Negative base numbers
You are encouraged to solve this task according to the task description, using any language you may know.
Negative base numbers are an alternate way to encode numbers without the need for a minus sign. Various negative bases may be used including negadecimal (base -10), negabinary (-2) and negaternary (-3).[1][2]
- Task
- Encode the decimal number 10 as negabinary (expect 11110)
- Encode the decimal number 146 as negaternary (expect 21102)
- Encode the decimal number 15 as negadecimal (expect 195)
- In each of the above cases, convert the encoded number back to decimal.
- extra credit
- supply an integer, that when encoded to base -62 (or something "higher"), expresses the
name of the language being used (with correct capitalization). If the computer language has
non-alphanumeric characters, try to encode them into the negatory numerals, or use other
characters instead.
ALGOL 68
<lang algol68># Conversion to/from negative base numbers #
- Note - no checks for valid bases or digits bases -2 .. -63 are handled #
- A-Z represent the digits 11 .. 35, a-z represent the digits 36 .. 61 #
- the digit 62 is represented by a space #
- 1 2 3 4 5 6 #
- 01234567890123456789012345678901234567890123456789012 #
[]CHAR base digits = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz "[ AT 0 ];
- returns s decoded to an integer from the negative base b #
PRIO FROMNBASE = 9; OP FROMNBASE = ( STRING s, INT b )LONG INT:
BEGIN
LONG INT result := 0;
LONG INT base multiplier := 1;
FOR d pos FROM UPB s BY -1 TO LWB s DO
INT digit = IF s[ d pos ] = " "
THEN 62
ELIF s[ d pos ] >= "a"
THEN ( ABS s[ d pos ] + 36 ) - ABS "a"
ELIF s[ d pos ] >= "A"
THEN ( ABS s[ d pos ] + 10 ) - ABS "A"
ELSE ABS s[ d pos ] - ABS "0"
FI;
result +:= base multiplier * digit;
base multiplier *:= b
OD;
result
END # FROMNBASE # ;
OP FROMNBASE = ( CHAR c, INT b )LONG INT: STRING(c) FROMNBASE b;
- returns n encoded as a string in the negative base b #
PRIO TONBASE = 9; OP TONBASE = ( LONG INT n, INT b )STRING:
BEGIN
STRING result := "";
LONG INT v := n;
WHILE
INT d := SHORTEN IF v < 0 THEN - ( ABS v MOD b ) ELSE v MOD b FI;
v OVERAB b;
IF d < 0
THEN
d -:= b;
v +:= 1
FI;
base digits[ d ] +=: result;
v /= 0
DO SKIP OD;
result
END # TONBASE # ;
- tests the TONBASE and FROMNBASE operators #
PROC test n base = ( LONG INT number, INT base, STRING expected )VOID:
BEGIN
PROC expect = ( BOOL result )STRING: IF result THEN "" ELSE " INCORRECT" FI;
STRING encoded = number TONBASE base;
LONG INT decoded = encoded FROMNBASE base;
print( ( whole( number, 0 ), " encodes to: ", encoded ) );
print( ( " base ", whole( base, 0 ), expect( encoded = expected ), newline ) );
print( ( encoded, " decodes to: ", whole( decoded, 0 ) ) );
print( ( " base ", whole( base, 0 ), expect( decoded = number ), newline ) )
END # test n base # ;
test n base( 10, -2, "11110" ); test n base( 146, -3, "21102" ); test n base( 15, -10, "195" );
- The defining document for ALGOL 68 spells the name "Algol 68" on the cover, though inside it is "ALGOL 68" #
- at the risk of "judging a language by it's cover", we use "Algol 68" as the name here... #
test n base( - LONG 36492107981104, -63, "Algol 68" )</lang>
- Output:
10 encodes to: 11110 base -2 11110 decodes to: 10 base -2 146 encodes to: 21102 base -3 21102 decodes to: 146 base -3 15 encodes to: 195 base -10 195 decodes to: 15 base -10 -36492107981104 encodes to: Algol 68 base -63 Algol 68 decodes to: -36492107981104 base -63
C#
<lang csharp>using System; using System.Collections.Generic; using System.Linq; using System.Text;
namespace NegativeBaseNumbers {
class Program {
const string DIGITS = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz";
static string EncodeNegativeBase(long n, int b) {
if (b < -62 || b > -1) {
throw new ArgumentOutOfRangeException("b");
}
if (n == 0) {
return "0";
}
StringBuilder output = new StringBuilder();
long nn = n;
while (nn != 0) {
int rem = (int)(nn % b);
nn /= b;
if (rem < 0) {
nn++;
rem -= b;
}
output.Append(DIGITS[rem]);
}
return new string(output.ToString().Reverse().ToArray());
}
static long DecodeNegativeBase(string ns, int b) {
if (b < -62 || b > -1) {
throw new ArgumentOutOfRangeException("b");
}
if (ns == "0") {
return 0;
}
long total = 0;
long bb = 1;
for (int i = ns.Length - 1; i >= 0; i--) {
char c = ns[i];
total += DIGITS.IndexOf(c) * bb;
bb *= b;
}
return total;
}
static void Main(string[] args) {
List<Tuple<long, int>> nbl = new List<Tuple<long, int>>() {
new Tuple<long, int>(10,-2),
new Tuple<long, int>(146,-3),
new Tuple<long, int>(15,-10),
new Tuple<long, int>(-34025238427,-62),
};
foreach (var p in nbl) {
string ns = EncodeNegativeBase(p.Item1, p.Item2);
Console.WriteLine("{0,12} encoded in base {1,-3} = {2}", p.Item1, p.Item2, ns);
long n = DecodeNegativeBase(ns, p.Item2);
Console.WriteLine("{0,12} decoded in base {1,-3} = {2}", ns, p.Item2, n);
Console.WriteLine();
}
}
}
}</lang>
- Output:
10 encoded in base -2 = 11110
11110 decoded in base -2 = 10
146 encoded in base -3 = 21102
21102 decoded in base -3 = 146
15 encoded in base -10 = 195
195 decoded in base -10 = 15
-34025238427 encoded in base -62 = csharp
csharp decoded in base -62 = -34025238427
D
<lang D>import std.stdio;
immutable DIGITS = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz";
void main() {
driver(10, -2); driver(146, -3); driver(15, -10); driver(13, -62);
}
void driver(long n, int b) {
string ns = encodeNegBase(n, b);
writefln("%12d encoded in base %3d = %12s", n, b, ns);
long p = decodeNegBase(ns, b);
writefln("%12s decoded in base %3d = %12d", ns, b, p);
writeln;
}
string encodeNegBase(long n, int b) in {
import std.exception : enforce; enforce(b <= -1 && b >= -62);
} body {
if (n==0) return "0";
char[] output;
long nn = n;
while (nn != 0) {
int rem = nn % b;
nn /= b;
if (rem < 0) {
nn++;
rem -= b;
}
output ~= DIGITS[rem];
}
import std.algorithm : reverse;
reverse(output); return cast(string) output;
}
long decodeNegBase(string ns, int b) in {
import std.exception : enforce; enforce(b <= -1 && b >= -62);
} body {
if (ns == "0") return 0;
long total = 0;
long bb = 1;
foreach_reverse (c; ns) {
foreach(i,d; DIGITS) {
if (c==d) {
total += i * bb;
bb *= b;
break;
}
}
}
return total;
}</lang>
- Output:
10 encoded in base -2 = 11110
11110 decoded in base -2 = 10
146 encoded in base -3 = 21102
21102 decoded in base -3 = 146
15 encoded in base -10 = 195
195 decoded in base -10 = 15
13 encoded in base -62 = D
D decoded in base -62 = 13
Go
<lang go>package main
import (
"fmt" "log" "strings"
)
const digits = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz"
func reverse(bs []byte) []byte {
for i, j := 0, len(bs)-1; i < len(bs)/2; i, j = i+1, j-1 {
bs[i], bs[j] = bs[j], bs[i]
}
return bs
}
func encodeNegBase(n, b int64) (string, error) {
if b < -62 || b > -1 {
return "", fmt.Errorf("base must be between -62 and -1 inclusive")
}
if n == 0 {
return "0", nil
}
var out []byte
for n != 0 {
rem := n % b
n /= b
if rem < 0 {
n++
rem -= b
}
out = append(out, digits[int(rem)])
}
reverse(out)
return string(out), nil
}
func decodeNegBase(ns string, b int64) (int64, error) {
if b < -62 || b > -1 {
return 0, fmt.Errorf("base must be between -62 and -1 inclusive")
}
if ns == "0" {
return 0, nil
}
total := int64(0)
bb := int64(1)
bs := []byte(ns)
reverse(bs)
for _, c := range bs {
total += int64(strings.IndexByte(digits, c)) * bb
bb *= b
}
return total, nil
}
func main() {
numbers := []int64{10, 146, 15, -942, 1488588316238}
bases := []int64{-2, -3, -10, -62, -62}
for i := 0; i < len(numbers); i++ {
n := numbers[i]
b := bases[i]
ns, err := encodeNegBase(n, b)
if err != nil {
log.Fatal(err)
}
fmt.Printf("%13d encoded in base %-3d = %s\n", n, b, ns)
n, err = decodeNegBase(ns, b)
if err != nil {
log.Fatal(err)
}
fmt.Printf("%13s decoded in base %-3d = %d\n\n", ns, b, n)
}
}</lang>
- Output:
10 encoded in base -2 = 11110
11110 decoded in base -2 = 10
146 encoded in base -3 = 21102
21102 decoded in base -3 = 146
15 encoded in base -10 = 195
195 decoded in base -10 = 15
-942 encoded in base -62 = Go
Go decoded in base -62 = -942
1488588316238 encoded in base -62 = Rosetta
Rosetta decoded in base -62 = 1488588316238
Haskell
<lang haskell>import Data.Char (chr, ord) import Numeric (showIntAtBase)
-- The remainder and quotient of n/d, where the remainder is guaranteed to be -- non-negative. The divisor, d, is assumed to be negative. quotRemP :: Integral a => a -> a -> (a, a) quotRemP n d = let (q, r) = quotRem n d
in if r < 0 then (q+1, r-d) else (q, r)
-- Convert the number n to base b, where b is assumed to be less than zero. toNegBase :: Integral a => a -> a -> a toNegBase b n = let (q, r) = quotRemP n b
in if q == 0 then r else negate b * toNegBase b q + r
-- Convert n to a string, where n is assumed to be a base b number, with b less -- than zero. showAtBase :: (Integral a, Show a) => a -> a -> String showAtBase b n = showIntAtBase (abs b) charOf n ""
where charOf m | m < 10 = chr $ m + ord '0'
| m < 36 = chr $ m + ord 'a' - 10
| otherwise = '?'
-- Print a number in base b, where b is assumed to be less than zero. printAtBase :: (Integral a, Show a) => a -> a -> IO () printAtBase b = putStrLn . showAtBase b . toNegBase b
main :: IO () main = do
printAtBase (-2) 10 printAtBase (-3) 146 printAtBase (-10) 15 printAtBase (-16) 107 printAtBase (-36) 41371458</lang>
- Output:
$ ./negbase 11110 21102 195 1ab perl6
Java
<lang Java>import java.util.List; import java.util.Map; import java.util.Objects;
public class NegativeBaseNumbers {
private static final String DIGITS = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz";
private static String encodeNegBase(long n, int b) {
if (b < -62 || b > -1) throw new IllegalArgumentException("Parameter b is out of bounds");
if (n == 0) return "0";
StringBuilder out = new StringBuilder();
long nn = n;
while (nn != 0) {
int rem = (int) (nn % b);
nn /= b;
if (rem < 0) {
nn++;
rem -= b;
}
out.append(DIGITS.charAt(rem));
}
out.reverse();
return out.toString();
}
private static long decodeNegBase(String ns, int b) {
if (b < -62 || b > -1) throw new IllegalArgumentException("Parameter b is out of bounds");
if (Objects.equals(ns, "0")) return 0;
long total = 0;
long bb = 1;
for (int i = ns.length() - 1; i >= 0; i--) {
char c = ns.charAt(i);
total += DIGITS.indexOf(c) * bb;
bb *= b;
}
return total;
}
public static void main(String[] args) {
List<Map.Entry<Long, Integer>> nbl = List.of(
Map.entry(10L, -2),
Map.entry(146L, -3),
Map.entry(15L, -10),
Map.entry(-4393346L, -62)
);
for (Map.Entry<Long, Integer> p : nbl) {
String ns = encodeNegBase(p.getKey(), p.getValue());
System.out.printf("%12d encoded in base %-3d = %s\n", p.getKey(), p.getValue(), ns);
long n = decodeNegBase(ns, p.getValue());
System.out.printf("%12s decoded in base %-3d = %d\n\n", ns, p.getValue(), n);
}
}
}</lang>
- Output:
10 encoded in base -2 = 11110
11110 decoded in base -2 = 10
146 encoded in base -3 = 21102
21102 decoded in base -3 = 146
15 encoded in base -10 = 195
195 decoded in base -10 = 15
-4393346 encoded in base -62 = Java
Java decoded in base -62 = -4393346
jq
If your jq does not have `trunc/0` then use this: <lang jq>def trunc: if . >= 0 then floor else -(-(.)|trunc) end;</lang> <lang jq>def negbase($b):
if ($b >= 0) then error("negbase requires negative base")
elif . == 0 then "0"
else {n: ., ans: ""}
| until(.n == 0;
.r = ((.n % $b))
| .n = ((.n / $b) | trunc)
| if .r < 0 then .n += 1 | .r -= $b else . end
| .ans = (.r|tostring) + .ans )
| .ans
end ;
def sigma(stream): reduce stream as $x (null; .+$x);
def invnegbase($b):
(explode | reverse | map(. - 48)) as $s | sigma( range(0; $s|length) | ($s[.] * pow($b; .)));
def testset: {
"11110": [10, -2], "21102": [146, -3], "195": [15, -10] };
def test: testset | to_entries[] | .value[0] as $n | .value[1] as $base | ($n|negbase($base)) as $neg | ($neg | invnegbase($base)) as $invneg | [.key == $neg, $n == $invneg]
test</lang>
- Output:
<lang jq>[true,true] [true,true] [true,true]</lang>
Julia
<lang julia> function negbase(n, b)
if n == 0 return "0" end
out = IOBuffer()
while n != 0
n, r = divrem(n, b)
if r < 0
n += 1
r -= b
end
print(out, r)
end
return reverse(String(out))
end
invnegbase(nst, b) = sum((ch - '0') * b ^ (i - 1) for (i, ch) in enumerate(reverse(nst)))
testset = Dict(
(10, -2) => "11110", (143, -3) => "21102", (15, -10) => "195")
for ((num, base), rst) in testset
encoded = negbase(num, base)
decoded = invnegbase(encoded, base)
println("\nencode $num in base $base:\n-> expected: $rst\n-> resulted: $encoded\n-> decoded: $decoded")
end</lang>
- Output:
encode 10 in base -2: -> expected: 11110 -> resulted: 11110 -> decoded: 10 encode 15 in base -10: -> expected: 195 -> resulted: 195 -> decoded: 15 encode 143 in base -3: -> expected: 21102 -> resulted: 21112 -> decoded: 143
Kotlin
<lang scala>// version 1.1.2
const val DIGITS = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz"
fun encodeNegBase(n: Long, b: Int): String {
require(b in -62 .. -1)
if (n == 0L) return "0"
val out = mutableListOf<Char>()
var nn = n
while (nn != 0L) {
var rem = (nn % b).toInt()
nn /= b
if (rem < 0) {
nn++
rem -= b
}
out.add(DIGITS[rem])
}
out.reverse()
return out.joinToString("")
}
fun decodeNegBase(ns: String, b: Int): Long {
require(b in -62 .. -1)
if (ns == "0") return 0
var total = 0L
var bb = 1L
for (c in ns.reversed()) {
total += DIGITS.indexOf(c) * bb
bb *= b
}
return total
}
fun main(args:Array<String>) {
val nbl = listOf(10L to -2, 146L to -3, 15L to -10, -17596769891 to -62)
for (p in nbl) {
val ns = encodeNegBase(p.first, p.second)
System.out.printf("%12d encoded in base %-3d = %s\n", p.first, p.second, ns)
val n = decodeNegBase(ns, p.second)
System.out.printf("%12s decoded in base %-3d = %d\n\n", ns, p.second, n)
}
}</lang>
- Output:
10 encoded in base -2 = 11110
11110 decoded in base -2 = 10
146 encoded in base -3 = 21102
21102 decoded in base -3 = 146
15 encoded in base -10 = 195
195 decoded in base -10 = 15
-17596769891 encoded in base -62 = Kotlin
Kotlin decoded in base -62 = -17596769891
Mathematica
<lang Mathematica>EncodeBase[number_,base_]:=Module[{
out={},
rem,n=number,b=base
},
While[
n!=0,
{n,rem}={Floor[Divide[n,b]],Mod[n,b]};
If[
rem<0,
n+=1;
rem-=b
];
PrependTo[out,rem]
];
out
]; DecodeBase[list_,base_]:=Total[list*base^Range[Length[list]-1,0,-1]];
Print[EncodeNegBase[10,-2],"=",DecodeBase[EncodeNegBase[10,-2],-2]]; Print[EncodeNegBase[146,-3],"=",DecodeBase[EncodeNegBase[146,-3],-3]]; Print[EncodeNegBase[15,-10],"=",DecodeBase[EncodeNegBase[15,-10],-10]];</lang>
{1,1,1,1,0}=10
{2,1,1,0,2}=146
{1,9,5}=15
Extra Credit: <lang Mathematica>DecodeBase[ToCharacterCode[$Version], -126](*ascii 1-byte encoding*) DecodeBase[ToCharacterCode[$Version], -(2^16 - 1)](*2-byte encoding*)</lang>
805433247971592164648901981307140864173502418954511864100981464890629926293823767730118860531000284192172723837 1402171866107096793294662824351970913227448502019658754262020315290937172496459102043149157175108006798078612200544768242812932737034448604720533372980987964929785521161173764118702296122915325508945150191589743829011670147956776269027485433441427722106
ooRexx
<lang oorexx>/* REXX ---------------------------------------------------------------
- Adapt for ooRexx (use of now invalid variable names)
- and make it work for base -63 (Algol example)
- --------------------------------------------------------------------*/
Numeric Digits 20 digits='0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz ' txt=' converted to base ' n= 10; b= -2; nb=nBase(n,b); Say right(n,20) txt right(b,3) '---->' nb ok() n= 146; b= -3; nb=nBase(n,b); Say right(n,20) txt right(b,3) '---->' nb ok() n= 15; b=-10; nb=nBase(n,b); Say right(n,20) txt right(b,3) '---->' nb ok() n= -15; b=-10; nb=nBase(n,b); Say right(n,20) txt right(b,3) '---->' nb ok() n= 0; b= -5; nb=nBase(n,b); Say right(n,20) txt right(b,3) '---->' nb ok() n=-6284695; b=-62; nb=nBase(n,b); Say right(n,20) txt right(b,3) '---->' nb ok() n=-36492107981104; b=-63;nb=nBase(n,b); Say right(n,20) txt right(b,3) '---->' nb ok() Exit
nBase: Procedure Expose digits /*---------------------------------------------------------------------
- convert x (base 10) to result (base r)
- --------------------------------------------------------------------*/
Parse arg x,r
result=
Do While x\==0 /*keep Processing while X isn't zero.*/
z=x//r
x=x%r /*calculate remainder; calculate int ÷.*/
If z<0 Then Do
z=z-r /*subtract a negative R from Z ?--+ */
x=x+1 /*bump X by one. ¦ */
End
result=substr(digits,z+1,1)result /*prepend the new digit */
End
If result= Then result=0
Return result
pBase: Procedure Expose digits; /*---------------------------------------------------------------------
- convert x (base r) to result (base 10)
- --------------------------------------------------------------------*/
Parse arg x,r; result=0; p=0 len=length(x) Do j=len by -1 For len /*Process each of the X's digits */ v=pos(substr(x,j,1),digits)-1 /*use digit's numeric value. */ result=result+v*r**p; /*add it to result */ p=p+1 /*bump power by 1 */ End Return result
ok: /*---------------------------------------------------------------------
- check back conversion
- --------------------------------------------------------------------*/
back=pBase(nb,b) If back\=n Then r='Error backward conversion results in' back Else r='ok' Return r</lang>
- Output:
10 converted to base -2 ----> 11110 ok
146 converted to base -3 ----> 21102 ok
15 converted to base -10 ----> 195 ok
-15 converted to base -10 ----> 25 ok
0 converted to base -5 ----> 0 ok
-6284695 converted to base -62 ----> Rexx ok
-36492107981104 converted to base -63 ----> Algol 68 ok
Perl 6
Perl 6 provides built-in methods / routines base and parse-base to convert to and from bases 2 through 36. We'll just shadow the core routines with versions that accept negative bases.
As a stretch goal, rather than implement something that can "Spell the name of the language with correct capitalization" (which is silly, trivial, and has absolutely nothing to do with negative base numbers,) we'll implement routines that correctly work with any Real number, not just integers.
The Real candidate has a 'precision' parameter, default -15, (15 places after the decimal point,) to limit the length of the fractional portion of the converted value. Change it if you need or want different precision. The Integer only routine is kept here as a multi-dispatch candidate since it is potentially much faster than the Real capable routine. The dispatcher will automatically use the most appropriate (fastest) routine.
Note that the parse-base routine will handle 'illegal' negative negative-base values without blowing up.
<lang perl6>multi sub base ( Int $value is copy, Int $radix where -37 < * < -1) {
my $result;
while $value {
my $r = $value mod $radix;
$value div= $radix;
if $r < 0 {
$value++;
$r -= $radix
}
$result ~= $r.base(-$radix);
}
flip $result || ~0;
}
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 parse-base (Str $str, Int $radix where -37 < * < -1) {
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) * $radix ** -($^k+1) } if $frac;
$fraction + [+] $whole.flip.comb.kv.map: { $^v.parse-base(-$radix) * $radix ** $^k }
}
- TESTING
for <4 -4 0 -7 10 -2 146 -3 15 -10 -19 -10 107 -16
227.65625 -16 2.375 -4 -1.3e2 -8 41371457.268272761 -36> -> $v, $r {
my $nbase = $v.&base($r, :precision(-5));
printf "%20s.&base\(%3d\) = %-11s : %13s.&parse-base\(%3d\) = %s\n",
$v, $r, $nbase, "'$nbase'", $r, $nbase.&parse-base($r);
}
- 'Illegal' negative-base value
say q| '-21'.&parse-base(-10) = |, '-21'.&parse-base(-10);</lang>
- Output:
4.&base( -4) = 130 : '130'.&parse-base( -4) = 4
0.&base( -7) = 0 : '0'.&parse-base( -7) = 0
10.&base( -2) = 11110 : '11110'.&parse-base( -2) = 10
146.&base( -3) = 21102 : '21102'.&parse-base( -3) = 146
15.&base(-10) = 195 : '195'.&parse-base(-10) = 15
-19.&base(-10) = 21 : '21'.&parse-base(-10) = -19
107.&base(-16) = 1AB : '1AB'.&parse-base(-16) = 107
227.65625.&base(-16) = 124.68 : '124.68'.&parse-base(-16) = 227.65625
2.375.&base( -4) = 3.32 : '3.32'.&parse-base( -4) = 2.375
-1.3e2.&base( -8) = 1616 : '1616'.&parse-base( -8) = -130
41371457.268272761.&base(-36) = PERL6.ROCKS : 'PERL6.ROCKS'.&parse-base(-36) = 41371457.268272761
'-21'.&parse-base(-10) = 19
Python
<lang python>#!/bin/python from __future__ import print_function
def EncodeNegBase(n, b): #Converts from decimal if n == 0: return "0" out = [] while n != 0: n, rem = divmod(n, b) if rem < 0: n += 1 rem -= b out.append(rem) return "".join(map(str, out[::-1]))
def DecodeNegBase(nstr, b): #Converts to decimal if nstr == "0": return 0
total = 0 for i, ch in enumerate(nstr[::-1]): total += int(ch) * b**i return total
if __name__=="__main__":
print ("Encode 10 as negabinary (expect 11110)") result = EncodeNegBase(10, -2) print (result) if DecodeNegBase(result, -2) == 10: print ("Converted back to decimal") else: print ("Error converting back to decimal")
print ("Encode 146 as negaternary (expect 21102)") result = EncodeNegBase(146, -3) print (result) if DecodeNegBase(result, -3) == 146: print ("Converted back to decimal") else: print ("Error converting back to decimal")
print ("Encode 15 as negadecimal (expect 195)") result = EncodeNegBase(15, -10) print (result) if DecodeNegBase(result, -10) == 15: print ("Converted back to decimal") else: print ("Error converting back to decimal")</lang>
- Output:
Encode 10 as negabinary (expect 11110) 11110 Converted back to decimal Encode 146 as negaternary (expect 21102) 21102 Converted back to decimal Encode 15 as negadecimal (expect 195) 195 Converted back to decimal
Racket
<lang racket>#lang racket
(define all-digits (string->list "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz-"))
(define d->i-map (for/hash ((i (in-naturals)) (d all-digits)) (values d i)))
(define max-base (length all-digits))
(define (q/r n d)
(let-values (((q r) (quotient/remainder n d)))
(if (negative? r)
(values (+ q 1) (- r d))
(values q r))))
(define (negabase-convertors base)
(when (not (integer? base)) (raise "Non-integer base."))
(when (not (<= 2 (abs base) max-base)) (raise (format "(abs base) must be inside [2 ~a] interval." max-base)))
(values
(let ((q/r_base (curryr q/r base)))
(λ (num)
(define (checked->base num dig)
(match num
[0 (apply string dig)]
[(app q/r_base num/ rst) (checked->base num/ (cons (list-ref all-digits rst) dig))]))
(if (integer? num)
(checked->base num (if (zero? num) '(#\0) '()))
(raise "Non-integer number."))))
(λ (dig)
(define (loop digs acc)
(match digs [(list) acc] [(cons a d) (loop d (+ (* acc base) (hash-ref d->i-map a)))]))
(loop (string->list dig) 0))))
(define-values (->negabinary negabinary->) (negabase-convertors -2))
(define-values (->negaternary negaternary->) (negabase-convertors -3))
(define-values (->negadecimal negadecimal->) (negabase-convertors -10))
(define-values (->nega63ary nega63ary->) (negabase-convertors (- max-base)))
(module+ main
(->negaternary 146) (->negabinary 10) (->negadecimal 15) (->nega63ary -26238001742))
(module+ test
(require rackunit)
;; tests from wikipedia page (check-equal? (call-with-values (λ () (q/r 146 -3)) cons) '(-48 . 2)) (check-equal? (call-with-values (λ () (q/r -48 -3)) cons) '(16 . 0)) (check-equal? (call-with-values (λ () (q/r 16 -3)) cons) '(-5 . 1)) (check-equal? (call-with-values (λ () (q/r -5 -3)) cons) '(2 . 1)) (check-equal? (call-with-values (λ () (q/r 2 -3)) cons) '(0 . 2))
(define-values (->hexadecimal hexadecimal->) (negabase-convertors 16)) (check-equal? (->hexadecimal 31) "1F"))</lang>
- Output:
"21102" "11110" "195" "Racket"
REXX
Both REXX versions use a type of assert (a function call of OK) that converts the numbers in the
negative base back to the original number in base ten (and issues an error message if not correct).
version 1 (up to base -10)
<lang rexx>/*REXX pgm converts & displays a base ten integer to a negative base number (up to -10).*/ @=' converted to base '; numeric digits 300 /*be able to handle ginormous numbers. */ n= 10; b= -2; nb=nBase(n,b); say right(n, 20) @ right(b, 3) '────►' nb ok() n=146; b= -3; nb=nBase(n,b); say right(n, 20) @ right(b, 3) '────►' nb ok() n= 15; b=-10; nb=nBase(n,b); say right(n, 20) @ right(b, 3) '────►' nb ok() n=-15; b=-10; nb=nBase(n,b); say right(n, 20) @ right(b, 3) '────►' nb ok() n= 0; b= -5; nb=nBase(n,b); say right(n, 20) @ right(b, 3) '────►' nb ok() exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ nBase: procedure; parse arg x,r; $= /*obtain args; $ is the integer result.*/
if r<-10 | r>-2 then do; say 'base' r "must be in range: -2 ───► -10"; exit 13; end
do while x\==0 /*keep processing while X isn't zero.*/
z=x//r; x=x%r /*calculate remainder; calculate int ÷.*/
if z<0 then do; z=z-r /*subtract a negative R from Z ◄──┐ */
x=x+1 /*bump X by one. │ */
end /* Funny "add" ►───┘ */
$=z || $ /*prepend new digit (numeral) to result*/
end /*while*/
return $
/*──────────────────────────────────────────────────────────────────────────────────────*/ ok: ?=; if pBase(nb, b)\=n then ?=' ◄──error in negative base calculation'; return ? /*──────────────────────────────────────────────────────────────────────────────────────*/ pBase: procedure; parse arg x,r; p=0; $=0 /*obtain args; $ is the integer result.*/
if r<-10 | r>-2 then do; say 'base' r "must be in range: -2 ───► -10"; exit 13; end
L=length(x); do j=L by -1 for L /*process each of the numerals in X. */
$=$ + substr(x,j,1)*r**p /*add value of a numeral to $ (result).*/
p=p+1 /*bump the power by 1. */
end /*j*/ /* [↓] process the number "bottom-up".*/
return $</lang>
- output when using the default inputs:
10 converted to base -2 ────► 11110
146 converted to base -3 ────► 21102
15 converted to base -10 ────► 195
-15 converted to base -10 ────► 25
0 converted to base -5 ────► 0
version 2 (up to base -71)
This REXX version supports up to negative base ─71, but it may not be compatible to other programming examples
because the symbols (glyphs or numerals) used herein may not be in the same exact order. The symbols represented
in this REXX program should be able to represent any programming language used on Rosetta Code.
<lang rexx>/*REXX pgm converts & displays a base ten integer to a negative base number (up to -71).*/
@=' converted to base '; numeric digits 300 /*be able to handle ginormous numbers. */
n= 10; b= -2; nb=nBase(n,b); say right(n,20) @ right(b,3) '────►' nb ok()
n= 146; b= -3; nb=nBase(n,b); say right(n,20) @ right(b,3) '────►' nb ok()
n= 15; b=-10; nb=nBase(n,b); say right(n,20) @ right(b,3) '────►' nb ok()
n= -15; b=-10; nb=nBase(n,b); say right(n,20) @ right(b,3) '────►' nb ok()
n= 0; b= -5; nb=nBase(n,b); say right(n,20) @ right(b,3) '────►' nb ok()
n=-6284695; b=-62; nb=nBase(n,b); say right(n,20) @ right(b,3) '────►' nb ok()
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
_Base: !='0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz /*+-!éáµ' /*sym*/
parse arg $; m=length(!); L=length(x); p=0
if r<-m | r>-2 then do; say 'base' r "must be in range: -2 ───► -"m; exit 13; end
return
/*──────────────────────────────────────────────────────────────────────────────────────*/ nBase: procedure; parse arg x,r; call _Base /*get args; $ will be integer result. */
do while x\==0 /*keep processing while X isn't zero.*/
z=x//r; x= x%r /*calculate remainder; calculate int ÷.*/
if z<0 then do; z= z-r /*subtract a negative R from Z ◄──┐ */
x= x+1 /*bump X by one. │ */
end /* Funny "add" ►───┘ */
$=substr(!, z+1, 1)$ /*prepend the new numeral to the result*/
end /*while*/
return $
/*──────────────────────────────────────────────────────────────────────────────────────*/ ok: if pBase(nb,b)\=n then return ' ◄──error in negative base calculation'; return /*──────────────────────────────────────────────────────────────────────────────────────*/ pBase: procedure; parse arg x,r; call _Base 0 /*get args; $ will be integer result. */
do j=L by -1 for L /*process each of the numerals in X. */
v=pos( substr(x,j,1), !) - 1 /*use base R for the numeral's value.*/
$=$ + v * r**p; p= p+1 /*add it to $ (result); bump power by 1*/
end /*j*/ /* [↑] process the number "bottom-up".*/
return $</lang>
- output when using the default inputs:
10 converted to base -2 ────► 11110
146 converted to base -3 ────► 21102
15 converted to base -10 ────► 195
-15 converted to base -10 ────► 25
0 converted to base -5 ────► 0
-6284695 converted to base -62 ────► Rexx
Ring
<lang ring>
- Project : Negative base numbers
negs = [[146,-3],[21102,-3],[10,-2],[11110,-2],[15,-10],[195,-10]] for n = 1 to len(negs) step 2
enc = encodeNegBase(negs[n][1],negs[n][2])
encstr = showarray(enc)
dec = decodeNegBase(negs[n+1][1],negs[n+1][2])
see "" + negs[n][1] + " encoded in base " + negs[n][2] + " = " + encstr + nl
see "" + negs[n+1][1] + " decoded in base " + negs[n+1][2] + " = " + dec + nl
next
func encodeNegBase(n,b)
out = []
while n != 0
rem = (n%b)
if rem < 0
rem = rem - b
ok
n = ceil(n/b)
rem = fabs(rem)
add(out,rem)
end
out = reverse(out)
return out
func decodeNegBase(n,b)
out = 0
n = string(n)
for nr = len(n) to 1 step -1
out = out + n[nr]*pow(b,len(n)-nr)
next
return out
func showarray(vect)
svect = ""
for n = 1 to len(vect)
svect = svect + vect[n]
next
return svect
</lang> Output:
146 encoded in base -3 = 21102 21102 decoded in base -3 = 146 10 encoded in base -2 = 11110 11110 decoded in base -2 = 10 15 encoded in base -10 = 195 195 decoded in base -10 = 15
Scala
<lang scala>object NegativeBase {
val digits = ('0' to '9') ++ ('a' to 'z') ++ ('A' to 'Z')
def intToStr(n: Int, b: Int): String = {
def _fromInt(n: Int): List[Int] = {
if (n == 0) {
Nil
} else {
val r = n % b
val rp = if (r < 0) r + b else r
val m = -(n - rp)/b
rp :: _fromInt(m)
}
}
_fromInt(n).map(digits).reverse.mkString
}
def strToInt(s: String, b: Int): Int = {
s.map(digits.indexOf).foldRight((0, 1)){ case (x, (sum, pow)) =>
(sum + x * pow, pow * -b)
}._1
}
}</lang>
<lang scala>def testConversion(b: Int)(n: Int, s: String): Unit = {
println(s"$n in base -$b = ${NegativeBase.intToStr(n, b)}")
println(s"$s from base -$b = ${NegativeBase.strToInt(s, b)}")
}
testConversion(2)(10, "11110") testConversion(3)(146, "21102") testConversion(10)(15, "195") testConversion(62)(795099356, "Scala")</lang>
- Output:
10 in base -2 = 11110 11110 from base -2 = 10 146 in base -3 = 21102 21102 from base -3 = 146 15 in base -10 = 195 195 from base -10 = 15 795099356 in base -62 = Scala Scala from base -62 = 795099356
Sidef
<lang ruby>func EncodeNegBase(Num n, Num b { .~~ (-36 .. -2) }) {
var out = []
var r = 0
while (n) {
(n, r) = divmod(n, b)
if (r < 0) {
n += 1
r -= b
}
out << r.base(-b)
}
return (out.join.flip || "0")
}
func DecodeNegBase(Str s, Num b { .~~ (-36 .. -2) }) {
var total = 0
for i,c in (s.flip.chars.kv) {
total += (Num(c, -b) * b**i)
}
return total
}
say (" 10 in base -2: ", EncodeNegBase(10, -2)) say ("146 in base -3: ", EncodeNegBase(146, -3)) say (" 15 in base -10: ", EncodeNegBase(15, -10))
say '-'*25
say ("11110 from base -2: ", DecodeNegBase("11110", -2)) say ("21102 from base -3: ", DecodeNegBase("21102", -3)) say (" 195 from base -10: ", DecodeNegBase("195", -10))
say '-'*25
- Extra
say ("25334424 in base -31: ", EncodeNegBase(25334424, -31)) say ("sidef from base -31: ", DecodeNegBase("sidef", -31))</lang>
- Output:
10 in base -2: 11110 146 in base -3: 21102 15 in base -10: 195 ------------------------- 11110 from base -2: 10 21102 from base -3: 146 195 from base -10: 15 ------------------------- 25334424 in base -31: sidef sidef from base -31: 25334424
zkl
<lang zkl>fcn toNBase(n,radix){
var [const] cs=[0..9].chain(["a".."z"]).pump(String); //"0123456789abcd..z"
_assert_(-37 < radix < -1,"invalid radix");
digits:=List();
while(n){ reg r;
n,r=n.divr(radix); // C compiler semantics
if(r<0){ n+=1; r-=radix; }
digits.append(r);
}
digits.reverse().pump(String,cs.get);
}
fcn toInt(str,radix){ // the toInt(radix) method radix is 2..36
str.reduce('wrap(s,d,rdx){ s*radix + d.toInt(rdx); },0,radix.abs());
}</lang> <lang zkl>ns:=T( T(10,-2), T(146,-3), T(15,-10), T(107,-16), T(41371458,-36), T(44661,-36) ); results:=ns.pump(List,Void.Xplode,toNBase); foreach nb,r in (ns.zip(results)){
_,b:=nb;
println("%10d.base(%3d) = \"%s\" --> %d".fmt(nb.xplode(),r,toInt(r,b)));
}</lang>
- Output:
10.base( -2) = "11110" --> 10
146.base( -3) = "21102" --> 146
15.base(-10) = "195" --> 15
107.base(-16) = "1ab" --> 107
41371458.base(-36) = "perl6" --> 41371458
44661.base(-36) = "zkl" --> 44661
References
- ↑ Negabinary on Wolfram Mathworld
- ↑ Negative base on Wikipedia