Imaginary base numbers
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 named.
Task: Write a set of procedures (functions, subroutines, however they are referred to in your language) to convert base 10 numbers to imaginary and back. At a minimum, support quater-imaginary (base 2i).
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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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 here. 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 8 decimal, um... radicimal(?) places of precision here.
<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
-> $v, $r {
my $ibase = $v.&base($r); printf "%20s.&base\(%2si\) = %-10s : %12s.&parse-base\(%2si\) = %s\n",
$v, $r.im, $ibase, "'$ibase'", $r.im, $ibase.&parse-base($r).round(1e-8).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
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