != 0) { int rem = re % -4; re /= -4; if (rem < 0) { rem = 4 + rem; re++; } ss << rem << 0; } if (im != 0) { double f = (std::complex<double>(0.0, c.imag()) / std::complex<double>(0.0, 2.0)).real(); im = (int)ceil(f); f = -4.0 * (f - im); size_t index = 1; while (im != 0) { int rem = im % -4; im /= -4; if (rem < 0) { rem = 4 + rem; im++; } if (index < ss.str().length()) { ss.str()[index] = (char)(rem + 48); } else { ss << 0 << rem; } index += 2; } fi = (int)f; }
auto r = ss.str(); std::reverse(r.begin(), r.end()); ss.str(""); ss.clear(); ss << r; if (fi != -1) ss << '.' << fi; r = ss.str(); r.erase(r.begin(), std::find_if(r.begin(), r.end(), [](char c) { return c != '0'; })); if (r[0] == '.')r = "0" + r; return QuaterImaginary(r);
}
int main() {
using namespace std;
for (int i = 1; i <= 16; i++) { complex<double> c1(i, 0); QuaterImaginary qi = toQuaterImaginary(c1); complex<double> c2 = qi.toComplex(); cout << setw(8) << c1 << " -> " << setw(8) << qi << " -> " << setw(8) << c2 << " "; c1 = -c1; qi = toQuaterImaginary(c1); c2 = qi.toComplex(); cout << setw(8) << c1 << " -> " << setw(8) << qi << " -> " << setw(8) << c2 << endl; } cout << endl;
for (int i = 1; i <= 16; i++) { complex<double> c1(0, i); QuaterImaginary qi = toQuaterImaginary(c1); complex<double> c2 = qi.toComplex(); cout << setw(8) << c1 << " -> " << setw(8) << qi << " -> " << setw(8) << c2 << " "; c1 = -c1; qi = toQuaterImaginary(c1); c2 = qi.toComplex(); cout << setw(8) << c1 << " -> " << setw(8) << qi << " -> " << setw(8) << c2 << endl; }
return 0;
}</lang>
- Output:
(1,0) -> 1 -> (1,0) (-1,-0) -> 103 -> (-1,0) (2,0) -> 2 -> (2,0) (-2,-0) -> 102 -> (-2,0) (3,0) -> 3 -> (3,0) (-3,-0) -> 101 -> (-3,0) (4,0) -> 10300 -> (4,0) (-4,-0) -> 100 -> (-4,0) (5,0) -> 10301 -> (5,0) (-5,-0) -> 203 -> (-5,0) (6,0) -> 10302 -> (6,0) (-6,-0) -> 202 -> (-6,0) (7,0) -> 10303 -> (7,0) (-7,-0) -> 201 -> (-7,0) (8,0) -> 10200 -> (8,0) (-8,-0) -> 200 -> (-8,0) (9,0) -> 10201 -> (9,0) (-9,-0) -> 303 -> (-9,0) (10,0) -> 10202 -> (10,0) (-10,-0) -> 302 -> (-10,0) (11,0) -> 10203 -> (11,0) (-11,-0) -> 301 -> (-11,0) (12,0) -> 10100 -> (12,0) (-12,-0) -> 300 -> (-12,0) (13,0) -> 10101 -> (13,0) (-13,-0) -> 1030003 -> (-13,0) (14,0) -> 10102 -> (14,0) (-14,-0) -> 1030002 -> (-14,0) (15,0) -> 10103 -> (15,0) (-15,-0) -> 1030001 -> (-15,0) (16,0) -> 10000 -> (16,0) (-16,-0) -> 1030000 -> (-16,0) (0,1) -> 10.2 -> (0,1) (-0,-1) -> 0.2 -> (0,-1) (0,2) -> 10.0 -> (0,2) (-0,-2) -> 1030.0 -> (0,-2) (0,3) -> 20.2 -> (0,3) (-0,-3) -> 1030.2 -> (0,-3) (0,4) -> 20.0 -> (0,4) (-0,-4) -> 1020.0 -> (0,-4) (0,5) -> 30.2 -> (0,5) (-0,-5) -> 1020.2 -> (0,-5) (0,6) -> 30.0 -> (0,6) (-0,-6) -> 1010.0 -> (0,-6) (0,7) -> 103000.2 -> (0,7) (-0,-7) -> 1010.2 -> (0,-7) (0,8) -> 103000.0 -> (0,8) (-0,-8) -> 1000.0 -> (0,-8) (0,9) -> 103010.2 -> (0,9) (-0,-9) -> 1000.2 -> (0,-9) (0,10) -> 103010.0 -> (0,10) (-0,-10) -> 2030.0 -> (0,-10) (0,11) -> 103020.2 -> (0,11) (-0,-11) -> 2030.2 -> (0,-11) (0,12) -> 103020.0 -> (0,12) (-0,-12) -> 2020.0 -> (0,-12) (0,13) -> 103030.2 -> (0,13) (-0,-13) -> 2020.2 -> (0,-13) (0,14) -> 103030.0 -> (0,14) (-0,-14) -> 2010.0 -> (0,-14) (0,15) -> 102000.2 -> (0,15) (-0,-15) -> 2010.2 -> (0,-15) (0,16) -> 102000.0 -> (0,16) (-0,-16) -> 2000.0 -> (0,-16)
C#
<lang csharp>using System; using System.Linq; using System.Text;
namespace ImaginaryBaseNumbers {
class Complex { private double real, imag;
public Complex(int r, int i) { real = r; imag = i; }
public Complex(double r, double i) { real = r; imag = i; }
public static Complex operator -(Complex self) => new Complex(-self.real, -self.imag);
public static Complex operator +(Complex rhs, Complex lhs) => new Complex(rhs.real + lhs.real, rhs.imag + lhs.imag);
public static Complex operator -(Complex rhs, Complex lhs) => new Complex(rhs.real - lhs.real, rhs.imag - lhs.imag);
public static Complex operator *(Complex rhs, Complex lhs) => new Complex( rhs.real * lhs.real - rhs.imag * lhs.imag, rhs.real * lhs.imag + rhs.imag * lhs.real );
public static Complex operator *(Complex rhs, double lhs) => new Complex(rhs.real * lhs, rhs.imag * lhs);
public static Complex operator /(Complex rhs, Complex lhs) => rhs * lhs.Inv();
public Complex Inv() { double denom = real * real + imag * imag; return new Complex(real / denom, -imag / denom); }
public QuaterImaginary ToQuaterImaginary() { if (real == 0.0 && imag == 0.0) return new QuaterImaginary("0"); int re = (int)real; int im = (int)imag; int fi = -1; StringBuilder sb = new StringBuilder(); while (re != 0) { int rem = re % -4; re /= -4; if (rem < 0) { rem = 4 + rem; re++; } sb.Append(rem); sb.Append(0); } if (im != 0) { double f = (new Complex(0.0, imag) / new Complex(0.0, 2.0)).real; im = (int)Math.Ceiling(f); f = -4.0 * (f - im); int index = 1; while (im != 0) { int rem = im % -4; im /= -4; if (rem < 0) { rem = 4 + rem; im++; } if (index < sb.Length) { sb[index] = (char)(rem + 48); } else { sb.Append(0); sb.Append(rem); } index += 2; } fi = (int)f; } string reverse = new string(sb.ToString().Reverse().ToArray()); sb.Length = 0; sb.Append(reverse); if (fi != -1) sb.AppendFormat(".{0}", fi); string s = sb.ToString().TrimStart('0'); if (s[0] == '.') s = "0" + s; return new QuaterImaginary(s); }
public override string ToString() { double real2 = (real == -0.0) ? 0.0 : real; // get rid of negative zero double imag2 = (imag == -0.0) ? 0.0 : imag; // ditto if (imag2 == 0.0) { return string.Format("{0}", real2); } if (real2 == 0.0) { return string.Format("{0}i", imag2); } if (imag2 > 0.0) { return string.Format("{0} + {1}i", real2, imag2); } return string.Format("{0} - {1}i", real2, -imag2); } }
class QuaterImaginary { internal static Complex twoI = new Complex(0.0, 2.0); internal static Complex invTwoI = twoI.Inv();
private string b2i;
public QuaterImaginary(string b2i) { if (b2i == "" || !b2i.All(c => "0123.".IndexOf(c) > -1) || b2i.Count(c => c == '.') > 1) { throw new Exception("Invalid Base 2i number"); } this.b2i = b2i; }
public Complex ToComplex() { int pointPos = b2i.IndexOf("."); int posLen = (pointPos != -1) ? pointPos : b2i.Length; Complex sum = new Complex(0.0, 0.0); Complex prod = new Complex(1.0, 0.0); for (int j = 0; j < posLen; j++) { double k = (b2i[posLen - 1 - j] - '0'); if (k > 0.0) { sum += prod * k; } prod *= twoI; } if (pointPos != -1) { prod = invTwoI; for (int j = posLen + 1; j < b2i.Length; j++) { double k = (b2i[j] - '0'); if (k > 0.0) { sum += prod * k; } prod *= invTwoI; } }
return sum; }
public override string ToString() { return b2i; } }
class Program { static void Main(string[] args) { for (int i = 1; i <= 16; i++) { Complex c1 = new Complex(i, 0); QuaterImaginary qi = c1.ToQuaterImaginary(); Complex c2 = qi.ToComplex(); Console.Write("{0,4} -> {1,8} -> {2,4} ", c1, qi, c2); c1 = -c1; qi = c1.ToQuaterImaginary(); c2 = qi.ToComplex(); Console.WriteLine("{0,4} -> {1,8} -> {2,4}", c1, qi, c2); } Console.WriteLine(); for (int i = 1; i <= 16; i++) { Complex c1 = new Complex(0, i); QuaterImaginary qi = c1.ToQuaterImaginary(); Complex c2 = qi.ToComplex(); Console.Write("{0,4} -> {1,8} -> {2,4} ", c1, qi, c2); c1 = -c1; qi = c1.ToQuaterImaginary(); c2 = qi.ToComplex(); Console.WriteLine("{0,4} -> {1,8} -> {2,4}", c1, qi, c2); } } }
}</lang>
- Output:
1 -> 1 -> 1 -1 -> 103 -> -1 2 -> 2 -> 2 -2 -> 102 -> -2 3 -> 3 -> 3 -3 -> 101 -> -3 4 -> 10300 -> 4 -4 -> 100 -> -4 5 -> 10301 -> 5 -5 -> 203 -> -5 6 -> 10302 -> 6 -6 -> 202 -> -6 7 -> 10303 -> 7 -7 -> 201 -> -7 8 -> 10200 -> 8 -8 -> 200 -> -8 9 -> 10201 -> 9 -9 -> 303 -> -9 10 -> 10202 -> 10 -10 -> 302 -> -10 11 -> 10203 -> 11 -11 -> 301 -> -11 12 -> 10100 -> 12 -12 -> 300 -> -12 13 -> 10101 -> 13 -13 -> 1030003 -> -13 14 -> 10102 -> 14 -14 -> 1030002 -> -14 15 -> 10103 -> 15 -15 -> 1030001 -> -15 16 -> 10000 -> 16 -16 -> 1030000 -> -16 1i -> 10.2 -> 1i -1i -> 0.2 -> -1i 2i -> 10.0 -> 2i -2i -> 1030.0 -> -2i 3i -> 20.2 -> 3i -3i -> 1030.2 -> -3i 4i -> 20.0 -> 4i -4i -> 1020.0 -> -4i 5i -> 30.2 -> 5i -5i -> 1020.2 -> -5i 6i -> 30.0 -> 6i -6i -> 1010.0 -> -6i 7i -> 103000.2 -> 7i -7i -> 1010.2 -> -7i 8i -> 103000.0 -> 8i -8i -> 1000.0 -> -8i 9i -> 103010.2 -> 9i -9i -> 1000.2 -> -9i 10i -> 103010.0 -> 10i -10i -> 2030.0 -> -10i 11i -> 103020.2 -> 11i -11i -> 2030.2 -> -11i 12i -> 103020.0 -> 12i -12i -> 2020.0 -> -12i 13i -> 103030.2 -> 13i -13i -> 2020.2 -> -13i 14i -> 103030.0 -> 14i -14i -> 2010.0 -> -14i 15i -> 102000.2 -> 15i -15i -> 2010.2 -> -15i 16i -> 102000.0 -> 16i -16i -> 2000.0 -> -16i
D
<lang D>import std.algorithm; import std.array; import std.complex; import std.conv; import std.format; import std.math; import std.stdio; import std.string;
Complex!double inv(Complex!double v) {
auto denom = v.re*v.re + v.im*v.im; return v.conj / denom;
}
QuaterImaginary toQuaterImaginary(Complex!double v) {
if (v.re == 0.0 && v.im == 0.0) return QuaterImaginary("0"); auto re = v.re.to!int; auto im = v.im.to!int; auto fi = -1; auto sb = appender!(char[]); while (re != 0) { auto rem = re % -4; re /= -4; if (rem < 0) { rem = 4 + rem; re++; } sb.formattedWrite("%d", rem); sb.put("0"); } if (im != 0) { auto f = (complex(0.0, v.im) / complex(0.0, 2.0)).re; im = f.ceil.to!int; f = -4.0 * (f - im.to!double); auto index = 1; while (im != 0) { auto rem = im % -4; im /= -4; if (rem < 0) { rem = 4 + rem; im++; } if (index < sb.data.length) { sb.data[index] = cast(char)(rem + '0'); } else { sb.put("0"); sb.formattedWrite("%d", rem); } index += 2; } fi = f.to!int; } sb.data.reverse; if (fi != -1) sb.formattedWrite(".%d", fi); int i; while (i < sb.data.length && sb.data[i] == '0') { i++; } auto s = sb.data[i..$].idup; if (s[0] == '.') s = "0" ~ s; return QuaterImaginary(s);
}
struct QuaterImaginary {
private string b2i;
this(string b2i) { if (b2i == "" || b2i.count('.') > 1) { throw new Exception("Invalid Base 2i number"); } foreach (c; b2i) { if (!canFind("0123.", c)) { throw new Exception("Invalid Base 2i number"); } } this.b2i = b2i; }
T opCast(T : Complex!double)() { auto pointPos = b2i.indexOf('.'); size_t posLen; if (pointPos != -1) { posLen = pointPos; } else { posLen = b2i.length; } auto sum = complex(0.0, 0.0); auto prod = complex(1.0, 0.0); foreach (j; 0..posLen) { auto k = (b2i[posLen - 1 - j] - '0').to!double; if (k > 0.0) { sum += prod * k; } prod *= twoI; } if (pointPos != -1) { prod = invTwoI; foreach (j; posLen+1..b2i.length) { auto k = (b2i[j] - '0').to!double; if (k > 0.0) { sum += prod * k; } prod *= invTwoI; } } return sum; }
void toString(scope void delegate(const(char)[]) sink, FormatSpec!char fmt) const { if (fmt.spec == 's') { for (int i=0; i<fmt.width-b2i.length; ++i) { sink(" "); } } sink(b2i); }
enum twoI = complex(0.0, 2.0); enum invTwoI = twoI.inv;
}
unittest {
import std.exception; assertThrown!Exception(QuaterImaginary("")); assertThrown!Exception(QuaterImaginary("1.2.3")); assertThrown!Exception(QuaterImaginary("a")); assertThrown!Exception(QuaterImaginary("4")); assertThrown!Exception(QuaterImaginary(" "));
}
void main() {
foreach (i; 1..17) { auto c1 = complex(i, 0); auto qi = c1.toQuaterImaginary; auto c2 = cast(Complex!double) qi; writef("%4s -> %8s -> %4s ", c1.re, qi, c2.re); c1 = -c1; qi = c1.toQuaterImaginary(); c2 = cast(Complex!double) qi; writefln("%4s -> %8s -> %4s", c1.re, qi, c2.re); } writeln; foreach (i; 1..17) { auto c1 = complex(0, i); auto qi = c1.toQuaterImaginary; auto c2 = qi.to!(Complex!double); writef("%4si -> %8s -> %4si ", c1.im, qi, c2.im); c1 = -c1; qi = c1.toQuaterImaginary(); c2 = cast(Complex!double) qi; writefln("%4si -> %8s -> %4si", c1.im, qi, c2.im); }
}</lang>
- Output:
1 -> 1 -> 1 -1 -> 103 -> -1 2 -> 2 -> 2 -2 -> 102 -> -2 3 -> 3 -> 3 -3 -> 101 -> -3 4 -> 10300 -> 4 -4 -> 100 -> -4 5 -> 10301 -> 5 -5 -> 203 -> -5 6 -> 10302 -> 6 -6 -> 202 -> -6 7 -> 10303 -> 7 -7 -> 201 -> -7 8 -> 10200 -> 8 -8 -> 200 -> -8 9 -> 10201 -> 9 -9 -> 303 -> -9 10 -> 10202 -> 10 -10 -> 302 -> -10 11 -> 10203 -> 11 -11 -> 301 -> -11 12 -> 10100 -> 12 -12 -> 300 -> -12 13 -> 10101 -> 13 -13 -> 1030003 -> -13 14 -> 10102 -> 14 -14 -> 1030002 -> -14 15 -> 10103 -> 15 -15 -> 1030001 -> -15 16 -> 10000 -> 16 -16 -> 1030000 -> -16 1i -> 10.2 -> 1i -1i -> 0.2 -> -1i 2i -> 10.0 -> 2i -2i -> 1030.0 -> -2i 3i -> 20.2 -> 3i -3i -> 1030.2 -> -3i 4i -> 20.0 -> 4i -4i -> 1020.0 -> -4i 5i -> 30.2 -> 5i -5i -> 1020.2 -> -5i 6i -> 30.0 -> 6i -6i -> 1010.0 -> -6i 7i -> 103000.2 -> 7i -7i -> 1010.2 -> -7i 8i -> 103000.0 -> 8i -8i -> 1000.0 -> -8i 9i -> 103010.2 -> 9i -9i -> 1000.2 -> -9i 10i -> 103010.0 -> 10i -10i -> 2030.0 -> -10i 11i -> 103020.2 -> 11i -11i -> 2030.2 -> -11i 12i -> 103020.0 -> 12i -12i -> 2020.0 -> -12i 13i -> 103030.2 -> 13i -13i -> 2020.2 -> -13i 14i -> 103030.0 -> 14i -14i -> 2010.0 -> -14i 15i -> 102000.2 -> 15i -15i -> 2010.2 -> -15i 16i -> 102000.0 -> 16i -16i -> 2000.0 -> -16i
Go
... though a bit shorter as Go has support for complex numbers built into the language. <lang go>package main
import (
"fmt" "math" "strconv" "strings"
)
const (
twoI = 2.0i invTwoI = 1.0 / twoI
)
type quaterImaginary struct {
b2i string
}
func reverse(s string) string {
r := []rune(s) for i, j := 0, len(r)-1; i < len(r)/2; i, j = i+1, j-1 { r[i], r[j] = r[j], r[i] } return string(r)
}
func newQuaterImaginary(b2i string) quaterImaginary {
b2i = strings.TrimSpace(b2i) _, err := strconv.ParseFloat(b2i, 64) if err != nil { panic("invalid Base 2i number") } return quaterImaginary{b2i}
}
func toComplex(q quaterImaginary) complex128 {
pointPos := strings.Index(q.b2i, ".") var posLen int if pointPos != -1 { posLen = pointPos } else { posLen = len(q.b2i) } sum := 0.0i prod := complex(1.0, 0.0) for j := 0; j < posLen; j++ { k := float64(q.b2i[posLen-1-j] - '0') if k > 0.0 { sum += prod * complex(k, 0.0) } prod *= twoI } if pointPos != -1 { prod = invTwoI for j := posLen + 1; j < len(q.b2i); j++ { k := float64(q.b2i[j] - '0') if k > 0.0 { sum += prod * complex(k, 0.0) } prod *= invTwoI } } return sum
}
func (q quaterImaginary) String() string {
return q.b2i
}
// only works properly if 'real' and 'imag' are both integral func toQuaterImaginary(c complex128) quaterImaginary {
if c == 0i { return quaterImaginary{"0"} } re := int(real(c)) im := int(imag(c)) fi := -1 var sb strings.Builder for re != 0 { rem := re % -4 re /= -4 if rem < 0 { rem += 4 re++ } sb.WriteString(strconv.Itoa(rem)) sb.WriteString("0") } if im != 0 { f := real(complex(0.0, imag(c)) / 2.0i) im = int(math.Ceil(f)) f = -4.0 * (f - float64(im)) index := 1 for im != 0 { rem := im % -4 im /= -4 if rem < 0 { rem += 4 im++ } if index < sb.Len() { bs := []byte(sb.String()) bs[index] = byte(rem + 48) sb.Reset() sb.Write(bs) } else { sb.WriteString("0") sb.WriteString(strconv.Itoa(rem)) } index += 2 } fi = int(f) } s := reverse(sb.String()) if fi != -1 { s = fmt.Sprintf("%s.%d", s, fi) } s = strings.TrimLeft(s, "0") if s[0] == '.' { s = "0" + s } return newQuaterImaginary(s)
}
func main() {
for i := 1; i <= 16; i++ { c1 := complex(float64(i), 0.0) qi := toQuaterImaginary(c1) c2 := toComplex(qi) fmt.Printf("%4.0f -> %8s -> %4.0f ", real(c1), qi, real(c2)) c1 = -c1 qi = toQuaterImaginary(c1) c2 = toComplex(qi) fmt.Printf("%4.0f -> %8s -> %4.0f\n", real(c1), qi, real(c2)) } fmt.Println() for i := 1; i <= 16; i++ { c1 := complex(0.0, float64(i)) qi := toQuaterImaginary(c1) c2 := toComplex(qi) fmt.Printf("%3.0fi -> %8s -> %3.0fi ", imag(c1), qi, imag(c2)) c1 = -c1 qi = toQuaterImaginary(c1) c2 = toComplex(qi) fmt.Printf("%3.0fi -> %8s -> %3.0fi\n", imag(c1), qi, imag(c2)) }
}</lang>
- Output:
1 -> 1 -> 1 -1 -> 103 -> -1 2 -> 2 -> 2 -2 -> 102 -> -2 3 -> 3 -> 3 -3 -> 101 -> -3 4 -> 10300 -> 4 -4 -> 100 -> -4 5 -> 10301 -> 5 -5 -> 203 -> -5 6 -> 10302 -> 6 -6 -> 202 -> -6 7 -> 10303 -> 7 -7 -> 201 -> -7 8 -> 10200 -> 8 -8 -> 200 -> -8 9 -> 10201 -> 9 -9 -> 303 -> -9 10 -> 10202 -> 10 -10 -> 302 -> -10 11 -> 10203 -> 11 -11 -> 301 -> -11 12 -> 10100 -> 12 -12 -> 300 -> -12 13 -> 10101 -> 13 -13 -> 1030003 -> -13 14 -> 10102 -> 14 -14 -> 1030002 -> -14 15 -> 10103 -> 15 -15 -> 1030001 -> -15 16 -> 10000 -> 16 -16 -> 1030000 -> -16 1i -> 10.2 -> 1i -1i -> 0.2 -> -1i 2i -> 10.0 -> 2i -2i -> 1030.0 -> -2i 3i -> 20.2 -> 3i -3i -> 1030.2 -> -3i 4i -> 20.0 -> 4i -4i -> 1020.0 -> -4i 5i -> 30.2 -> 5i -5i -> 1020.2 -> -5i 6i -> 30.0 -> 6i -6i -> 1010.0 -> -6i 7i -> 103000.2 -> 7i -7i -> 1010.2 -> -7i 8i -> 103000.0 -> 8i -8i -> 1000.0 -> -8i 9i -> 103010.2 -> 9i -9i -> 1000.2 -> -9i 10i -> 103010.0 -> 10i -10i -> 2030.0 -> -10i 11i -> 103020.2 -> 11i -11i -> 2030.2 -> -11i 12i -> 103020.0 -> 12i -12i -> 2020.0 -> -12i 13i -> 103030.2 -> 13i -13i -> 2020.2 -> -13i 14i -> 103030.0 -> 14i -14i -> 2010.0 -> -14i 15i -> 102000.2 -> 15i -15i -> 2010.2 -> -15i 16i -> 102000.0 -> 16i -16i -> 2000.0 -> -16i
Haskell
<lang Haskell>import Data.Maybe import Data.List import Data.Complex import Data.Char
base = 0 :+ 2
quotRemPositive :: (Integral a) => a -> a -> (a, a) quotRemPositive a b=
let (q, r) = quotRem a b in if (r < 0) then (1 + q, (floor $ realPart (- base^^2)) + r) else (q, r)
digitToIntQI :: Char -> Int digitToIntQI c = if (isDigit c) then (digitToInt c) else ((ord c) - (ord 'a') + 10)
shiftRight :: [Char] -> [Char] shiftRight n = let (l, h) = (last n, init n) in
if (l == '0') then h else h ++ "." ++ [l]
intToDigitQI :: Int -> Char intToDigitQI i = if (elem i [0..9]) then (intToDigit i) else (chr (i - 10 + (ord 'a')))
fromQItoComplex :: (RealFloat a) => [Char] -> Complex a -> Complex a fromQItoComplex num b =
let dot = fromMaybe (length num) (elemIndex '.' num) in fst $ foldl (\(acc, indx) x -> (acc + (fromIntegral $ digitToIntQI x)*(b^^(dot - indx)), indx + 1)) (0, 1) (delete '.' num)
euclidEr :: Int -> Int -> [Int] -> [Int] euclidEr a b l =
if (a == 0) then l else let (q, r) = quotRemPositive a b in euclidEr q b (0:r:l)
fromIntToQI :: Int -> [Int] fromIntToQI 0 = [0] fromIntToQI x = tail (euclidEr x (floor $ realPart (base^^2)) [])
getCuid :: Integral a => Complex a -> a getCuid c = (imagPart c)*(floor $ imagPart (-base))
qizip :: Complex Int -> [Int] qizip c = let (r, i) = (fromIntToQI (realPart c) ++ [0], fromIntToQI (getCuid c)) in
let m = min (length r) (length i) in (take ((length r) - m) r) ++ (take ((length i) - m) i) ++ reverse (zipWith (+) (take m (reverse r)) (take m (reverse i)))
fromComplexToQI :: Complex Int -> [Char] fromComplexToQI c = shiftRight (map intToDigitQI (qizip c))
main = print (fromComplexToQI (35 :+ 23)) >>
print (fromQItoComplex "10.2" base)
</lang>
- Output:
"121003.2" 0.0 :+ 1.0
With base = 8i (you may choose any base):
"3z.8" 0.0 :+ 7.75
Java
<lang Java>public class ImaginaryBaseNumber {
private static class Complex { private Double real, imag;
public Complex(double r, double i) { this.real = r; this.imag = i; }
public Complex(int r, int i) { this.real = (double) r; this.imag = (double) i; }
public Complex add(Complex rhs) { return new Complex( real + rhs.real, imag + rhs.imag ); }
public Complex times(Complex rhs) { return new Complex( real * rhs.real - imag * rhs.imag, real * rhs.imag + imag * rhs.real ); }
public Complex times(double rhs) { return new Complex( real * rhs, imag * rhs ); }
public Complex inv() { double denom = real * real + imag * imag; return new Complex( real / denom, -imag / denom ); }
public Complex unaryMinus() { return new Complex(-real, -imag); }
public Complex divide(Complex rhs) { return this.times(rhs.inv()); }
// only works properly if 'real' and 'imag' are both integral public QuaterImaginary toQuaterImaginary() { if (real == 0.0 && imag == 0.0) return new QuaterImaginary("0"); int re = real.intValue(); int im = imag.intValue(); int fi = -1; StringBuilder sb = new StringBuilder(); while (re != 0) { int rem = re % -4; re /= -4; if (rem < 0) { rem += 4; re++; } sb.append(rem); sb.append(0); } if (im != 0) { Double f = new Complex(0.0, imag).divide(new Complex(0.0, 2.0)).real; im = ((Double) Math.ceil(f)).intValue(); f = -4.0 * (f - im); int index = 1; while (im != 0) { int rem = im % -4; im /= -4; if (rem < 0) { rem += 4; im++; } if (index < sb.length()) { sb.setCharAt(index, (char) (rem + 48)); } else { sb.append(0); sb.append(rem); } index += 2; } fi = f.intValue(); } sb.reverse(); if (fi != -1) sb.append(".").append(fi); while (sb.charAt(0) == '0') sb.deleteCharAt(0); if (sb.charAt(0) == '.') sb.insert(0, '0'); return new QuaterImaginary(sb.toString()); }
@Override public String toString() { double real2 = real == -0.0 ? 0.0 : real; // get rid of negative zero double imag2 = imag == -0.0 ? 0.0 : imag; // ditto String result = imag2 >= 0.0 ? String.format("%.0f + %.0fi", real2, imag2) : String.format("%.0f - %.0fi", real2, -imag2); result = result.replace(".0 ", " ").replace(".0i", "i").replace(" + 0i", ""); if (result.startsWith("0 + ")) result = result.substring(4); if (result.startsWith("0 - ")) result = result.substring(4); return result; } }
private static class QuaterImaginary { private static final Complex TWOI = new Complex(0.0, 2.0); private static final Complex INVTWOI = TWOI.inv();
private String b2i;
public QuaterImaginary(String b2i) { if (b2i.equals("") || !b2i.chars().allMatch(c -> "0123.".indexOf(c) > -1) || b2i.chars().filter(c -> c == '.').count() > 1) { throw new RuntimeException("Invalid Base 2i number"); } this.b2i = b2i; }
public Complex toComplex() { int pointPos = b2i.indexOf("."); int posLen = pointPos != -1 ? pointPos : b2i.length(); Complex sum = new Complex(0, 0); Complex prod = new Complex(1, 0);
for (int j = 0; j < posLen; ++j) { double k = b2i.charAt(posLen - 1 - j) - '0'; if (k > 0.0) sum = sum.add(prod.times(k)); prod = prod.times(TWOI); } if (pointPos != -1) { prod = INVTWOI; for (int j = posLen + 1; j < b2i.length(); ++j) { double k = b2i.charAt(j) - '0'; if (k > 0.0) sum = sum.add(prod.times(k)); prod = prod.times(INVTWOI); } }
return sum; }
@Override public String toString() { return b2i; } }
public static void main(String[] args) { String fmt = "%4s -> %8s -> %4s"; for (int i = 1; i <= 16; ++i) { Complex c1 = new Complex(i, 0); QuaterImaginary qi = c1.toQuaterImaginary(); Complex c2 = qi.toComplex(); System.out.printf(fmt + " ", c1, qi, c2); c1 = c2.unaryMinus(); qi = c1.toQuaterImaginary(); c2 = qi.toComplex(); System.out.printf(fmt, c1, qi, c2); System.out.println(); } System.out.println(); for (int i = 1; i <= 16; ++i) { Complex c1 = new Complex(0, i); QuaterImaginary qi = c1.toQuaterImaginary(); Complex c2 = qi.toComplex(); System.out.printf(fmt + " ", c1, qi, c2); c1 = c2.unaryMinus(); qi = c1.toQuaterImaginary(); c2 = qi.toComplex(); System.out.printf(fmt, c1, qi, c2); System.out.println(); } }
}</lang>
- Output:
1 -> 1 -> 1 -1 -> 103 -> -1 2 -> 2 -> 2 -2 -> 102 -> -2 3 -> 3 -> 3 -3 -> 101 -> -3 4 -> 10300 -> 4 -4 -> 100 -> -4 5 -> 10301 -> 5 -5 -> 203 -> -5 6 -> 10302 -> 6 -6 -> 202 -> -6 7 -> 10303 -> 7 -7 -> 201 -> -7 8 -> 10200 -> 8 -8 -> 200 -> -8 9 -> 10201 -> 9 -9 -> 303 -> -9 10 -> 10202 -> 10 -10 -> 302 -> -10 11 -> 10203 -> 11 -11 -> 301 -> -11 12 -> 10100 -> 12 -12 -> 300 -> -12 13 -> 10101 -> 13 -13 -> 1030003 -> -13 14 -> 10102 -> 14 -14 -> 1030002 -> -14 15 -> 10103 -> 15 -15 -> 1030001 -> -15 16 -> 10000 -> 16 -16 -> 1030000 -> -16 1i -> 10.2 -> 1i 1i -> 0.2 -> 1i 2i -> 10.0 -> 2i 2i -> 1030.0 -> 2i 3i -> 20.2 -> 3i 3i -> 1030.2 -> 3i 4i -> 20.0 -> 4i 4i -> 1020.0 -> 4i 5i -> 30.2 -> 5i 5i -> 1020.2 -> 5i 6i -> 30.0 -> 6i 6i -> 1010.0 -> 6i 7i -> 103000.2 -> 7i 7i -> 1010.2 -> 7i 8i -> 103000.0 -> 8i 8i -> 1000.0 -> 8i 9i -> 103010.2 -> 9i 9i -> 1000.2 -> 9i 10i -> 103010.0 -> 10i 10i -> 2030.0 -> 10i 11i -> 103020.2 -> 11i 11i -> 2030.2 -> 11i 12i -> 103020.0 -> 12i 12i -> 2020.0 -> 12i 13i -> 103030.2 -> 13i 13i -> 2020.2 -> 13i 14i -> 103030.0 -> 14i 14i -> 2010.0 -> 14i 15i -> 102000.2 -> 15i 15i -> 2010.2 -> 15i 16i -> 102000.0 -> 16i 16i -> 2000.0 -> 16i
Julia
<lang julia>import Base.show, Base.parse, Base.+, Base.-, Base.*, Base./, Base.^
function inbase4(charvec::Vector)
if (!all(x -> x in ['-', '0', '1', '2', '3', '.'], charvec)) || ((x = findlast(x -> x == '-', charvec)) != nothing && x > findfirst(x -> x != '-', charvec)) || ((x = findall(x -> x == '.', charvec)) != nothing && length(x) > 1) return false end true
end inbase4(s::String) = inbase4(split(s, ""))
abstract type ImaginaryBaseNumber <: Number end
struct QuaterImaginary <: ImaginaryBaseNumber
cvector::Vector{Char} isnegative::Bool
end
function QuaterImaginary(charvec::Vector{Char})
isneg = false if !inbase4(charvec) throw("Constructor vector for QuaterImaginary ($charvec) is not base 2i") elseif (i = length(findall(x -> x == '-', charvec))) > 0 isneg = (-1) ^ i == -1 end while length(charvec) > 1 && charvec[1] == '0' && charvec[2] != '.' popfirst!(charvec) end if (i = findfirst(x -> x == '.', charvec)) != nothing while length(charvec) > 3 && charvec[end] == '0' && charvec[end-1] != '.' pop!(charvec) end end if charvec[1] == '.' pushfirst!(charvec, '0') end if charvec[end] == '.' pop!(charvec) end QuaterImaginary(filter!(x -> x in ['0', '1', '2', '3', '.'], charvec), isneg)
end
function QuaterImaginary(s::String = "0")
if match(r"^-?[0123\.]+$", s) == nothing throw("String constructor argument <$s> for QuaterImaginary is not base 2i") end QuaterImaginary([s[i] for i in 1:length(s)])
end
show(io::IO, qim::QuaterImaginary) = print(io, qim.isnegative ? "-" : "", join(qim.cvector, ""))
function parse(QuaterImaginary, x::Complex)
sb = Vector{Char}() rea, ima = Int(floor(real(x))), Int(floor(imag(x))) if floor(real(x)) != rea || floor(imag(x)) != ima throw("Non-integer real and complex portions of complex numbers are not supported for QuaterImaginary") elseif rea == 0 == ima return QuaterImaginary(['0']) else fi = -1 while rea != 0 rea, rem = divrem(rea, -4) if rem < 0 rem += 4 rea += 1 end push!(sb, Char(rem + '0'), '0') end if ima != 0 f = real((ima * im)/(2im)) ima = Int(ceil(f)) f = -4.0 * (f - ima) idx = 1 while ima != 0 ima, rem = divrem(ima, -4) if rem < 0 rem += 4 ima += 1 end if idx < length(sb) sb[idx + 1] = Char(rem + '0') else push!(sb, '0', Char(rem + '0')) end idx += 2 end fi = Int(floor(f)) end sb = reverse(sb) if fi != -1 push!(sb, '.') append!(sb, map(x -> x[1], split(string(fi), ""))) end end QuaterImaginary(sb)
end
function parse(Complex, qim::QuaterImaginary)
pointpos = ((x = indexin('.', qim.cvector))[1] == nothing) ? -1 : x[1] poslen = (pointpos != -1) ? pointpos : length(qim.cvector) + 1 qsum = 0.0 + 0.0im prod = 1.0 + 0.0im for j in 1:poslen-1 k = Float64(qim.cvector[poslen - j] - '0') if k > 0.0 qsum += prod * k end prod *= 2im end if pointpos != -1 prod = inv(2im) for j in poslen+1:length(qim.cvector) k = Float64(qim.cvector[j] - '0') if k > 0.0 qsum += prod * k end prod *= inv(2im) end end qsum
end
function testquim()
function printcqc(c) q = parse(QuaterImaginary, Complex(c)) c2 = parse(Complex, q) if imag(c2) == 0 c2 = Int(c2) end print(lpad(c, 10), " -> ", lpad(q, 10), " -> ", lpad(c2, 12)) end for i in 1:16 printcqc(i) print(" ") printcqc(-i) println() end println() for i in 1:16 c1 = Complex(0, i) printcqc(c1) print(" ") printcqc(-c1) println() end
end
QuaterImaginary(c::Complex) = parse(QuaterImaginary, c) Complex(q::QuaterImaginary) = parse(Complex, q)
+(q1::QuaterImaginary, q2::QuaterImaginary) = QuaterImaginary(Complex(q1) + Complex(q2)) +(q1::Complex, q2::QuaterImaginary) = q1 + Complex(q2) +(q1::QuaterImaginary, q2::Complex) = Complex(q1) + q2 -(q1::QuaterImaginary, q2::QuaterImaginary) = QuaterImaginary(Complex(q1) - Complex(q2)) -(q1::Complex, q2::QuaterImaginary) = q1 - Complex(q2) -(q1::QuaterImaginary, q2::Complex) = Complex(q1) - q2
- (q1::QuaterImaginary, q2::QuaterImaginary) = QuaterImaginary(Complex(q1) * Complex(q2))
- (q1::Complex, q2::QuaterImaginary) = q1 * Complex(q2)
- (q1::QuaterImaginary, q2::Complex) = Complex(q1) * q2
/(q1::QuaterImaginary, q2::QuaterImaginary) = QuaterImaginary(Complex(q1) / Complex(q2)) /(q1::Complex, q2::QuaterImaginary) = q1 / Complex(q2) /(q1::QuaterImaginary, q2::Complex) = Complex(q1) / q2 ^(q1::QuaterImaginary, q2::QuaterImaginary) = QuaterImaginary(Complex(q1) ^ Complex(q2)) ^(q1::Complex, q2::QuaterImaginary) = q1 ^ Complex(q2) ^(q1::QuaterImaginary, q2::Complex) = Complex(q1) ^ q2
testquim()
</lang>
- Output:
1 -> 1 -> 1 -1 -> 103 -> -1 2 -> 2 -> 2 -2 -> 102 -> -2 3 -> 3 -> 3 -3 -> 101 -> -3 4 -> 10300 -> 4 -4 -> 100 -> -4 5 -> 10301 -> 5 -5 -> 203 -> -5 6 -> 10302 -> 6 -6 -> 202 -> -6 7 -> 10303 -> 7 -7 -> 201 -> -7 8 -> 10200 -> 8 -8 -> 200 -> -8 9 -> 10201 -> 9 -9 -> 303 -> -9 10 -> 10202 -> 10 -10 -> 302 -> -10 11 -> 10203 -> 11 -11 -> 301 -> -11 12 -> 10100 -> 12 -12 -> 300 -> -12 13 -> 10101 -> 13 -13 -> 1030003 -> -13 14 -> 10102 -> 14 -14 -> 1030002 -> -14 15 -> 10103 -> 15 -15 -> 1030001 -> -15 16 -> 10000 -> 16 -16 -> 1030000 -> -160 + 1im -> 10.2 -> 0.0 + 1.0im 0 - 1im -> 0.2 -> 0.0 - 1.0im 0 + 2im -> 10.0 -> 0.0 + 2.0im 0 - 2im -> 1030.0 -> 0.0 - 2.0im 0 + 3im -> 20.2 -> 0.0 + 3.0im 0 - 3im -> 1030.2 -> 0.0 - 3.0im 0 + 4im -> 20.0 -> 0.0 + 4.0im 0 - 4im -> 1020.0 -> 0.0 - 4.0im 0 + 5im -> 30.2 -> 0.0 + 5.0im 0 - 5im -> 1020.2 -> 0.0 - 5.0im 0 + 6im -> 30.0 -> 0.0 + 6.0im 0 - 6im -> 1010.0 -> 0.0 - 6.0im 0 + 7im -> 103000.2 -> 0.0 + 7.0im 0 - 7im -> 1010.2 -> 0.0 - 7.0im 0 + 8im -> 103000.0 -> 0.0 + 8.0im 0 - 8im -> 1000.0 -> 0.0 - 8.0im 0 + 9im -> 103010.2 -> 0.0 + 9.0im 0 - 9im -> 1000.2 -> 0.0 - 9.0im 0 + 10im -> 103010.0 -> 0.0 + 10.0im 0 - 10im -> 2030.0 -> 0.0 - 10.0im 0 + 11im -> 103020.2 -> 0.0 + 11.0im 0 - 11im -> 2030.2 -> 0.0 - 11.0im 0 + 12im -> 103020.0 -> 0.0 + 12.0im 0 - 12im -> 2020.0 -> 0.0 - 12.0im 0 + 13im -> 103030.2 -> 0.0 + 13.0im 0 - 13im -> 2020.2 -> 0.0 - 13.0im 0 + 14im -> 103030.0 -> 0.0 + 14.0im 0 - 14im -> 2010.0 -> 0.0 - 14.0im 0 + 15im -> 102000.2 -> 0.0 + 15.0im 0 - 15im -> 2010.2 -> 0.0 - 15.0im 0 + 16im -> 102000.0 -> 0.0 + 16.0im 0 - 16im -> 2000.0 -> 0.0 - 16.0im
Kotlin
The following deals with conversions to and from quater-imaginary only.
As the JDK lacks a complex number class, I've included a very basic one in the program. <lang scala>// version 1.2.10
import kotlin.math.ceil
class Complex(val real: Double, val imag: Double) {
constructor(r: Int, i: Int) : this(r.toDouble(), i.toDouble())
operator fun plus(other: Complex) = Complex(real + other.real, imag + other.imag)
operator fun times(other: Complex) = Complex( real * other.real - imag * other.imag, real * other.imag + imag * other.real )
operator fun times(other: Double) = Complex(real * other, imag * other)
fun inv(): Complex { val denom = real * real + imag * imag return Complex(real / denom, -imag / denom) }
operator fun unaryMinus() = Complex(-real, -imag)
operator fun minus(other: Complex) = this + (-other)
operator fun div(other: Complex) = this * other.inv()
// only works properly if 'real' and 'imag' are both integral fun toQuaterImaginary(): QuaterImaginary { if (real == 0.0 && imag == 0.0) return QuaterImaginary("0") var re = real.toInt() var im = imag.toInt() var fi = -1 val sb = StringBuilder() while (re != 0) { var rem = re % -4 re /= -4 if (rem < 0) { rem = 4 + rem re++ } sb.append(rem) sb.append(0) } if (im != 0) { var f = (Complex(0.0, imag) / Complex(0.0, 2.0)).real im = ceil(f).toInt() f = -4.0 * (f - im.toDouble()) var index = 1 while (im != 0) { var rem = im % -4 im /= -4 if (rem < 0) { rem = 4 + rem im++ } if (index < sb.length) { sb[index] = (rem + 48).toChar() } else { sb.append(0) sb.append(rem) } index += 2 } fi = f.toInt() } sb.reverse() if (fi != -1) sb.append(".$fi") var s = sb.toString().trimStart('0') if (s.startsWith(".")) s = "0$s" return QuaterImaginary(s) }
override fun toString(): String { val real2 = if (real == -0.0) 0.0 else real // get rid of negative zero val imag2 = if (imag == -0.0) 0.0 else imag // ditto var result = if (imag2 >= 0.0) "$real2 + ${imag2}i" else "$real2 - ${-imag2}i" result = result.replace(".0 ", " ").replace(".0i", "i").replace(" + 0i", "") if (result.startsWith("0 + ")) result = result.drop(4) if (result.startsWith("0 - ")) result = "-" + result.drop(4) return result }
}
class QuaterImaginary(val b2i: String) {
init { if (b2i == "" || !b2i.all { it in "0123." } || b2i.count { it == '.'} > 1 ) throw RuntimeException("Invalid Base 2i number") }
fun toComplex(): Complex { val pointPos = b2i.indexOf(".") var posLen = if (pointPos != -1) pointPos else b2i.length var sum = Complex(0.0, 0.0) var prod = Complex(1.0, 0.0) for (j in 0 until posLen) { val k = (b2i[posLen - 1 - j] - '0').toDouble() if (k > 0.0) sum += prod * k prod *= twoI } if (pointPos != -1) { prod = invTwoI for (j in posLen + 1 until b2i.length) { val k = (b2i[j] - '0').toDouble() if (k > 0.0) sum += prod * k prod *= invTwoI } } return sum }
override fun toString() = b2i
companion object { val twoI = Complex(0.0, 2.0) val invTwoI = twoI.inv() }
}
fun main(args: Array<String>) {
val fmt = "%4s -> %8s -> %4s" for (i in 1..16) { var c1 = Complex(i, 0) var qi = c1.toQuaterImaginary() var c2 = qi.toComplex() print("$fmt ".format(c1, qi, c2)) c1 = -c1 qi = c1.toQuaterImaginary() c2 = qi.toComplex() println(fmt.format(c1, qi, c2)) } println() for (i in 1..16) { var c1 = Complex(0, i) var qi = c1.toQuaterImaginary() var c2 = qi.toComplex() print("$fmt ".format(c1, qi, c2)) c1 = -c1 qi = c1.toQuaterImaginary() c2 = qi.toComplex() println(fmt.format(c1, qi, c2)) }
}</lang>
- Output:
1 -> 1 -> 1 -1 -> 103 -> -1 2 -> 2 -> 2 -2 -> 102 -> -2 3 -> 3 -> 3 -3 -> 101 -> -3 4 -> 10300 -> 4 -4 -> 100 -> -4 5 -> 10301 -> 5 -5 -> 203 -> -5 6 -> 10302 -> 6 -6 -> 202 -> -6 7 -> 10303 -> 7 -7 -> 201 -> -7 8 -> 10200 -> 8 -8 -> 200 -> -8 9 -> 10201 -> 9 -9 -> 303 -> -9 10 -> 10202 -> 10 -10 -> 302 -> -10 11 -> 10203 -> 11 -11 -> 301 -> -11 12 -> 10100 -> 12 -12 -> 300 -> -12 13 -> 10101 -> 13 -13 -> 1030003 -> -13 14 -> 10102 -> 14 -14 -> 1030002 -> -14 15 -> 10103 -> 15 -15 -> 1030001 -> -15 16 -> 10000 -> 16 -16 -> 1030000 -> -16 1i -> 10.2 -> 1i -1i -> 0.2 -> -1i 2i -> 10.0 -> 2i -2i -> 1030.0 -> -2i 3i -> 20.2 -> 3i -3i -> 1030.2 -> -3i 4i -> 20.0 -> 4i -4i -> 1020.0 -> -4i 5i -> 30.2 -> 5i -5i -> 1020.2 -> -5i 6i -> 30.0 -> 6i -6i -> 1010.0 -> -6i 7i -> 103000.2 -> 7i -7i -> 1010.2 -> -7i 8i -> 103000.0 -> 8i -8i -> 1000.0 -> -8i 9i -> 103010.2 -> 9i -9i -> 1000.2 -> -9i 10i -> 103010.0 -> 10i -10i -> 2030.0 -> -10i 11i -> 103020.2 -> 11i -11i -> 2030.2 -> -11i 12i -> 103020.0 -> 12i -12i -> 2020.0 -> -12i 13i -> 103030.2 -> 13i -13i -> 2020.2 -> -13i 14i -> 103030.0 -> 14i -14i -> 2010.0 -> -14i 15i -> 102000.2 -> 15i -15i -> 2010.2 -> -15i 16i -> 102000.0 -> 16i -16i -> 2000.0 -> -16i
Modula-2
<lang modula2>MODULE ImaginaryBase; FROM FormatString IMPORT FormatString; FROM RealMath IMPORT round; FROM Terminal IMPORT WriteString,WriteLn,ReadChar;
(* Helper *) TYPE
String = ARRAY[0..10] OF CHAR; StringBuilder = RECORD buf : String; ptr : CARDINAL; END;
PROCEDURE ToChar(n : INTEGER) : CHAR; BEGIN
CASE n OF 0 : RETURN '0' | 1 : RETURN '1' | 2 : RETURN '2' | 3 : RETURN '3' | 4 : RETURN '4' | 5 : RETURN '5' | 6 : RETURN '6' | 7 : RETURN '7' | 8 : RETURN '8' | 9 : RETURN '9' ELSE RETURN '-' END
END ToChar;
PROCEDURE AppendChar(VAR sb : StringBuilder; c : CHAR); BEGIN
sb.buf[sb.ptr] := c; INC(sb.ptr); sb.buf[sb.ptr] := 0C
END AppendChar;
PROCEDURE AppendInt(VAR sb : StringBuilder; n : INTEGER); BEGIN
sb.buf[sb.ptr] := ToChar(n); INC(sb.ptr); sb.buf[sb.ptr] := 0C
END AppendInt;
PROCEDURE Ceil(r : REAL) : REAL; VAR t : REAL; BEGIN
t := FLOAT(INT(r)); IF r - t > 0.0 THEN t := t + 1.0 END; RETURN t
END Ceil;
PROCEDURE Modulus(q,d : INTEGER) : INTEGER; VAR t : INTEGER; BEGIN
t := q / d; RETURN q - d * t
END Modulus;
PROCEDURE PrependInt(VAR sb : StringBuilder; n : INTEGER); VAR i : CARDINAL; BEGIN
i := sb.ptr; INC(sb.ptr); sb.buf[sb.ptr] := 0C; WHILE i > 0 DO sb.buf[i] := sb.buf[i-1]; DEC(i) END; sb.buf[0] := ToChar(n)
END PrependInt;
PROCEDURE Reverse(VAR str : String); VAR
i,j : CARDINAL; c : CHAR;
BEGIN
IF str[0] = 0C THEN RETURN END; i := 0; WHILE str[i] # 0C DO INC(i) END; DEC(i); j := 0; WHILE i > j DO c := str[i]; str[i] := str[j]; str[j] := c; DEC(i); INC(j) END
END Reverse;
PROCEDURE TrimStart(VAR str : String; c : CHAR); VAR i : CARDINAL; BEGIN
WHILE str[0] = c DO i := 0; WHILE str[i] # 0C DO str[i] := str[i+1]; INC(i) END END
END TrimStart;
PROCEDURE WriteInteger(n : INTEGER); VAR buf : ARRAY[0..15] OF CHAR; BEGIN
FormatString("%i", buf, n); WriteString(buf)
END WriteInteger;
(* Imaginary *) TYPE
Complex = RECORD real,imag : REAL; END; QuaterImaginary = RECORD b2i : String; END;
PROCEDURE ComplexMul(lhs,rhs : Complex) : Complex; BEGIN
RETURN Complex{ rhs.real * lhs.real - rhs.imag * lhs.imag, rhs.real * lhs.imag + rhs.imag * lhs.real }
END ComplexMul;
PROCEDURE ComplexMulR(lhs : Complex; rhs : REAL) : Complex; BEGIN
RETURN Complex{lhs.real * rhs, lhs.imag * rhs}
END ComplexMulR;
PROCEDURE ComplexInv(c : Complex) : Complex; VAR denom : REAL; BEGIN
denom := c.real * c.real + c.imag * c.imag; RETURN Complex{c.real / denom, -c.imag / denom}
END ComplexInv;
PROCEDURE ComplexDiv(lhs,rhs : Complex) : Complex; BEGIN
RETURN ComplexMul(lhs, ComplexInv(rhs))
END ComplexDiv;
PROCEDURE ComplexNeg(c : Complex) : Complex; BEGIN
RETURN Complex{-c.real, -c.imag}
END ComplexNeg;
PROCEDURE ComplexSum(lhs,rhs : Complex) : Complex; BEGIN
RETURN Complex{lhs.real + rhs.real, lhs.imag + rhs.imag}
END ComplexSum;
PROCEDURE WriteComplex(c : Complex); VAR buf : ARRAY[0..15] OF CHAR; BEGIN
IF c.imag = 0.0 THEN WriteInteger(INT(c.real)) ELSIF c.real = 0.0 THEN WriteInteger(INT(c.imag)); WriteString("i") ELSIF c.imag > 0.0 THEN WriteInteger(INT(c.real)); WriteString(" + "); WriteInteger(INT(c.imag)); WriteString("i") ELSE WriteInteger(INT(c.real)); WriteString(" - "); WriteInteger(INT(-c.imag)); WriteString("i") END
END WriteComplex;
PROCEDURE ToQuaterImaginary(c : Complex) : QuaterImaginary; VAR
re,im,fi,rem,index : INTEGER; f : REAL; t : Complex; sb : StringBuilder;
BEGIN
IF (c.real = 0.0) AND (c.imag = 0.0) THEN RETURN QuaterImaginary{"0"} END; re := INT(c.real); im := INT(c.imag); fi := -1; sb := StringBuilder{"", 0}; WHILE re # 0 DO rem := Modulus(re, -4); re := re / (-4); IF rem < 0 THEN rem := 4 + rem; INC(re) END; AppendInt(sb, rem); AppendInt(sb, 0) END; IF im # 0 THEN t := ComplexDiv(Complex{0.0, c.imag}, Complex{0.0, 2.0}); f := t.real; im := INT(Ceil(f)); f := -4.0 * (f - FLOAT(im)); index := 1; WHILE im # 0 DO rem := Modulus(im, -4); im := im / (-4); IF rem < 0 THEN rem := 4 + rem; INC(im) END; IF index < INT(sb.ptr) THEN sb.buf[index] := ToChar(rem) ELSE AppendInt(sb, 0); AppendInt(sb, rem) END; index := index + 2; END; fi := INT(f) END; Reverse(sb.buf); IF fi # -1 THEN AppendChar(sb, '.'); AppendInt(sb, fi) END; TrimStart(sb.buf, '0'); IF sb.buf[0] = '.' THEN PrependInt(sb, 0) END; RETURN QuaterImaginary{sb.buf}
END ToQuaterImaginary;
PROCEDURE ToComplex(qi : QuaterImaginary) : Complex; VAR
j,pointPos,posLen,b2iLen : INTEGER; k : REAL; sum,prod : Complex;
BEGIN
pointPos := 0; WHILE (qi.b2i[pointPos] # 0C) AND (qi.b2i[pointPos] # '.') DO INC(pointPos) END; IF qi.b2i[pointPos] # '.' THEN pointPos := -1; posLen := 0; WHILE qi.b2i[posLen] # 0C DO INC(posLen) END ELSE posLen := pointPos END;
sum := Complex{0.0, 0.0}; prod := Complex{1.0, 0.0}; FOR j:=0 TO posLen - 1 DO k := FLOAT(ORD(qi.b2i[posLen - 1 - j]) - ORD('0')); IF k > 0.0 THEN sum := ComplexSum(sum, ComplexMulR(prod, k)) END; prod := ComplexMul(prod, Complex{0.0, 2.0}) END; IF pointPos # -1 THEN prod := ComplexInv(Complex{0.0, 2.0}); b2iLen := 0; WHILE qi.b2i[b2iLen] # 0C DO INC(b2iLen) END; FOR j:=posLen + 1 TO b2iLen - 1 DO k := FLOAT(ORD(qi.b2i[j]) - ORD('0')); IF k > 0.0 THEN sum := ComplexSum(sum, ComplexMulR(prod, k)) END; prod := ComplexMul(prod, ComplexInv(Complex{0.0, 2.0})) END END; RETURN sum
END ToComplex;
(* Main *) VAR
c1,c2 : Complex; qi : QuaterImaginary; i : INTEGER;
BEGIN
FOR i:=1 TO 16 DO c1 := Complex{FLOAT(i), 0.0}; WriteComplex(c1); WriteString(" -> "); qi := ToQuaterImaginary(c1); WriteString(qi.b2i); WriteString(" -> "); c2 := ToComplex(qi); WriteComplex(c2); WriteString(" "); c1 := ComplexNeg(c1); WriteComplex(c1); WriteString(" -> "); qi := ToQuaterImaginary(c1); WriteString(qi.b2i); WriteString(" -> "); c2 := ToComplex(qi); WriteComplex(c2); WriteLn END; WriteLn;
FOR i:=1 TO 16 DO c1 := Complex{0.0, FLOAT(i)}; WriteComplex(c1); WriteString(" -> "); qi := ToQuaterImaginary(c1); WriteString(qi.b2i); WriteString(" -> "); c2 := ToComplex(qi); WriteComplex(c2); WriteString(" ");
c1 := ComplexNeg(c1); WriteComplex(c1); WriteString(" -> "); qi := ToQuaterImaginary(c1); WriteString(qi.b2i); WriteString(" -> "); c2 := ToComplex(qi); WriteComplex(c2); WriteLn END;
ReadChar
END ImaginaryBase.</lang>
- Output:
1 -> 1 -> 1 -1 -> 103 -> -1 2 -> 2 -> 2 -2 -> 102 -> -2 3 -> 3 -> 3 -3 -> 101 -> -3 4 -> 10300 -> 4 -4 -> 100 -> -4 5 -> 10301 -> 5 -5 -> 203 -> -5 6 -> 10302 -> 6 -6 -> 202 -> -6 7 -> 10303 -> 7 -7 -> 201 -> -7 8 -> 10200 -> 8 -8 -> 200 -> -8 9 -> 10201 -> 9 -9 -> 303 -> -9 10 -> 10202 -> 10 -10 -> 302 -> -10 11 -> 10203 -> 11 -11 -> 301 -> -11 12 -> 10100 -> 12 -12 -> 300 -> -12 13 -> 10101 -> 13 -13 -> 1030003 -> -13 14 -> 10102 -> 14 -14 -> 1030002 -> -14 15 -> 10103 -> 15 -15 -> 1030001 -> -15 16 -> 10000 -> 16 -16 -> 1030000 -> -16 1i -> 10.2 -> 1i -1i -> 0.2 -> -1i 2i -> 10.0 -> 2i -2i -> 1030.0 -> -2i 3i -> 20.2 -> 3i -3i -> 1030.2 -> -3i 4i -> 20.0 -> 4i -4i -> 1020.0 -> -4i 5i -> 30.2 -> 5i -5i -> 1020.2 -> -5i 6i -> 30.0 -> 6i -6i -> 1010.0 -> -6i 7i -> 103000.2 -> 7i -7i -> 1010.2 -> -7i 8i -> 103000.0 -> 8i -8i -> 1000.0 -> -8i 9i -> 103010.2 -> 9i -9i -> 1000.2 -> -9i 10i -> 103010.0 -> 10i -10i -> 2030.0 -> -10i 11i -> 103020.2 -> 11i -11i -> 2030.2 -> -11i 12i -> 103020.0 -> 12i -12i -> 2020.0 -> -12i 13i -> 103030.2 -> 13i -13i -> 2020.2 -> -13i 14i -> 103030.0 -> 14i -14i -> 2010.0 -> -14i 15i -> 102000.2 -> 15i -15i -> 2010.2 -> -15i 16i -> 102000.0 -> 16i -16i -> 2000.0 -> -16i
Perl 6
These are generalized imaginary-base conversion routines. They only work for imaginary bases, not complex. (Any real portion of the radix must be zero.) Theoretically they could be made to work for any imaginary base; in practice, they are limited to integer bases from -6i to -2i and 2i to 6i. Bases -1i and 1i exist but require special handling and are not supported. Bases larger than 6i (or -6i) require digits outside of base 36 to express them, so aren't as standardized, are implementation dependent and are not supported. Note that imaginary number coefficients are stored as floating point numbers in Perl 6 so some rounding error may creep in during calculations. The precision these conversion routines use is configurable; we are using 6 decimal, um... radicimal(?) places of precision here.
Implements minimum, extra kudos and stretch goals.
<lang perl6>multi sub base ( Real $num, Int $radix where -37 < * < -1, :$precision = -15 ) {
return '0' unless $num; my $value = $num; my $result = ; my $place = 0; my $upper-bound = 1 / (-$radix + 1); my $lower-bound = $radix * $upper-bound;
$value = $num / $radix ** ++$place until $lower-bound <= $value < $upper-bound;
while ($value or $place > 0) and $place > $precision { my $digit = ($radix * $value - $lower-bound).Int; $value = $radix * $value - $digit; $result ~= '.' unless $place or $result.contains: '.'; $result ~= $digit == -$radix ?? ($digit-1).base(-$radix)~'0' !! $digit.base(-$radix); $place-- } $result
}
multi sub base (Numeric $num, Complex $radix where *.re == 0, :$precision = -8 ) {
die "Base $radix out of range" unless -6 <= $radix.im <= -2 or 2 <= $radix.im <= 6; my ($re, $im) = $num.Complex.reals; my ($re-wh, $re-fr) = $re.&base( -$radix.im².Int, :precision($precision) ).split: '.'; my ($im-wh, $im-fr) = ($im/$radix.im).&base( -$radix.im².Int, :precision($precision) ).split: '.'; $_ //= for $re-fr, $im-fr;
sub zip (Str $a, Str $b) { my $l = '0' x ($a.chars - $b.chars).abs; ([~] flat ($a~$l).comb Z flat ($b~$l).comb).subst(/ '0'+ $ /, ) || '0' }
my $whole = flip zip $re-wh.flip, $im-wh.flip; my $fraction = zip $im-fr, $re-fr; $fraction eq 0 ?? "$whole" !! "$whole.$fraction"
}
multi sub parse-base (Str $str, Complex $radix where *.re == 0) {
return -1 * $str.substr(1).&parse-base($radix) if $str.substr(0,1) eq '-'; my ($whole, $frac) = $str.split: '.'; my $fraction = 0; $fraction = [+] $frac.comb.kv.map: { $^v.parse-base($radix.im².Int) * $radix ** -($^k+1) } if $frac; $fraction + [+] $whole.flip.comb.kv.map: { $^v.parse-base($radix.im².Int) * $radix ** $^k }
}
- TESTING
for 0, 2i, 1, 2i, 5, 2i, -13, 2i, 9i, 2i, -3i, 2i, 7.75-7.5i, 2i, .25, 2i, # base 2i tests
5+5i, 2i, 5+5i, 3i, 5+5i, 4i, 5+5i, 5i, 5+5i, 6i, # same value, positive imaginary bases 5+5i, -2i, 5+5i, -3i, 5+5i, -4i, 5+5i, -5i, 5+5i, -6i, # same value, negative imaginary bases 227.65625+10.859375i, 4i, # larger test value 31433.3487654321-2902.4480452675i, 6i # heh -> $v, $r {
my $ibase = $v.&base($r, :precision(-6)); printf "%33s.&base\(%2si\) = %-11s : %13s.&parse-base\(%2si\) = %s\n",
$v, $r.im, $ibase, "'$ibase'", $r.im, $ibase.&parse-base($r).round(1e-10).narrow;
}</lang>
- Output:
0.&base( 2i) = 0 : '0'.&parse-base( 2i) = 0 1.&base( 2i) = 1 : '1'.&parse-base( 2i) = 1 5.&base( 2i) = 10301 : '10301'.&parse-base( 2i) = 5 -13.&base( 2i) = 1030003 : '1030003'.&parse-base( 2i) = -13 0+9i.&base( 2i) = 103010.2 : '103010.2'.&parse-base( 2i) = 0+9i -0-3i.&base( 2i) = 1030.2 : '1030.2'.&parse-base( 2i) = 0-3i 7.75-7.5i.&base( 2i) = 11210.31 : '11210.31'.&parse-base( 2i) = 7.75-7.5i 0.25.&base( 2i) = 1.03 : '1.03'.&parse-base( 2i) = 0.25 5+5i.&base( 2i) = 10331.2 : '10331.2'.&parse-base( 2i) = 5+5i 5+5i.&base( 3i) = 25.3 : '25.3'.&parse-base( 3i) = 5+5i 5+5i.&base( 4i) = 25.C : '25.C'.&parse-base( 4i) = 5+5i 5+5i.&base( 5i) = 15 : '15'.&parse-base( 5i) = 5+5i 5+5i.&base( 6i) = 15.6 : '15.6'.&parse-base( 6i) = 5+5i 5+5i.&base(-2i) = 11321.2 : '11321.2'.&parse-base(-2i) = 5+5i 5+5i.&base(-3i) = 1085.6 : '1085.6'.&parse-base(-3i) = 5+5i 5+5i.&base(-4i) = 10F5.4 : '10F5.4'.&parse-base(-4i) = 5+5i 5+5i.&base(-5i) = 10O5 : '10O5'.&parse-base(-5i) = 5+5i 5+5i.&base(-6i) = 5.U : '5.U'.&parse-base(-6i) = 5+5i 227.65625+10.859375i.&base( 4i) = 10234.5678 : '10234.5678'.&parse-base( 4i) = 227.65625+10.859375i 31433.3487654321-2902.4480452675i.&base( 6i) = PERL6.ROCKS : 'PERL6.ROCKS'.&parse-base( 6i) = 31433.3487654321-2902.4480452675i
Phix
<lang Phix>include complex.e
function base2(atom num, integer radix, precision = -8)
if radix<-36 or radix>-2 then throw("radix out of range (-2..-36)") end if sequence result if num=0 then result = {"0",""} else integer place = 0 result = "" atom v = num atom upper_bound = 1/(1-radix), lower_bound = radix*upper_bound while not(lower_bound <= v) or not(v < upper_bound) do place += 1 v = num/power(radix,place) end while while (v or place > 0) and (place > precision) do integer digit = floor(radix*v - lower_bound) v = (radix*v - digit) if place=0 and not find('.',result) then result &= '.' end if result &= digit+iff(digit>9?'a'-10:'0') place -= 1 end while integer dot = find('.',result) if dot then result = trim_tail(result,'0') result = {result[1..dot-1],result[dot+1..$]} else result = {result,""} end if end if return result
end function
function zip(string a, string b)
integer ld = length(a)-length(b) if ld!=0 then if ld>0 then b &= repeat('0',ld) else a &= repeat('0',abs(ld)) end if end if string res = "" for i=1 to length(a) do res &= a[i]&b[i] end for res = trim_tail(res,'0') if res="" then res = "0" end if return res
end function
function base(complexn num, integer radix, precision = -8)
integer absrad = abs(radix), radix2 = -power(radix,2) if absrad<2 or absrad>6 then throw("base radix out of range") end if atom {re, im} = {complex_real(num), complex_imag(num)} string {re_wh, re_fr} = base2(re, radix2, precision), {im_wh, im_fr} = base2(im/radix, radix2, precision) string whole = reverse(zip(reverse(re_wh), reverse(im_wh))), fraction = zip(im_fr, re_fr) if fraction!="0" then whole &= '.'&fraction end if return whole
end function
function parse_base(string str, integer radix)
complexn fraction = 0
integer dot = find('.',str) if dot then string fr = str[dot+1..$] for i=1 to length(fr) do integer c = fr[i] c -= iff(c>='a'?'a'-10:'0') fraction = complex_add(fraction,complex_mul(c,complex_power({0,radix},-i))) end for str = str[1..dot-1] end if
str = reverse(str) for i=1 to length(str) do integer c = str[i] c -= iff(c>='a'?'a'-10:'0') fraction = complex_add(fraction,complex_mul(c,complex_power({0,radix},(i-1)))) end for
return fraction
end function
constant tests = {{0,2},{1,2},{5,2},{-13,2},{{0,9},2},{{0,-3},2},{{7.75,-7.5}, 2},{.25, 2}, -- base 2i tests
{{5,5}, 2},{{5,5}, 3},{{5,5}, 4},{{5,5}, 5},{{5,5}, 6}, -- same value, positive imaginary bases {{5,5},-2},{{5,5},-3},{{5,5},-4},{{5,5},-5},{{5,5},-6}, -- same value, negative imaginary bases {{227.65625,10.859375},4}, -- larger test value {{-579.8225308641975744,-5296.406378600824},6}} -- phix.rules
-- matches output of Sidef and Perl6: for t=1 to length(tests) do
{complexn v, integer r} = tests[t] string ibase = base(v,r), strv = complex_sprint(v), strb = complex_sprint(parse_base(ibase, r)) printf(1,"base(%20s, %2di) = %-10s : parse_base(%12s, %2di) = %s\n", {strv, r, ibase, '"'&ibase&'"', r, strb})
end for
-- matches output of Kotlin, Java, Go, D, and C#: for ri=1 to 2 do -- real then imag
for i=1 to 16 do complexn c = iff(ri=1?i:{0,i}), nc = complex_neg(c) string sc = complex_sprint(c), snc = complex_sprint(nc), ib = base(c,2), inb = base(nc,2), rc = complex_sprint(parse_base(ib,2)), rnc = complex_sprint(parse_base(inb,2)) printf(1,"%4s -> %8s -> %4s %4s -> %8s -> %4s\n", {sc, ib, rc, snc, inb, rnc }) end for puts(1,"\n")
end for</lang>
- Output:
Matches the output of Sidef and Perl6, except for the final line:
base( -579.823-5296.41i, 6i) = phix.rules : parse_base("phix.rules", 6i) = -579.823-5296.41i
Also matches the output of Kotlin, Java, Go, D, and C#, except the even entries in the second half, eg:
2i -> 10 -> 2i -2i -> 1030 -> -2i
instead of
2i -> 10.0 -> 2i -2i -> 1030.0 -> -2i
ie the unnecessary trailing ".0" are trimmed. (see talk page)
Python
<lang python>import math import re
def inv(c):
denom = c.real * c.real + c.imag * c.imag return complex(c.real / denom, -c.imag / denom)
class QuaterImaginary:
twoI = complex(0, 2) invTwoI = inv(twoI)
def __init__(self, str): if not re.match("^[0123.]+$", str) or str.count('.') > 1: raise Exception('Invalid base 2i number') self.b2i = str
def toComplex(self): pointPos = self.b2i.find('.') posLen = len(self.b2i) if (pointPos < 0) else pointPos sum = complex(0, 0) prod = complex(1, 0) for j in xrange(0, posLen): k = (ord(self.b2i[posLen - 1 - j]) - ord('0')) if k > 0: sum = sum + prod * k prod = prod * QuaterImaginary.twoI if pointPos != -1: prod = QuaterImaginary.invTwoI for j in xrange(posLen + 1, len(self.b2i)): k = (ord(self.b2i[j]) - ord('0')) if k > 0: sum = sum + prod * k prod = prod * QuaterImaginary.invTwoI return sum
def __str__(self): return str(self.b2i)
def toQuaterImaginary(c):
if c.real == 0.0 and c.imag == 0.0: return QuaterImaginary("0")
re = int(c.real) im = int(c.imag) fi = -1 ss = "" while re != 0: rem = re % -4 re = re / -4 if rem < 0: rem = 4 + rem re = re + 1 ss = ss + str(rem) + '0' if im != 0: f = (complex(0, c.imag) / complex(0, 2)).real im = int(math.ceil(f)) f = -4 * (f - im) index = 1 while im != 0: rem = im % -4 im = im / -4 if rem < 0: rem = 4 + rem im = im + 1 if index < len(ss): ss[index] = chr(rem + 48) else: ss = ss + '0' + str(rem) index = index + 2 fi = int(f) ss = ss[::-1] if fi != -1: ss = ss + '.' + str(fi) ss = ss.lstrip('0') if ss[0] == '.': ss = '0' + ss return QuaterImaginary(ss)
for i in xrange(1,17):
c1 = complex(i, 0) qi = toQuaterImaginary(c1) c2 = qi.toComplex() print "{0:8} -> {1:>8} -> {2:8} ".format(c1, qi, c2),
c1 = -c1 qi = toQuaterImaginary(c1) c2 = qi.toComplex() print "{0:8} -> {1:>8} -> {2:8}".format(c1, qi, c2)
for i in xrange(1,17):
c1 = complex(0, i) qi = toQuaterImaginary(c1) c2 = qi.toComplex() print "{0:8} -> {1:>8} -> {2:8} ".format(c1, qi, c2),
c1 = -c1 qi = toQuaterImaginary(c1) c2 = qi.toComplex() print "{0:8} -> {1:>8} -> {2:8}".format(c1, qi, c2)
print "done" </lang>
- Output:
(1+0j) -> 1 -> (1+0j) (-1-0j) -> 103 -> (-1+0j) (2+0j) -> 2 -> (2+0j) (-2-0j) -> 102 -> (-2+0j) (3+0j) -> 3 -> (3+0j) (-3-0j) -> 101 -> (-3+0j) (4+0j) -> 10300 -> (4+0j) (-4-0j) -> 100 -> (-4+0j) (5+0j) -> 10301 -> (5+0j) (-5-0j) -> 203 -> (-5+0j) (6+0j) -> 10302 -> (6+0j) (-6-0j) -> 202 -> (-6+0j) (7+0j) -> 10303 -> (7+0j) (-7-0j) -> 201 -> (-7+0j) (8+0j) -> 10200 -> (8+0j) (-8-0j) -> 200 -> (-8+0j) (9+0j) -> 10201 -> (9+0j) (-9-0j) -> 303 -> (-9+0j) (10+0j) -> 10202 -> (10+0j) (-10-0j) -> 302 -> (-10+0j) (11+0j) -> 10203 -> (11+0j) (-11-0j) -> 301 -> (-11+0j) (12+0j) -> 10100 -> (12+0j) (-12-0j) -> 300 -> (-12+0j) (13+0j) -> 10101 -> (13+0j) (-13-0j) -> 1030003 -> (-13+0j) (14+0j) -> 10102 -> (14+0j) (-14-0j) -> 1030002 -> (-14+0j) (15+0j) -> 10103 -> (15+0j) (-15-0j) -> 1030001 -> (-15+0j) (16+0j) -> 10000 -> (16+0j) (-16-0j) -> 1030000 -> (-16+0j) 1j -> 10.2 -> 1j (-0-1j) -> 0.2 -> -1j 2j -> 10.0 -> 2j (-0-2j) -> 1030.0 -> -2j 3j -> 20.2 -> 3j (-0-3j) -> 1030.2 -> -3j 4j -> 20.0 -> 4j (-0-4j) -> 1020.0 -> -4j 5j -> 30.2 -> 5j (-0-5j) -> 1020.2 -> -5j 6j -> 30.0 -> 6j (-0-6j) -> 1010.0 -> -6j 7j -> 103000.2 -> 7j (-0-7j) -> 1010.2 -> -7j 8j -> 103000.0 -> 8j (-0-8j) -> 1000.0 -> -8j 9j -> 103010.2 -> 9j (-0-9j) -> 1000.2 -> -9j 10j -> 103010.0 -> 10j (-0-10j) -> 2030.0 -> -10j 11j -> 103020.2 -> 11j (-0-11j) -> 2030.2 -> -11j 12j -> 103020.0 -> 12j (-0-12j) -> 2020.0 -> -12j 13j -> 103030.2 -> 13j (-0-13j) -> 2020.2 -> -13j 14j -> 103030.0 -> 14j (-0-14j) -> 2010.0 -> -14j 15j -> 102000.2 -> 15j (-0-15j) -> 2010.2 -> -15j 16j -> 102000.0 -> 16j (-0-16j) -> 2000.0 -> -16j done
Sidef
<lang ruby>func base (Number num, Number radix { _ ~~ (-36 .. -2) }, precision = -15) -> String {
num || return '0'
var place = 0 var result = var value = num var upper_bound = 1/(-radix + 1) var lower_bound = radix*upper_bound
while (!(lower_bound <= value) || !(value < upper_bound)) { value = num/(radix**++place) }
while ((value || (place > 0)) && (place > precision)) { var digit = (radix*value - lower_bound -> int) value = (radix*value - digit) result += '.' if (!place && !result.contains('.')) result += ((digit == -radix) ? (digit-1 -> base(-radix) + '0') : digit.base(-radix)) place-- }
return result
}
func base (Number num, Number radix { .re == 0 }, precision = -8) -> String {
(radix.im.abs ~~ 2..6) || die "Base #{radix} out of range"
var (re, im) = (num.re, num.im) var (re_wh, re_fr=) = base(re, -radix.im**2, precision).split('.')... var (im_wh, im_fr=) = base(im/radix.im, -radix.im**2, precision).split('.')...
func zip (String a, String b) { var l = ('0' * abs(a.len - b.len)) chars(a+l) ~Z chars(b+l) -> flat.join.sub(/0+\z/, ) || '0' }
var whole = zip(re_wh.flip, im_wh.flip).flip var fraction = zip(im_fr, re_fr) fraction == '0' ? whole : "#{whole}.#{fraction}"
}
func parse_base (String str, Number radix { .re == 0 }) -> Number {
if (str.char(0) == '-') { return (-1 * parse_base(str.substr(1), radix)) }
var (whole, frac=) = str.split('.')...
var fraction = frac.chars.map_kv {|k,v| Number(v, radix.im**2) * radix**-(k+1) }.sum
fraction += whole.flip.chars.map_kv {|k,v| Number(v, radix.im**2) * radix**k }.sum
return fraction
}
var tests = [0, 2i, 1, 2i, 5, 2i, -13, 2i, 9i, 2i, -3i, 2i, 7.75-7.5i, 2i, .25, 2i, # base 2i tests
5+5i, 2i, 5+5i, 3i, 5+5i, 4i, 5+5i, 5i, 5+5i, 6i, # same value, positive imaginary bases 5+5i, -2i, 5+5i, -3i, 5+5i, -4i, 5+5i, -5i, 5+5i, -6i, # same value, negative imaginary bases 227.65625+10.859375i, 4i] # larger test value
tests.each_slice(2, {|v,r|
var ibase = base(v, r) printf("base(%20s, %2si) = %-10s : parse_base(%12s, %2si) = %s\n", v, r.im, ibase, "'#{ibase}'", r.im, parse_base(ibase, r).round(-8))
})</lang>
- Output:
base( 0, 2i) = 0 : parse_base( '0', 2i) = 0 base( 1, 2i) = 1 : parse_base( '1', 2i) = 1 base( 5, 2i) = 10301 : parse_base( '10301', 2i) = 5 base( -13, 2i) = 1030003 : parse_base( '1030003', 2i) = -13 base( 9i, 2i) = 103010.2 : parse_base( '103010.2', 2i) = 9i base( -3i, 2i) = 1030.2 : parse_base( '1030.2', 2i) = -3i base( 7.75-7.5i, 2i) = 11210.31 : parse_base( '11210.31', 2i) = 7.75-7.5i base( 0.25, 2i) = 1.03 : parse_base( '1.03', 2i) = 0.25 base( 5+5i, 2i) = 10331.2 : parse_base( '10331.2', 2i) = 5+5i base( 5+5i, 3i) = 25.3 : parse_base( '25.3', 3i) = 5+5i base( 5+5i, 4i) = 25.c : parse_base( '25.c', 4i) = 5+5i base( 5+5i, 5i) = 15 : parse_base( '15', 5i) = 5+5i base( 5+5i, 6i) = 15.6 : parse_base( '15.6', 6i) = 5+5i base( 5+5i, -2i) = 11321.2 : parse_base( '11321.2', -2i) = 5+5i base( 5+5i, -3i) = 1085.6 : parse_base( '1085.6', -3i) = 5+5i base( 5+5i, -4i) = 10f5.4 : parse_base( '10f5.4', -4i) = 5+5i base( 5+5i, -5i) = 10o5 : parse_base( '10o5', -5i) = 5+5i base( 5+5i, -6i) = 5.u : parse_base( '5.u', -6i) = 5+5i base(227.65625+10.859375i, 4i) = 10234.5678 : parse_base('10234.5678', 4i) = 227.65625+10.859375i