# Zeckendorf arithmetic

(Redirected from Zeckendorf arithmatic)
Zeckendorf arithmetic
You are encouraged to solve this task according to the task description, using any language you may know.

This task is a total immersion zeckendorf task; using decimal numbers will attract serious disapprobation.

The task is to implement addition, subtraction, multiplication, and division using Zeckendorf number representation. Optionally provide decrement, increment and comparitive operation functions.

Addition

Like binary 1 + 1 = 10, note carry 1 left. There the similarity ends. 10 + 10 = 101, note carry 1 left and 1 right. 100 + 100 = 1001, note carry 1 left and 2 right, this is the general case.

Occurrences of 11 must be changed to 100. Occurrences of 111 may be changed from the right by replacing 11 with 100, or from the left converting 111 to 100 + 100;

Subtraction

10 - 1 = 1. The general rule is borrow 1 right carry 1 left. eg:

```  abcde
10100 -
1000
_____
100  borrow 1 from a leaves 100
+ 100  add the carry
_____
1001
```

A larger example:

```  abcdef
100100 -
1000
______
1*0100 borrow 1 from b
+ 100 add the carry
______
1*1001

Sadly we borrowed 1 from b which didn't have it to lend. So now b borrows from a:

1001
+ 1000 add the carry
____
10100
```
Multiplication

Here you teach your computer its zeckendorf tables. eg. 101 * 1001:

```  a = 1 * 101 = 101
b = 10 * 101 = a + a = 10000
c = 100 * 101 = b + a = 10101
d = 1000 * 101 = c + b = 101010

1001 = d + a therefore 101 * 1001 =

101010
+ 101
______
1000100
```
Division

Lets try 1000101 divided by 101, so we can use the same table used for multiplication.

```  1000101 -
101010 subtract d (1000 * 101)
_______
1000 -
101 b and c are too large to subtract, so subtract a
____
1 so 1000101 divided by 101 is d + a (1001) remainder 1
```

Efficient algorithms for Zeckendorf arithmetic is interesting. The sections on addition and subtraction are particularly relevant for this task.

## 11l

Translation of: Python
```T Zeckendorf
Int dLen
dVal = 0

F (x = ‘0’)
V q = 1
V i = x.len - 1
.dLen = i I/ 2
L i >= 0
.dVal = .dVal + (x[i].code - ‘0’.code) * q
q = q * 2
i = i - 1

F a(n)
V i = n
L
I .dLen < i
.dLen = i
V j = (.dVal >> (i * 2)) [&] 3
I j == 0 | j == 1
R
I j == 2
I (.dVal >> ((i + 1) * 2) [&] 1) != 1
R
.dVal = .dVal + (1 << (i * 2 + 1))
R
I j == 3
V temp = 3 << (i * 2)
temp = temp (+) -1
.dVal = .dVal [&] temp
.b((i + 1) * 2)
i = i + 1

F b(pos)
I pos == 0
.inc()
R
I (.dVal >> pos) [&] 1 == 0
.dVal = .dVal + (1 << pos)
.a(Int(pos / 2))
I pos > 1
.a(Int(pos / 2) - 1)
E
V temp = 1 << pos
temp = temp (+) -1
.dVal = .dVal [&] temp
.b(pos + 1)
.b(pos - (I pos > 1 {2} E 1))

F c(pos)
I (.dVal >> pos) [&] 1 == 1
V temp = 1 << pos
temp = temp (+) -1
.dVal = .dVal [&] temp
R
.c(pos + 1)
I pos > 0
.b(pos - 1)
E
.inc()

F inc() -> N
.dVal = .dVal + 1
.a(0)

F +(rhs)
V copy = (.)
V rhs_dVal = rhs.dVal
V limit = (rhs.dLen + 1) * 2
L(gn) 0 .< limit
I ((rhs_dVal >> gn) [&] 1) == 1
copy.b(gn)
R copy

F -(rhs)
V copy = (.)
V rhs_dVal = rhs.dVal
V limit = (rhs.dLen + 1) * 2
L(gn) 0 .< limit
I (rhs_dVal >> gn) [&] 1 == 1
copy.c(gn)
L (((copy.dVal >> ((copy.dLen * 2) [&] 31)) [&] 3) == 0) | (copy.dLen == 0)
copy.dLen = copy.dLen - 1
R copy

F *(rhs)
V na = copy(rhs)
V nb = copy(rhs)
V nr = Zeckendorf()
V dVal = .dVal
L(i) 0 .< (.dLen + 1) * 2
I ((dVal >> i) [&] 1) > 0
nr = nr + nb
V nt = copy(nb)
nb = nb + na
na = copy(nt)
R nr

F String()
V dig = [‘00’, ‘01’, ‘10’]
V dig1 = [‘’, ‘1’, ‘10’]

I .dVal == 0
R ‘0’
V idx = (.dVal >> ((.dLen * 2) [&] 31)) [&] 3
String sb = dig1[idx]
V i = .dLen - 1
L i >= 0
idx = (.dVal >> (i * 2)) [&] 3
sb ‘’= dig[idx]
i = i - 1
R sb

print(‘Addition:’)
V g = Zeckendorf(‘10’)
g = g + Zeckendorf(‘10’)
print(g)
g = g + Zeckendorf(‘10’)
print(g)
g = g + Zeckendorf(‘1001’)
print(g)
g = g + Zeckendorf(‘1000’)
print(g)
g = g + Zeckendorf(‘10101’)
print(g)
print()

print(‘Subtraction:’)
g = Zeckendorf(‘1000’)
g = g - Zeckendorf(‘101’)
print(g)
g = Zeckendorf(‘10101010’)
g = g - Zeckendorf(‘1010101’)
print(g)
print()

print(‘Multiplication:’)
g = Zeckendorf(‘1001’)
g = g * Zeckendorf(‘101’)
print(g)
g = Zeckendorf(‘101010’)
g = g + Zeckendorf(‘101’)
print(g)```
Output:
```Addition:
101
1001
10101
100101
1010000

Subtraction:
1
1000000

Multiplication:
1000100
1000100
```

## C

Translation of: D
```#include <stdbool.h>
#include <stdio.h>
#include <string.h>

int inv(int a) {
return a ^ -1;
}

struct Zeckendorf {
int dVal, dLen;
};

void a(struct Zeckendorf *self, int n) {
void b(struct Zeckendorf *, int); // forward declare

int i = n;
while (true) {
if (self->dLen < i) self->dLen = i;
int j = (self->dVal >> (i * 2)) & 3;
switch (j) {
case 0:
case 1:
return;
case 2:
if (((self->dVal >> ((i + 1) * 2)) & 1) != 1) return;
self->dVal += 1 << (i * 2 + 1);
return;
case 3:
self->dVal = self->dVal & inv(3 << (i * 2));
b(self, (i + 1) * 2);
break;
default:
break;
}
i++;
}
}

void b(struct Zeckendorf *self, int pos) {
void increment(struct Zeckendorf *); // forward declare

if (pos == 0) {
increment(self);
return;
}
if (((self->dVal >> pos) & 1) == 0) {
self->dVal += 1 << pos;
a(self, pos / 2);
if (pos > 1) a(self, pos / 2 - 1);
} else {
self->dVal = self->dVal & inv(1 << pos);
b(self, pos + 1);
b(self, pos - (pos > 1 ? 2 : 1));
}
}

void c(struct Zeckendorf *self, int pos) {
if (((self->dVal >> pos) & 1) == 1) {
self->dVal = self->dVal & inv(1 << pos);
return;
}
c(self, pos + 1);
if (pos > 0) {
b(self, pos - 1);
} else {
increment(self);
}
}

struct Zeckendorf makeZeckendorf(char *x) {
struct Zeckendorf z = { 0, 0 };
int i = strlen(x) - 1;
int q = 1;

z.dLen = i / 2;
while (i >= 0) {
z.dVal += (x[i] - '0') * q;
q *= 2;
i--;
}

return z;
}

void increment(struct Zeckendorf *self) {
self->dVal++;
a(self, 0);
}

void addAssign(struct Zeckendorf *self, struct Zeckendorf rhs) {
int gn;
for (gn = 0; gn < (rhs.dLen + 1) * 2; gn++) {
if (((rhs.dVal >> gn) & 1) == 1) {
b(self, gn);
}
}
}

void subAssign(struct Zeckendorf *self, struct Zeckendorf rhs) {
int gn;
for (gn = 0; gn < (rhs.dLen + 1) * 2; gn++) {
if (((rhs.dVal >> gn) & 1) == 1) {
c(self, gn);
}
}
while ((((self->dVal >> self->dLen * 2) & 3) == 0) || (self->dLen == 0)) {
self->dLen--;
}
}

void mulAssign(struct Zeckendorf *self, struct Zeckendorf rhs) {
struct Zeckendorf na = rhs;
struct Zeckendorf nb = rhs;
struct Zeckendorf nr = makeZeckendorf("0");
struct Zeckendorf nt;
int i;

for (i = 0; i < (self->dLen + 1) * 2; i++) {
if (((self->dVal >> i) & 1) > 0) addAssign(&nr, nb);
nt = nb;
addAssign(&nb, na);
na = nt;
}

*self = nr;
}

void printZeckendorf(struct Zeckendorf z) {
static const char *const dig[3] = { "00", "01", "10" };
static const char *const dig1[3] = { "", "1", "10" };

if (z.dVal == 0) {
printf("0");
return;
} else {
int idx = (z.dVal >> (z.dLen * 2)) & 3;
int i;

printf(dig1[idx]);
for (i = z.dLen - 1; i >= 0; i--) {
idx = (z.dVal >> (i * 2)) & 3;
printf(dig[idx]);
}
}
}

int main() {
struct Zeckendorf g;

printf("Addition:\n");
g = makeZeckendorf("10");
addAssign(&g, makeZeckendorf("10"));
printZeckendorf(g);
printf("\n");
addAssign(&g, makeZeckendorf("10"));
printZeckendorf(g);
printf("\n");
addAssign(&g, makeZeckendorf("1001"));
printZeckendorf(g);
printf("\n");
addAssign(&g, makeZeckendorf("1000"));
printZeckendorf(g);
printf("\n");
addAssign(&g, makeZeckendorf("10101"));
printZeckendorf(g);
printf("\n\n");

printf("Subtraction:\n");
g = makeZeckendorf("1000");
subAssign(&g, makeZeckendorf("101"));
printZeckendorf(g);
printf("\n");
g = makeZeckendorf("10101010");
subAssign(&g, makeZeckendorf("1010101"));
printZeckendorf(g);
printf("\n\n");

printf("Multiplication:\n");
g = makeZeckendorf("1001");
mulAssign(&g, makeZeckendorf("101"));
printZeckendorf(g);
printf("\n");
g = makeZeckendorf("101010");
addAssign(&g, makeZeckendorf("101"));
printZeckendorf(g);
printf("\n");

return 0;
}
```
Output:
```Addition:
101
1001
10101
100101
1010000

Subtraction:
1
1000000

Multiplication:
1000100
1000100```

## C#

Translation of: Java
```using System;
using System.Text;

namespace ZeckendorfArithmetic {
class Zeckendorf : IComparable<Zeckendorf> {
private static readonly string[] dig = { "00", "01", "10" };
private static readonly string[] dig1 = { "", "1", "10" };

private int dVal = 0;
private int dLen = 0;

public Zeckendorf() : this("0") {
// empty
}

public Zeckendorf(string x) {
int q = 1;
int i = x.Length - 1;
dLen = i / 2;
while (i >= 0) {
dVal += (x[i] - '0') * q;
q *= 2;
i--;
}
}

private void A(int n) {
int i = n;
while (true) {
if (dLen < i) dLen = i;
int j = (dVal >> (i * 2)) & 3;
switch (j) {
case 0:
case 1:
return;
case 2:
if (((dVal >> ((i + 1) * 2)) & 1) != 1) return;
dVal += 1 << (i * 2 + 1);
return;
case 3:
int temp = 3 << (i * 2);
temp ^= -1;
dVal = dVal & temp;
B((i + 1) * 2);
break;
}
i++;
}
}

private void B(int pos) {
if (pos == 0) {
Inc();
return;
}
if (((dVal >> pos) & 1) == 0) {
dVal += 1 << pos;
A(pos / 2);
if (pos > 1) A(pos / 2 - 1);
}
else {
int temp = 1 << pos;
temp ^= -1;
dVal = dVal & temp;
B(pos + 1);
B(pos - (pos > 1 ? 2 : 1));
}
}

private void C(int pos) {
if (((dVal >> pos) & 1) == 1) {
int temp = 1 << pos;
temp ^= -1;
dVal = dVal & temp;
return;
}
C(pos + 1);
if (pos > 0) {
B(pos - 1);
}
else {
Inc();
}
}

public Zeckendorf Inc() {
dVal++;
A(0);
return this;
}

public Zeckendorf Copy() {
Zeckendorf z = new Zeckendorf {
dVal = dVal,
dLen = dLen
};
return z;
}

public void PlusAssign(Zeckendorf other) {
for (int gn = 0; gn < (other.dLen + 1) * 2; gn++) {
if (((other.dVal >> gn) & 1) == 1) {
B(gn);
}
}
}

public void MinusAssign(Zeckendorf other) {
for (int gn = 0; gn < (other.dLen + 1) * 2; gn++) {
if (((other.dVal >> gn) & 1) == 1) {
C(gn);
}
}
while ((((dVal >> dLen * 2) & 3) == 0) || (dLen == 0)) {
dLen--;
}
}

public void TimesAssign(Zeckendorf other) {
Zeckendorf na = other.Copy();
Zeckendorf nb = other.Copy();
Zeckendorf nt;
Zeckendorf nr = new Zeckendorf();
for (int i = 0; i < (dLen + 1) * 2; i++) {
if (((dVal >> i) & 1) > 0) {
nr.PlusAssign(nb);
}
nt = nb.Copy();
nb.PlusAssign(na);
na = nt.Copy();
}
dVal = nr.dVal;
dLen = nr.dLen;
}

public int CompareTo(Zeckendorf other) {
return dVal.CompareTo(other.dVal);
}

public override string ToString() {
if (dVal == 0) {
return "0";
}

int idx = (dVal >> (dLen * 2)) & 3;
StringBuilder sb = new StringBuilder(dig1[idx]);
for (int i = dLen - 1; i >= 0; i--) {
idx = (dVal >> (i * 2)) & 3;
sb.Append(dig[idx]);
}
return sb.ToString();
}
}

class Program {
static void Main(string[] args) {
Console.WriteLine("Addition:");
Zeckendorf g = new Zeckendorf("10");
g.PlusAssign(new Zeckendorf("10"));
Console.WriteLine(g);
g.PlusAssign(new Zeckendorf("10"));
Console.WriteLine(g);
g.PlusAssign(new Zeckendorf("1001"));
Console.WriteLine(g);
g.PlusAssign(new Zeckendorf("1000"));
Console.WriteLine(g);
g.PlusAssign(new Zeckendorf("10101"));
Console.WriteLine(g);
Console.WriteLine();

Console.WriteLine("Subtraction:");
g = new Zeckendorf("1000");
g.MinusAssign(new Zeckendorf("101"));
Console.WriteLine(g);
g = new Zeckendorf("10101010");
g.MinusAssign(new Zeckendorf("1010101"));
Console.WriteLine(g);
Console.WriteLine();

Console.WriteLine("Multiplication:");
g = new Zeckendorf("1001");
g.TimesAssign(new Zeckendorf("101"));
Console.WriteLine(g);
g = new Zeckendorf("101010");
g.PlusAssign(new Zeckendorf("101"));
Console.WriteLine(g);
}
}
}
```
Output:
```Addition:
101
1001
10101
100101
1010000

Subtraction:
1
1000000

Multiplication:
1000100
1000100```

## C++

Works with: C++11
```// For a class N which implements Zeckendorf numbers:
// I define an increment operation ++()
// I define a comparison operation <=(other N)
// I define an addition operation +=(other N)
// I define a subtraction operation -=(other N)
// Nigel Galloway October 28th., 2012
#include <iostream>
enum class zd {N00,N01,N10,N11};
class N {
private:
int dVal = 0, dLen;
void _a(int i) {
for (;; i++) {
if (dLen < i) dLen = i;
switch ((zd)((dVal >> (i*2)) & 3)) {
case zd::N00: case zd::N01: return;
case zd::N10: if (((dVal >> ((i+1)*2)) & 1) != 1) return;
dVal += (1 << (i*2+1)); return;
case zd::N11: dVal &= ~(3 << (i*2)); _b((i+1)*2);
}}}
void _b(int pos) {
if (pos == 0) {++*this; return;}
if (((dVal >> pos) & 1) == 0) {
dVal += 1 << pos;
_a(pos/2);
if (pos > 1) _a((pos/2)-1);
} else {
dVal &= ~(1 << pos);
_b(pos + 1);
_b(pos - ((pos > 1)? 2:1));
}}
void _c(int pos) {
if (((dVal >> pos) & 1) == 1) {dVal &= ~(1 << pos); return;}
_c(pos + 1);
if (pos > 0) _b(pos - 1); else ++*this;
return;
}
public:
N(char const* x = "0") {
int i = 0, q = 1;
for (; x[i] > 0; i++);
for (dLen = --i/2; i >= 0; i--) {dVal+=(x[i]-48)*q; q*=2;
}}
const N& operator++() {dVal += 1; _a(0); return *this;}
const N& operator+=(const N& other) {
for (int GN = 0; GN < (other.dLen + 1) * 2; GN++) if ((other.dVal >> GN) & 1 == 1) _b(GN);
return *this;
}
const N& operator-=(const N& other) {
for (int GN = 0; GN < (other.dLen + 1) * 2; GN++) if ((other.dVal >> GN) & 1 == 1) _c(GN);
for (;((dVal >> dLen*2) & 3) == 0 or dLen == 0; dLen--);
return *this;
}
const N& operator*=(const N& other) {
N Na = other, Nb = other, Nt, Nr;
for (int i = 0; i <= (dLen + 1) * 2; i++) {
if (((dVal >> i) & 1) > 0) Nr += Nb;
Nt = Nb; Nb += Na; Na = Nt;
}
return *this = Nr;
}
const bool operator<=(const N& other) const {return dVal <= other.dVal;}
friend std::ostream& operator<<(std::ostream&, const N&);
};
N operator "" N(char const* x) {return N(x);}
std::ostream &operator<<(std::ostream &os, const N &G) {
const static std::string dig[] {"00","01","10"}, dig1[] {"","1","10"};
if (G.dVal == 0) return os << "0";
os << dig1[(G.dVal >> (G.dLen*2)) & 3];
for (int i = G.dLen-1; i >= 0; i--) os << dig[(G.dVal >> (i*2)) & 3];
return os;
}
```

### Testing

The following tests addtition:

```int main(void) {
N G;
G = 10N;
G += 10N;
std::cout << G << std::endl;
G += 10N;
std::cout << G << std::endl;
G += 1001N;
std::cout << G << std::endl;
G += 1000N;
std::cout << G << std::endl;
G += 10101N;
std::cout << G << std::endl;
return 0;
}
```
Output:
```101
1001
10101
100101
1010000
```

The following tests subtraction:

```int main(void) {
N G;
G = 1000N;
G -= 101N;
std::cout << G << std::endl;
G = 10101010N;
G -= 1010101N;
std::cout << G << std::endl;
return 0;
}
```
Output:
```1
1000000
```

The following tests multiplication:

```int main(void) {
N G = 1001N;
G *= 101N;
std::cout << G << std::endl;

G = 101010N;
G += 101N;
std::cout << G << std::endl;
return 0;
}
```
Output:
```1000100
1000100
```

## D

Translation of: Kotlin
```import std.stdio;

int inv(int a) {
return a ^ -1;
}

class Zeckendorf {
private int dVal;
private int dLen;

private void a(int n) {
auto i = n;
while (true) {
if (dLen < i) dLen = i;
auto j = (dVal >> (i * 2)) & 3;
switch(j) {
case 0:
case 1:
return;
case 2:
if (((dVal >> ((i + 1) * 2)) & 1) != 1) return;
dVal += 1 << (i * 2 + 1);
return;
case 3:
dVal = dVal & (3 << (i * 2)).inv();
b((i + 1) * 2);
break;
default:
assert(false);
}
i++;
}
}

private void b(int pos) {
if (pos == 0) {
this++;
return;
}
if (((dVal >> pos) & 1) == 0) {
dVal += 1 << pos;
a(pos / 2);
if (pos > 1) a(pos / 2 - 1);
} else {
dVal = dVal & (1 << pos).inv();
b(pos + 1);
b(pos - (pos > 1 ? 2 : 1));
}
}

private void c(int pos) {
if (((dVal >> pos) & 1) == 1) {
dVal = dVal & (1 << pos).inv();
return;
}
c(pos + 1);
if (pos > 0) {
b(pos - 1);
} else {
++this;
}
}

this(string x = "0") {
int q = 1;
int i = x.length - 1;
dLen = i / 2;
while (i >= 0) {
dVal += (x[i] - '0') * q;
q *= 2;
i--;
}
}

auto opUnary(string op : "++")() {
dVal += 1;
a(0);
return this;
}

void opOpAssign(string op : "+")(Zeckendorf rhs) {
foreach (gn; 0..(rhs.dLen + 1) * 2) {
if (((rhs.dVal >> gn) & 1) == 1) {
b(gn);
}
}
}

void opOpAssign(string op : "-")(Zeckendorf rhs) {
foreach (gn; 0..(rhs.dLen + 1) * 2) {
if (((rhs.dVal >> gn) & 1) == 1) {
c(gn);
}
}
while ((((dVal >> dLen * 2) & 3) == 0) || (dLen == 0)) {
dLen--;
}
}

void opOpAssign(string op : "*")(Zeckendorf rhs) {
auto na = rhs.dup;
auto nb = rhs.dup;
Zeckendorf nt;
auto nr = "0".Z;
foreach (i; 0..(dLen + 1) * 2) {
if (((dVal >> i) & 1) > 0) nr += nb;
nt = nb.dup;
nb += na;
na = nt.dup;
}
dVal = nr.dVal;
dLen = nr.dLen;
}

void toString(scope void delegate(const(char)[]) sink) const {
if (dVal == 0) {
sink("0");
return;
}
sink(dig1[(dVal >> (dLen * 2)) & 3]);
foreach_reverse (i; 0..dLen) {
sink(dig[(dVal >> (i * 2)) & 3]);
}
}

Zeckendorf dup() {
auto z = "0".Z;
z.dVal = dVal;
z.dLen = dLen;
return z;
}

enum dig = ["00", "01", "10"];
enum dig1 = ["", "1", "10"];
}

auto Z(string val) {
return new Zeckendorf(val);
}

void main() {
writeln("Addition:");
auto g = "10".Z;
g += "10".Z;
writeln(g);
g += "10".Z;
writeln(g);
g += "1001".Z;
writeln(g);
g += "1000".Z;
writeln(g);
g += "10101".Z;
writeln(g);
writeln();

writeln("Subtraction:");
g = "1000".Z;
g -= "101".Z;
writeln(g);
g = "10101010".Z;
g -= "1010101".Z;
writeln(g);
writeln();

writeln("Multiplication:");
g = "1001".Z;
g *= "101".Z;
writeln(g);
g = "101010".Z;
g += "101".Z;
writeln(g);
}
```
Output:
```Addition:
101
1001
10101
100101
1010000

Subtraction:
1
1000000

Multiplication:
1000100
1000100```

## Elena

Translation of: C++

ELENA 6.x :

```import extensions;

const dig = new string[]{"00","01","10"};
const dig1 = new string[]{"","1","10"};

sealed struct ZeckendorfNumber
{
int dVal;
int dLen;

clone()
= ZeckendorfNumber.newInternal(dVal,dLen);

cast n(string s)
{
int i := s.Length - 1;
int q := 1;

dLen := i / 2;
dVal := 0;

while (i >= 0)
{
dVal += ((intConvertor.convert(s[i]) - 48) * q);
q *= 2;

i -= 1
}
}

internal readContent(ref int val, ref int len)
{
val := dVal;
len := dLen;
}

private a(int n)
{
int i := n;

while (true)
{
if (dLen < i)
{
dLen := i
};

int v := (dVal \$shr (i * 2)) & 3;
v =>
0 { ^ self }
1 { ^ self }
2 {
ifnot ((dVal \$shr ((i + 1) * 2)).allMask:1)
{
^ self
};

dVal += (1 \$shl (i*2 + 1));

^ self
}
3 {
int tmp := 3 \$shl (i * 2);
tmp := tmp.bxor(-1);
dVal := dVal & tmp;

self.b((i+1)*2)
};

i += 1
}
}

inc()
{
dVal += 1;
self.a(0)
}

private b(int pos)
{
if (pos == 0) { ^ self.inc() };

ifnot((dVal \$shr pos).allMask:1)
{
dVal += (1 \$shl pos);
self.a(pos / 2);
if (pos > 1) { self.a((pos / 2) - 1) }
}
else
{
dVal := dVal & (1 \$shl pos).BInverted;
self.b(pos + 1);
int arg := pos - ((pos > 1) ? 2 : 1);
self.b(/*pos - ((pos > 1) ? 2 : 1)*/arg)
}
}

private c(int pos)
{
if ((dVal \$shr pos).allMask:1)
{
int tmp := 1 \$shl pos;
tmp := tmp.bxor(-1);

dVal := dVal & tmp;

^ self
};

self.c(pos + 1);

if (pos > 0)
{
self.b(pos - 1)
}
else
{
self.inc()
}
}

internal constructor sum(ZeckendorfNumber n, ZeckendorfNumber m)
{
int mVal := 0;
int mLen := 0;

n.readContent(ref int v, ref int l);
m.readContent(ref mVal, ref mLen);

dVal := v;
dLen := l;

for(int GN := 0, GN < (mLen + 1) * 2, GN += 1)
{
if ((mVal \$shr GN).allMask:1)
{
self.b(GN)
}
}
}

internal constructor difference(ZeckendorfNumber n, ZeckendorfNumber m)
{
int mVal := 0;
int mLen := 0;

n.readContent(ref int v, ref int l);
m.readContent(ref mVal, ref mLen);

dVal := v;
dLen := l;

for(int GN := 0, GN < (mLen + 1) * 2, GN += 1)
{
if ((mVal \$shr GN).allMask:1)
{
self.c(GN)
}
};

while (((dVal \$shr (dLen*2)) & 3) == 0 || dLen == 0)
{
dLen -= 1
}
}

internal constructor product(ZeckendorfNumber n, ZeckendorfNumber m)
{
n.readContent(ref int v, ref int l);

dVal := v;
dLen := l;

ZeckendorfNumber Na := m;
ZeckendorfNumber Nb := m;
ZeckendorfNumber Nr := 0n;
ZeckendorfNumber Nt := 0n;

for(int i := 0, i < (dLen + 1) * 2, i += 1)
{
if (((dVal \$shr i) & 1) > 0)
{
Nr += Nb
};
Nt := Nb;
Nb += Na;
Na := Nt
};

Nr.readContent(ref v, ref l);

dVal := v;
dLen := l;
}

internal constructor newInternal(int v, int l)
{
dVal := v;
dLen := l
}

string toPrintable()
{
if (dVal == 0)
{ ^ "0" };

string s := dig1[(dVal \$shr (dLen * 2)) & 3];
int i := dLen - 1;
while (i >= 0)
{
s := s + dig[(dVal \$shr (i * 2)) & 3];

i-=1
};

^ s
}

add(ZeckendorfNumber n)
= ZeckendorfNumber.sum(self, n);

subtract(ZeckendorfNumber n)
= ZeckendorfNumber.difference(self, n);

multiply(ZeckendorfNumber n)
= ZeckendorfNumber.product(self, n);
}

public program()
{
console.printLine("Addition:");
var n := 10n;

n += 10n;
console.printLine(n);
n += 10n;
console.printLine(n);
n += 1001n;
console.printLine(n);
n += 1000n;
console.printLine(n);
n += 10101n;
console.printLine(n);

console.printLine("Subtraction:");
n := 1000n;
n -= 101n;
console.printLine(n);
n := 10101010n;
n -= 1010101n;
console.printLine(n);

console.printLine("Multiplication:");
n := 1001n;
n *= 101n;
console.printLine(n);
n := 101010n;
n += 101n;
console.printLine(n)
}```
Output:
```Addition:
101
1001
10101
100101
1010000
Subtraction:
1
1000000
Multiplication:
1000100
1000100
```

## Go

Translation of: Kotlin
```package main

import (
"fmt"
"strings"
)

var (
dig  = [3]string{"00", "01", "10"}
dig1 = [3]string{"", "1", "10"}
)

type Zeckendorf struct{ dVal, dLen int }

func NewZeck(x string) *Zeckendorf {
z := new(Zeckendorf)
if x == "" {
x = "0"
}
q := 1
i := len(x) - 1
z.dLen = i / 2
for ; i >= 0; i-- {
z.dVal += int(x[i]-'0') * q
q *= 2
}
return z
}

func (z *Zeckendorf) a(i int) {
for ; ; i++ {
if z.dLen < i {
z.dLen = i
}
j := (z.dVal >> uint(i*2)) & 3
switch j {
case 0, 1:
return
case 2:
if ((z.dVal >> (uint(i+1) * 2)) & 1) != 1 {
return
}
z.dVal += 1 << uint(i*2+1)
return
case 3:
z.dVal &= ^(3 << uint(i*2))
z.b((i + 1) * 2)
}
}
}

func (z *Zeckendorf) b(pos int) {
if pos == 0 {
z.Inc()
return
}
if ((z.dVal >> uint(pos)) & 1) == 0 {
z.dVal += 1 << uint(pos)
z.a(pos / 2)
if pos > 1 {
z.a(pos/2 - 1)
}
} else {
z.dVal &= ^(1 << uint(pos))
z.b(pos + 1)
temp := 1
if pos > 1 {
temp = 2
}
z.b(pos - temp)
}
}

func (z *Zeckendorf) c(pos int) {
if ((z.dVal >> uint(pos)) & 1) == 1 {
z.dVal &= ^(1 << uint(pos))
return
}
z.c(pos + 1)
if pos > 0 {
z.b(pos - 1)
} else {
z.Inc()
}
}

func (z *Zeckendorf) Inc() {
z.dVal++
z.a(0)
}

func (z1 *Zeckendorf) PlusAssign(z2 *Zeckendorf) {
for gn := 0; gn < (z2.dLen+1)*2; gn++ {
if ((z2.dVal >> uint(gn)) & 1) == 1 {
z1.b(gn)
}
}
}

func (z1 *Zeckendorf) MinusAssign(z2 *Zeckendorf) {
for gn := 0; gn < (z2.dLen+1)*2; gn++ {
if ((z2.dVal >> uint(gn)) & 1) == 1 {
z1.c(gn)
}
}

for z1.dLen > 0 && ((z1.dVal>>uint(z1.dLen*2))&3) == 0 {
z1.dLen--
}
}

func (z1 *Zeckendorf) TimesAssign(z2 *Zeckendorf) {
na := z2.Copy()
nb := z2.Copy()
nr := new(Zeckendorf)
for i := 0; i <= (z1.dLen+1)*2; i++ {
if ((z1.dVal >> uint(i)) & 1) > 0 {
nr.PlusAssign(nb)
}
nt := nb.Copy()
nb.PlusAssign(na)
na = nt.Copy()
}
z1.dVal = nr.dVal
z1.dLen = nr.dLen
}

func (z *Zeckendorf) Copy() *Zeckendorf {
return &Zeckendorf{z.dVal, z.dLen}
}

func (z1 *Zeckendorf) Compare(z2 *Zeckendorf) int {
switch {
case z1.dVal < z2.dVal:
return -1
case z1.dVal > z2.dVal:
return 1
default:
return 0
}
}

func (z *Zeckendorf) String() string {
if z.dVal == 0 {
return "0"
}
var sb strings.Builder
sb.WriteString(dig1[(z.dVal>>uint(z.dLen*2))&3])
for i := z.dLen - 1; i >= 0; i-- {
sb.WriteString(dig[(z.dVal>>uint(i*2))&3])
}
return sb.String()
}

func main() {
fmt.Println("Addition:")
g := NewZeck("10")
g.PlusAssign(NewZeck("10"))
fmt.Println(g)
g.PlusAssign(NewZeck("10"))
fmt.Println(g)
g.PlusAssign(NewZeck("1001"))
fmt.Println(g)
g.PlusAssign(NewZeck("1000"))
fmt.Println(g)
g.PlusAssign(NewZeck("10101"))
fmt.Println(g)

fmt.Println("\nSubtraction:")
g = NewZeck("1000")
g.MinusAssign(NewZeck("101"))
fmt.Println(g)
g = NewZeck("10101010")
g.MinusAssign(NewZeck("1010101"))
fmt.Println(g)

fmt.Println("\nMultiplication:")
g = NewZeck("1001")
g.TimesAssign(NewZeck("101"))
fmt.Println(g)
g = NewZeck("101010")
g.PlusAssign(NewZeck("101"))
fmt.Println(g)
}
```
Output:
```Addition:
101
1001
10101
100101
1010000

Subtraction:
1
1000000

Multiplication:
1000100
1000100
```

## Haskell

We make Zeckendorf numbers first class citizens implementing instances of `Eq`, `Ord`, `Num`, `Enum`, `Real` and `Integral` classes. So everything that could be done with integral numbers is applicable with Zeckendorf numbers.

Addition and subtraction are done using cellular automata. Conversion from integers, multiplication and division are implemented via generalized Fibonacci series (Zeckendorf tables).

```{-# LANGUAGE LambdaCase #-}
import Data.List (find, mapAccumL)
import Control.Arrow (first, second)

-- Generalized Fibonacci series defined for any Num instance, and for Zeckendorf numbers as well.
-- Used to build Zeckendorf tables.
fibs :: Num a => a -> a -> [a]
fibs a b = res
where
res = a : b : zipWith (+) res (tail res)

data Fib = Fib { sign :: Int, digits :: [Int]}

-- smart constructor
mkFib s ds =
case dropWhile (==0) ds of
[] -> 0
ds -> Fib s (reverse ds)

-- Textual representation
instance Show Fib where
show (Fib s ds) = sig s ++ foldMap show (reverse ds)
where sig = \case { -1 -> "-"; s -> "" }

-- Equivalence relation
instance Eq Fib where
Fib sa a == Fib sb b = sa == sb && a == b

-- Order relation
instance Ord Fib where
a `compare` b =
sign a `compare` sign b <>
case find (/= 0) \$ alignWith (-) (digits a) (digits b) of
Nothing -> EQ
Just 1 -> if sign a > 0 then GT else LT
Just (-1) -> if sign a > 0 then LT else GT

-- Arithmetic
instance Num Fib where
negate (Fib s ds) = Fib (negate s) ds
abs (Fib s ds) = Fib 1 ds
signum (Fib s _) = fromIntegral s

fromInteger n =
case compare n 0 of
LT -> negate \$ fromInteger (- n)
EQ -> Fib 0 [0]
GT -> Fib 1 . reverse . fst \$ divModFib n 1

0 + a = a
a + 0 = a
a + b =
case (sign a, sign b) of
( 1, 1) -> res
(-1, 1) -> b - (-a)
( 1,-1) -> a - (-b)
(-1,-1) -> - ((- a) + (- b))
where
res = mkFib 1 . process \$ 0:0:c
c = alignWith (+) (digits a) (digits b)
-- use cellular automata
process =
runRight 3 r2 . runLeftR 3 r2 . runRightR 4 r1

0 - a = -a
a - 0 = a
a - b =
case (sign a, sign b) of
( 1, 1) -> res
(-1, 1) -> - ((-a) + b)
( 1,-1) -> a + (-b)
(-1,-1) -> - ((-a) - (-b))
where
res = case find (/= 0) c of
Just 1  -> mkFib 1 . process \$ c
Just (-1) -> - (b - a)
Nothing -> 0
c = alignWith (-) (digits a) (digits b)
-- use cellular automata
process =
runRight 3 r2 . runLeftR 3 r2 . runRightR 4 r1 . runRight 3 r3

0 * a = 0
a * 0 = 0
1 * a = a
a * 1 = a
a * b =
case (sign a, sign b) of
(1, 1) -> res
(-1, 1) -> - ((-a) * b)
( 1,-1) -> - (a * (-b))
(-1,-1) -> ((-a) * (-b))
where
-- use Zeckendorf table
table = fibs a (a + a)
res = sum \$ onlyOnes \$ zip (digits b) table
onlyOnes = map snd . filter ((==1) . fst)

-- Enumeration
instance Enum Fib where
toEnum = fromInteger . fromIntegral
fromEnum = fromIntegral . toInteger

instance Real Fib where
toRational = fromInteger . toInteger

-- Integral division
instance Integral Fib where
toInteger (Fib s ds) = signum (fromIntegral s) * res
where
res = sum (zipWith (*) (fibs 1 2) (fromIntegral <\$> ds))

quotRem 0 _ = (0, 0)
quotRem a 0 = error "divide by zero"
quotRem a b = case (sign a, sign b) of
(1, 1) -> first (mkFib 1) \$ divModFib a b
(-1, 1) -> second negate . first negate \$ quotRem (-a) b
( 1,-1) -> first negate \$ quotRem a (-b)
(-1,-1) -> second negate \$ quotRem (-a) (-b)

------------------------------------------------------------
-- helper funtions

-- general division using Zeckendorf table
divModFib :: (Ord a, Num c, Num a) => a -> a -> ([c], a)
divModFib a b = (q, r)
where
(r, q) = mapAccumL f a \$ reverse \$ takeWhile (<= a) table
table = fibs b (b+b)
f n x = if  n < x then (n, 0) else (n - x, 1)

-- application of rewriting rules
-- runs window from left to right
runRight n f = go
where
go []  = []
go lst = let (w, r) = splitAt n lst
(h: t) = f w
in h : go (t ++ r)

-- runs window from left to right and reverses the result
runRightR n f = go []
where
go res []  = res
go res lst = let (w, r) = splitAt n lst
(h: t) = f w
in go (h : res) (t ++ r)

-- runs reversed window and reverses the result
runLeftR n f = runRightR n (reverse . f . reverse)

-- rewriting rules from [C. Ahlbach et. all]
r1 = \case [0,3,0]   -> [1,1,1]
[0,2,0]   -> [1,0,1]
[0,1,2]   -> [1,0,1]
[0,2,1]   -> [1,1,0]
[x,0,2]   -> [x,1,0]
[x,0,3]   -> [x,1,1]
[0,1,2,0] -> [1,0,1,0]
[0,2,0,x] -> [1,0,0,x+1]
[0,3,0,x] -> [1,1,0,x+1]
[0,2,1,x] -> [1,1,0,x  ]
[0,1,2,x] -> [1,0,1,x  ]
l -> l

r2 = \case [0,1,1] -> [1,0,0]
l -> l

r3 = \case [1,-1]    -> [0,1]
[2,-1]    -> [1,1]
[1, 0, 0] -> [0,1,1]
[1,-1, 0] -> [0,0,1]
[1,-1, 1] -> [0,0,2]
[1, 0,-1] -> [0,1,0]
[2, 0, 0] -> [1,1,1]
[2,-1, 0] -> [1,0,1]
[2,-1, 1] -> [1,0,2]
[2, 0,-1] -> [1,1,0]
l -> l

alignWith :: (Int -> Int -> a) -> [Int] -> [Int] -> [a]
alignWith f a b = go [] a b
where
go res as [] = ((`f` 0) <\$> reverse as) ++ res
go res [] bs = ((0 `f`) <\$> reverse bs) ++ res
go res (a:as) (b:bs) = go (f a b : res) as bs
```
```λ> 15 :: Fib
100010

λ> 153 :: Fib
10000010001

λ> [1..13] :: [Fib]
[1,10,100,101,1000,1001,1010,10000,10001,10010,10100,10101,100000]

λ> 15 + 47 :: Fib
100001010

λ> toInteger it
62

λ> 15 - 47 :: Fib
-1010100

λ> toInteger it
-32

λ> 15 * 47 :: Fib
10001000001001

λ> toInteger it
705

λ> 47 `div` 15 :: Fib
100

λ> 47 `mod` 15 :: Fib
10```

## J

Loosely based on the perl implementation:
```zform=: {{ 10 |."1@(#.inv) y }} :. (10#.|."1) NB. use decimal numbers for representation
zinc=: {{  carry ({.,2}.])carry 1,y }}
zdec=: {{ (|.k\$0 1),y }.~k=. 1+y i.1 }}
zadd=: {{ x while. 1 e. y do. x=. zinc x [ y=. zdec y end. }}
zsub=: {{ x while. 1 e. y do. x=. zdec x [ y=. zdec y end. }} NB. intended for unsigned arithmetic
zmul=: {{ t=. 0 0 while. 1 e. y do. t=. t zadd x [ y=. zdec y end. }}
zdiv=: {{ t=. 0 0 while. x zge y do. t=. zinc t [ x=. x zsub y end. }} NB. discards remainder
carry=: {{
s=. 0
for_b. y do.
if. (1+b) = s=. s-_1^b do. y=. (-.b) (b_index-0,b)} y end.
end.
if. 2=s do. y,1 else. y end.
}}
zge=: {{ cmp=. x -/@,: y while. (#cmp)*0={:cmp do. cmp=. }:cmp end. 0<:{:cmp }}
```

For example, we use the decimal number 10100 to represent 11 in base 10, and 1010 would represent 7. We convert these numbers to an internal zeckendorf representation and add them, then convert the result back to decimal 101000 which represents 18 in base 10.

Task examples:
```   1 zadd&.zform 1
10
10 zadd&.zform 10
101
10100 zadd&.zform 1010
101000
10100 zsub&.zform 1010
101
10100 zmul&.zform 100101
10010010001
10100 zdiv&.zform 1010
1
10100 zdiv&.zform 1000
10
100001000001 zdiv&.zform 100010
100101
100001000001 zdiv&.zform 100101
100010
```

## Java

Translation of: Kotlin
Works with: Java version 9
```import java.util.List;

public class Zeckendorf implements Comparable<Zeckendorf> {
private static List<String> dig = List.of("00", "01", "10");
private static List<String> dig1 = List.of("", "1", "10");

private String x;
private int dVal = 0;
private int dLen = 0;

public Zeckendorf() {
this("0");
}

public Zeckendorf(String x) {
this.x = x;

int q = 1;
int i = x.length() - 1;
dLen = i / 2;
while (i >= 0) {
dVal += (x.charAt(i) - '0') * q;
q *= 2;
i--;
}
}

private void a(int n) {
int i = n;
while (true) {
if (dLen < i) dLen = i;
int j = (dVal >> (i * 2)) & 3;
switch (j) {
case 0:
case 1:
return;
case 2:
if (((dVal >> ((i + 1) * 2)) & 1) != 1) return;
dVal += 1 << (i * 2 + 1);
return;
case 3:
int temp = 3 << (i * 2);
temp ^= -1;
dVal = dVal & temp;
b((i + 1) * 2);
break;
}
i++;
}
}

private void b(int pos) {
if (pos == 0) {
Zeckendorf thiz = this;
thiz.inc();
return;
}
if (((dVal >> pos) & 1) == 0) {
dVal += 1 << pos;
a(pos / 2);
if (pos > 1) a(pos / 2 - 1);
} else {
int temp = 1 << pos;
temp ^= -1;
dVal = dVal & temp;
b(pos + 1);
b(pos - (pos > 1 ? 2 : 1));
}
}

private void c(int pos) {
if (((dVal >> pos) & 1) == 1) {
int temp = 1 << pos;
temp ^= -1;
dVal = dVal & temp;
return;
}
c(pos + 1);
if (pos > 0) {
b(pos - 1);
} else {
Zeckendorf thiz = this;
thiz.inc();
}
}

public Zeckendorf inc() {
dVal++;
a(0);
return this;
}

public void plusAssign(Zeckendorf other) {
for (int gn = 0; gn < (other.dLen + 1) * 2; gn++) {
if (((other.dVal >> gn) & 1) == 1) {
b(gn);
}
}
}

public void minusAssign(Zeckendorf other) {
for (int gn = 0; gn < (other.dLen + 1) * 2; gn++) {
if (((other.dVal >> gn) & 1) == 1) {
c(gn);
}
}
while ((((dVal >> dLen * 2) & 3) == 0) || (dLen == 0)) {
dLen--;
}
}

public void timesAssign(Zeckendorf other) {
Zeckendorf na = other.copy();
Zeckendorf nb = other.copy();
Zeckendorf nt;
Zeckendorf nr = new Zeckendorf();
for (int i = 0; i < (dLen + 1) * 2; i++) {
if (((dVal >> i) & 1) > 0) {
nr.plusAssign(nb);
}
nt = nb.copy();
nb.plusAssign(na);
na = nt.copy();
}
dVal = nr.dVal;
dLen = nr.dLen;
}

private Zeckendorf copy() {
Zeckendorf z = new Zeckendorf();
z.dVal = dVal;
z.dLen = dLen;
return z;
}

@Override
public int compareTo(Zeckendorf other) {
return ((Integer) dVal).compareTo(other.dVal);
}

@Override
public String toString() {
if (dVal == 0) {
return "0";
}

int idx = (dVal >> (dLen * 2)) & 3;
StringBuilder stringBuilder = new StringBuilder(dig1.get(idx));
for (int i = dLen - 1; i >= 0; i--) {
idx = (dVal >> (i * 2)) & 3;
stringBuilder.append(dig.get(idx));
}
return stringBuilder.toString();
}

public static void main(String[] args) {
System.out.println("Addition:");
Zeckendorf g = new Zeckendorf("10");
g.plusAssign(new Zeckendorf("10"));
System.out.println(g);
g.plusAssign(new Zeckendorf("10"));
System.out.println(g);
g.plusAssign(new Zeckendorf("1001"));
System.out.println(g);
g.plusAssign(new Zeckendorf("1000"));
System.out.println(g);
g.plusAssign(new Zeckendorf("10101"));
System.out.println(g);

System.out.println("\nSubtraction:");
g = new Zeckendorf("1000");
g.minusAssign(new Zeckendorf("101"));
System.out.println(g);
g = new Zeckendorf("10101010");
g.minusAssign(new Zeckendorf("1010101"));
System.out.println(g);

System.out.println("\nMultiplication:");
g = new Zeckendorf("1001");
g.timesAssign(new Zeckendorf("101"));
System.out.println(g);
g = new Zeckendorf("101010");
g.plusAssign(new Zeckendorf("101"));
System.out.println(g);
}
}
```
Output:
```Addition:
101
1001
10101
100101
1010000

Subtraction:
1
1000000

Multiplication:
1000100
1000100```

## Julia

Influenced by the format of the Tcl and Raku versions, but added other functionality.

```import Base.*, Base.+, Base.-, Base./, Base.show, Base.!=, Base.==, Base.<=, Base.<, Base.>, Base.>=, Base.divrem

const z0 = "0"
const z1 = "1"
const flipordered = (z1 < z0)

mutable struct Z s::String end
Z() = Z(z0)
Z(z::Z) = Z(z.s)

pairlen(x::Z, y::Z) = max(length(x.s), length(y.s))
tolen(x::Z, n::Int) = (s = x.s; while length(s) < n s = z0 * s end; s)

<(x::Z, y::Z) = (l = pairlen(x, y); flipordered ? tolen(x, l) > tolen(y, l) : tolen(x, l) < tolen(y, l))
>(x::Z, y::Z) = (l = pairlen(x, y); flipordered ? tolen(x, l) < tolen(y, l) : tolen(x, l) > tolen(y, l))
==(x::Z, y::Z) = (l = pairlen(x, y); tolen(x, l) == tolen(y, l))
<=(x::Z, y::Z) = (l = pairlen(x, y); flipordered ? tolen(x, l) >= tolen(y, l) : tolen(x, l) <= tolen(y, l))
>=(x::Z, y::Z) = (l = pairlen(x, y); flipordered ? tolen(x, l) <= tolen(y, l) : tolen(x, l) >= tolen(y, l))
!=(x::Z, y::Z) = (l = pairlen(x, y); tolen(x, l) != tolen(y, l))

function tocanonical(z::Z)
while occursin(z0 * z1 * z1, z.s)
z.s = replace(z.s, z0 * z1 * z1 => z1 * z0 * z0)
end
len = length(z.s)
if len > 1 && z.s[1:2] == z1 * z1
z.s = z1 * z0 * z0 * ((len > 2) ? z.s[3:end] : "")
end
while (len = length(z.s)) > 1 && string(z.s[1]) == z0
if len == 2
if z.s == z0 * z0
z.s = z0
elseif z.s == z0 * z1
z.s = z1
end
else
z.s = z.s[2:end]
end
end
z
end

function inc(z)
if z.s[end] == z0[1]
z.s = z.s[1:end-1] * z1[1]
elseif z.s[end] == z1[1]
if length(z.s) > 1
if z.s[end-1:end] == z0 * z1
z.s = z.s[1:end-2] * z1 * z0
end
else
z.s = z1 * z0
end
end
tocanonical(z)
end

function dec(z)
if z.s[end] == z1[1]
z.s = z.s[1:end-1] * z0
else
if (m = match(Regex(z1 * z0 * '+' * '\$'), z.s)) != nothing
len = length(m.match)
if iseven(len)
z.s = z.s[1:end-len] * (z0 * z1) ^ div(len, 2)
else
z.s = z.s[1:end-len] * (z0 * z1) ^ div(len, 2) * z0
end
end
end
tocanonical(z)
z
end

function +(x::Z, y::Z)
a = Z(x.s)
b = Z(y.s)
while b.s != z0
inc(a)
dec(b)
end
a
end

function -(x::Z, y::Z)
a = Z(x.s)
b = Z(y.s)
while b.s != z0
dec(a)
dec(b)
end
a
end

function *(x::Z, y::Z)
if (x.s == z0) || (y.s == z0)
return Z(z0)
elseif x.s == z1
return Z(y.s)
elseif y.s == z1
return Z(x.s)
end
a = Z(x.s)
b = Z(z1)
while b != y
c = Z(z0)
while c != x
inc(a)
inc(c)
end
inc(b)
end
a
end

function divrem(x::Z, y::Z)
if y.s == z0
throw("Zeckendorf division by 0")
elseif (y.s == z1) || (x.s == z0)
return Z(x.s)
end
a = Z(x.s)
b = Z(y.s)
c = Z(z0)
while a > b
a = a - b
inc(c)
end
tocanonical(c), tocanonical(a)
end

function /(x::Z, y::Z)
a, _ = divrem(x, y)
a
end

show(io::IO, z::Z) = show(io, parse(BigInt, tocanonical(z).s))

function zeckendorftest()
a = Z("10")
b = Z("1001")
c = Z("1000")
d = Z("10101")

println("Addition:")
x = a
println(x += a)
println(x += a)
println(x += b)
println(x += c)
println(x += d)

println("\nSubtraction:")
x = Z("1000")
println(x - Z("101"))
x = Z("10101010")
println(x - Z("1010101"))

println("\nMultiplication:")
x = Z("1001")
y = Z("101")
println(x * y)
println(Z("101010") * y)

println("\nDivision:")
x = Z("1000101")
y = Z("101")
println(x / y)
println(divrem(x, y))
end

zeckendorftest()
```
Output:
```
Addition:
101
1001
10101
100101
1010000

Subtraction:
1
1000000

Multiplication:
1000100
101000101

Division:
1001
(1001, 1)

```

## Kotlin

Translation of: C++
```// version 1.1.51

class Zeckendorf(x: String = "0") : Comparable<Zeckendorf> {

var dVal = 0
var dLen = 0

private fun a(n: Int) {
var i = n
while (true) {
if (dLen < i) dLen = i
val j = (dVal shr (i * 2)) and 3
when (j) {
0, 1 -> return

2 -> {
if (((dVal shr ((i + 1) * 2)) and 1) != 1) return
dVal += 1 shl (i * 2 + 1)
return
}

3 -> {
dVal = dVal and (3 shl (i * 2)).inv()
b((i + 1) * 2)
}
}
i++
}
}

private fun b(pos: Int) {
if (pos == 0) {
var thiz = this
++thiz
return
}
if (((dVal shr pos) and 1) == 0) {
dVal += 1 shl pos
a(pos / 2)
if (pos > 1) a(pos / 2 - 1)
}
else {
dVal = dVal and (1 shl pos).inv()
b(pos + 1)
b(pos - (if (pos > 1) 2 else 1))
}
}

private fun c(pos: Int) {
if (((dVal shr pos) and 1) == 1) {
dVal = dVal and (1 shl pos).inv()
return
}
c(pos + 1)
if (pos > 0) b(pos - 1) else { var thiz = this; ++thiz }
}

init {
var q = 1
var i = x.length - 1
dLen = i / 2
while (i >= 0) {
dVal += (x[i] - '0').toInt() * q
q *= 2
i--
}
}

operator fun inc(): Zeckendorf {
dVal += 1
a(0)
return this
}

operator fun plusAssign(other: Zeckendorf) {
for (gn in 0 until (other.dLen + 1) * 2) {
if (((other.dVal shr gn) and 1) == 1) b(gn)
}
}

operator fun minusAssign(other: Zeckendorf) {
for (gn in 0 until (other.dLen + 1) * 2) {
if (((other.dVal shr gn) and 1) == 1) c(gn)
}
while ((((dVal shr dLen * 2) and 3) == 0) || (dLen == 0)) dLen--
}

operator fun timesAssign(other: Zeckendorf) {
var na = other.copy()
var nb = other.copy()
var nt: Zeckendorf
var nr = "0".Z
for (i in 0..(dLen + 1) * 2) {
if (((dVal shr i) and 1) > 0) nr += nb
nt = nb.copy()
nb += na
na = nt.copy()
}
dVal = nr.dVal
dLen = nr.dLen
}

override operator fun compareTo(other: Zeckendorf) = dVal.compareTo(other.dVal)

override fun toString(): String {
if (dVal == 0) return "0"
val sb = StringBuilder(dig1[(dVal shr (dLen * 2)) and 3])
for (i in dLen - 1 downTo 0) {
sb.append(dig[(dVal shr (i * 2)) and 3])
}
return sb.toString()
}

fun copy(): Zeckendorf {
val z = "0".Z
z.dVal = dVal
z.dLen = dLen
return z
}

companion object {
val dig = listOf("00", "01", "10")
val dig1 = listOf("", "1", "10")
}
}

val String.Z get() = Zeckendorf(this)

fun main(args: Array<String>) {
println("Addition:")
var g = "10".Z
g += "10".Z
println(g)
g += "10".Z
println(g)
g += "1001".Z
println(g)
g += "1000".Z
println(g)
g += "10101".Z
println(g)
println("\nSubtraction:")
g = "1000".Z
g -= "101".Z
println(g)
g = "10101010".Z
g -= "1010101".Z
println(g)
println("\nMultiplication:")
g = "1001".Z
g *= "101".Z
println(g)
g = "101010".Z
g += "101".Z
println(g)
}
```
Output:
```Addition:
101
1001
10101
100101
1010000

Subtraction:
1
1000000

Multiplication:
1000100
1000100
```

## Nim

Translation of: Go
```type Zeckendorf = object
dVal: Natural
dLen: Natural

const
Dig = ["00", "01", "10"]
Dig1 = ["", "1", "10"]

# Forward references.
func b(z: var Zeckendorf; pos: Natural)
func inc(z: var Zeckendorf)

func a(z: var Zeckendorf; n: Natural) =
var i = n
while true:

if z.dLen < i: z.dLen = i
let j = z.dVal shr (i * 2) and 3

case j
of 0, 1:
return
of 2:
if (z.dVal shr ((i + 1) * 2) and 1) != 1: return
z.dVal += 1 shl (i * 2 + 1)
return
of 3:
z.dVal = z.dVal and not (3 shl (i * 2))
z.b((i + 1) * 2)
else:
assert(false)

inc i

func b(z: var Zeckendorf; pos: Natural) =
if pos == 0:
inc z
return

if (z.dVal shr pos and 1) == 0:
z.dVal += 1 shl pos
z.a(pos div 2)
if pos > 1: z.a(pos div 2 - 1)
else:
z.dVal = z.dVal and not(1 shl pos)
z.b(pos + 1)
z.b(pos - (if pos > 1: 2 else: 1))

func c(z: var Zeckendorf; pos: Natural) =
if (z.dVal shr pos and 1) == 1:
z.dVal = z.dVal and not(1 shl pos)
return

z.c(pos + 1)
if pos > 0:
z.b(pos - 1)
else:
inc z

func initZeckendorf(s = "0"): Zeckendorf =
var q = 1
var i = s.high
result.dLen = i div 2
while i >= 0:
result.dVal += (ord(s[i]) - ord('0')) * q
q *= 2
dec i

func inc(z: var Zeckendorf) =
inc z.dVal
z.a(0)

func `+=`(z1: var Zeckendorf; z2: Zeckendorf) =
for gn in 0 .. (2 * z2.dLen + 1):
if (z2.dVal shr gn and 1) == 1:
z1.b(gn)

func `-=`(z1: var Zeckendorf; z2: Zeckendorf) =
for gn in 0 .. (2 * z2.dLen + 1):
if (z2.dVal shr gn and 1) == 1:
z1.c(gn)

while z1.dLen > 0 and (z1.dVal shr (z1.dLen * 2) and 3) == 0:
dec z1.dLen

func `*=`(z1: var Zeckendorf; z2: Zeckendorf) =
var na, nb = z2
var nr: Zeckendorf
for i in 0 .. (z1.dLen + 1) * 2:
if (z1.dVal shr i and 1) > 0: nr += nb
let nt = nb
nb += na
na = nt
z1 = nr

func`\$`(z: var Zeckendorf): string =
if z.dVal == 0: return "0"
result.add Dig1[z.dVal shr (z.dLen * 2) and 3]
for i in countdown(z.dLen - 1, 0):
result.add Dig[z.dVal shr (i * 2) and 3]

when isMainModule:

var g: Zeckendorf

echo "Addition:"
g = initZeckendorf("10")
g += initZeckendorf("10")
echo g
g += initZeckendorf("10")
echo g
g += initZeckendorf("1001")
echo g
g += initZeckendorf("1000")
echo g
g += initZeckendorf("10101")
echo g

echo "\nSubtraction:"
g = initZeckendorf("1000")
g -= initZeckendorf("101")
echo g
g = initZeckendorf("10101010")
g -= initZeckendorf("1010101")
echo g

echo "\nMultiplication:"
g = initZeckendorf("1001")
g *= initZeckendorf("101")
echo g
g = initZeckendorf("101010")
g += initZeckendorf("101")
echo g
```
Output:
```Addition:
101
1001
10101
100101
1010000

Subtraction:
1
1000000

Multiplication:
1000100
1000100```

## Perl

```use v5.36;

package Zeckendorf;
use overload qw("" zstring + zadd - zsub ++ zinc -- zdec * zmul / zdiv ge zge);

sub new (\$class, \$value) {
bless \\$value, ref \$class || \$class;
}

sub zinc (\$self, \$, \$) {
local \$_ = \$\$self;
s/0\$/1/ or s/(?:^|0)1\$/10/;
1 while s/(?:^|0)11/100/;
\$\$self = \$self->new( s/^0+\B//r )
}

sub zdec (\$self, \$, \$) {
local \$_ = \$\$self;
1 while s/100(?=0*\$)/011/;
s/1\$/0/ || s/10\$/01/;
\$\$self = \$self->new( s/^0+\B//r )
}

sub zadd (\$self, \$other, \$) {
my (\$x, \$y) = map \$self->new(\$\$_), \$self, \$other;
\$x++, \$y-- while \$\$y;
\$x
}

sub zsub (\$self, \$other, \$) {
my (\$x, \$y) = map \$self->new(\$\$_), \$self, \$other;
\$x--, \$y-- while \$\$y;
\$x
}

sub zmul (\$self, \$other, \$) {
my (\$x, \$y) = map \$self->new(\$\$_), \$self, \$other;
my \$product = Zeckendorf->new(0);
\$product = \$product + \$x, \$y-- while \$y;
\$product
}

sub zdiv (\$self, \$other, \$) {
my (\$x, \$y) = map \$self->new(\$\$_), \$self, \$other;
my \$quotient = Zeckendorf->new(0);
\$quotient++, \$x = \$x - \$y while \$x ge \$y;
\$quotient
}

sub zge (\$self, \$other, \$) {
my \$l; \$l = length \$\$other if length \$other > (\$l = length \$\$self);
0 x (\$l - length \$\$self) . \$\$self ge 0 x (\$l - length \$\$other) . \$\$other;
}

sub asdecimal (\$self) {
my(\$aa, \$bb, \$n) = (1, 1, 0);
for ( reverse split '', \$\$self ) {
\$n += \$bb * \$_;
(\$aa, \$bb) = (\$bb, \$aa + \$bb);
}
\$n
}

sub fromdecimal (\$self, \$value) {
my \$z = \$self->new(0);
\$z++ for 1 .. \$value;
\$z
}

sub zstring { \${ shift() } }

package main;

for ( split /\n/, <<END ) # test cases
1 + 1
10 + 10
10100 + 1010
10100 - 1010
10100 * 1010
100010 * 100101
10100 / 1010
101000 / 1000
100001000001 / 100010
100001000001 / 100101
END
{
my (\$left, \$op, \$right) = split;
my (\$x, \$y) = map Zeckendorf->new(\$_), \$left, \$right;
my \$answer =
\$op eq '+' ? \$x + \$y :
\$op eq '-' ? \$x - \$y :
\$op eq '*' ? \$x * \$y :
\$op eq '/' ? \$x / \$y :
die "bad op <\$op>";
printf "%12s %s %-9s => %12s  in Zeckendorf\n", \$x, \$op, \$y, \$answer;
printf "%12d %s %-9d => %12d  in decimal\n\n",
\$x->asdecimal, \$op, \$y->asdecimal, \$answer->asdecimal;
}
```
Output:
```           1 + 1         =>           10  in Zeckendorf
1 + 1         =>            2  in decimal

10 + 10        =>          101  in Zeckendorf
2 + 2         =>            4  in decimal

10100 + 1010      =>       101000  in Zeckendorf
11 + 7         =>           18  in decimal

10100 - 1010      =>          101  in Zeckendorf
11 - 7         =>            4  in decimal

10100 * 1010      =>    101000001  in Zeckendorf
11 * 7         =>           77  in decimal

100010 * 100101    => 100001000001  in Zeckendorf
15 * 17        =>          255  in decimal

10100 / 1010      =>            1  in Zeckendorf
11 / 7         =>            1  in decimal

101000 / 1000      =>          100  in Zeckendorf
18 / 5         =>            3  in decimal

100001000001 / 100010    =>       100101  in Zeckendorf
255 / 15        =>           17  in decimal

100001000001 / 100101    =>       100010  in Zeckendorf
255 / 17        =>           15  in decimal```

## Phix

Uses a binary representation of Zeckendorf numbers, eg decimal 11 is stored as 0b10100, ie meaning 8+3, but actually 20 in decimal.
As such, they can be directly compared using the standard comparison operators, and printed quite trivially just by using the %b format.
They are however (and not all that surprisingly) pulled apart into individual bits for addition/subtraction, etc.
Does not handle negative numbers or anything >139583862445 (-ve probably doable but messy, >1.4e12 requires a total rewrite, probably using string representation).

```with javascript_semantics

sequence fib = {1,1}

function zeckendorf(atom n)
-- Same as Zeckendorf_number_representation#Phix
atom r = 0
while fib[\$]<n do
fib &= fib[\$] + fib[\$-1]
end while
integer k = length(fib)
while k>2 and n<fib[k] do
k -= 1
end while
for i=k to 2 by -1 do
integer c = n>=fib[i]
r += r+c
n -= c*fib[i]
end for
return r
end function

function decimal(object z)
-- Convert Zeckendorf number(s) to decimal
if sequence(z) then
sequence res = repeat(0,length(z))
for i=1 to length(z) do
res[i] = decimal(z[i])
end for
return res
end if
atom dec = 0, bit = 2
while z do
if and_bits(z,1) then
dec += fib[bit]
end if
bit += 1
if bit>length(fib) then
fib &= fib[\$] + fib[\$-1]
end if
z = floor(z/2)
end while
return dec
end function

function to_bits(integer x)
-- Simplified copy of int_to_bits(), but in reverse order,
-- and +ve only but (also only) as many bits as needed, and
-- ensures there are *two* trailing 0 (most significant)
if x<0 then ?9/0 end if     -- sanity/avoid infinite loop
sequence bits = {}
while 1 do
bits &= remainder(x,2)
if x=0 then exit end if
x = floor(x/2)
end while
bits &= 0 -- (since eg 101+101 -> 10000)
return bits
end function

function to_bits2(integer a,b)
-- Apply to_bits() to a and b, and pad to the same length
sequence sa = to_bits(a),
sb = to_bits(b)
integer diff = length(sa)-length(sb)
if diff!=0 then
if diff<0 then  sa &= repeat(0,-diff)
else  sb &= repeat(0,+diff)
end if
end if
return {sa,sb}
end function

function to_int(sequence bits)
-- Copy of bits_to_int(), but in reverse order (lsb last)
atom val = 0, p = 1
for i=length(bits) to 1 by -1 do
if bits[i] then
val += p
end if
p += p
end for
return val
end function

function zstr(object z)
if sequence(z) then
sequence res = repeat(0,length(z))
for i=1 to length(z) do
res[i] = zstr(z[i])
end for
return res
end if
return sprintf("%b",z)
end function

function rep(sequence res, integer ds, sequence was, wth)
-- helper for cleanup, validates replacements
integer de = ds+length(was)-1
if res[ds..de]!=was then ?9/0 end if
if length(was)!=length(wth) then ?9/0 end if
res = deep_copy(res)
res[ds..de] = wth
return res
end function

function zcleanup(sequence res)
-- (shared by zadd and zsub)
integer l = length(res)
res = deep_copy(res)
-- first stage, left to right, {020x -> 100x', 030x -> 110x', 021x->110x, 012x->101x}
for i=1 to l-3 do
sequence s3 = res[i..i+2]
if s3={0,2,0} then res[i..i+2] = {1,0,0} res[i+3] += 1
elsif s3={0,3,0} then res[i..i+2] = {1,1,0} res[i+3] += 1
elsif s3={0,2,1} then res[i..i+2] = {1,1,0}
elsif s3={0,1,2} then res[i..i+2] = {1,0,1}
end if
end for
-- first stage cleanup
if l>1 then
if res[l-1]=3 then      res = rep(res,l-2,{0,3,0},{1,1,1})      -- 030 -> 111
elsif res[l-1]=2 then
if res[l-2]=0 then  res = rep(res,l-2,{0,2,0},{1,0,1})      -- 020 -> 101
else  res = rep(res,l-3,{0,1,2,0},{1,0,1,0})  -- 0120 -> 1010
end if
end if
end if
if res[l]=3 then            res = rep(res,l-1,{0,3},{1,1})          -- 03 -> 11
elsif res[l]=2 then
if res[l-1]=0 then      res = rep(res,l-1,{0,2},{1,0})          -- 02 -> 10
else      res = rep(res,l-2,{0,1,2},{1,0,1})      -- 012 -> 101
end if
end if
-- second stage, pass 1, right to left, 011 -> 100
for i=length(res)-2 to 1 by -1 do
if res[i..i+2]={0,1,1} then res[i..i+2] = {1,0,0} end if
end for
-- second stage, pass 2, left to right, 011 -> 100
for i=1 to length(res)-2 do
if res[i..i+2]={0,1,1} then res[i..i+2] = {1,0,0} end if
end for
return to_int(res)
end function

function zadd(integer a, b)
sequence {sa,sb} = to_bits2(a,b)
return zcleanup(reverse(sq_add(sa,sb)))
end function

function zinc(integer a)
return zadd(a,0b1)
end function

function zsub(integer a, b)
sequence {sa,sb} = to_bits2(a,b)
sequence res = reverse(sq_sub(sa,sb))
-- (/not/ combined with the first pass of the add routine!)
for i=1 to length(res)-2 do
sequence s3 = res[i..i+2]
if s3={1, 0, 0} then res[i..i+2] = {0,1,1}
elsif s3={1,-1, 0} then res[i..i+2] = {0,0,1}
elsif s3={1,-1, 1} then res[i..i+2] = {0,0,2}
elsif s3={1, 0,-1} then res[i..i+2] = {0,1,0}
elsif s3={2, 0, 0} then res[i..i+2] = {1,1,1}
elsif s3={2,-1, 0} then res[i..i+2] = {1,0,1}
elsif s3={2,-1, 1} then res[i..i+2] = {1,0,2}
elsif s3={2, 0,-1} then res[i..i+2] = {1,1,0}
end if
end for
-- copied from PicoLisp: {1,-1} -> {0,1} and {2,-1} -> {1,1}
for i=1 to length(res)-1 do
sequence s2 = res[i..i+1]
if s2={1,-1} then res[i..i+1] = {0,1}
elsif s2={2,-1} then res[i..i+1] = {1,1}
end if
end for
if find(-1,res) then ?9/0 end if -- sanity check
return zcleanup(res)
end function

function zdec(integer a)
return zsub(a,0b1)
end function

function zmul(integer a, b)
sequence mult = {a,zadd(a,a)}   -- (as per task desc)
integer bits = 2
while bits<b do
mult = append(mult,zadd(mult[\$],mult[\$-1]))
bits *= 2
end while
integer res = 0,
bit = 1
while b do
if and_bits(b,1) then
res = zadd(res,mult[bit])
end if
b = floor(b/2)
bit += 1
end while
return res
end function

function zdiv(integer a, b)
sequence mult = {b,zadd(b,b)}
integer bits = 2
while mult[\$]<a do
mult = append(mult,zadd(mult[\$],mult[\$-1]))
bits *= 2
end while
integer res = 0
for i=length(mult) to 1 by -1 do
integer mi = mult[i]
if mi<=a then
res = zadd(res,bits)
a = zsub(a,mi)
if a=0 then exit end if
end if
bits = floor(bits/2)
end for
return {res,a} -- (a is the remainder)
end function

for i=0 to 20 do
integer zi = zeckendorf(i)
atom d = decimal(zi)
printf(1,"%2d: %7b (%d)\n",{i,zi,d})
end for

procedure test(atom a, string op, atom b, object res, string expected)
string zres = iff(atom(res)?zstr(res):join(zstr(res)," rem ")),
dres = sprintf(iff(atom(res)?"%d":"%d rem %d"),decimal(res)),
aka = sprintf("aka %d %s %d = %s",{decimal(a),op,decimal(b),dres}),
ok = iff(zres=expected?"":" *** ERROR ***!!")
printf(1,"%s %s %s = %s, %s %s\n",{zstr(a),op,zstr(b),zres,aka,ok})
end procedure

test(0b0,"+",0b0,zadd(0b0,0b0),"0")
test(0b101,"+",0b101,zadd(0b101,0b101),"10000")
test(0b10100,"-",0b1000,zsub(0b10100,0b1000),"1001")
test(0b100100,"-",0b1000,zsub(0b100100,0b1000),"10100")
test(0b1001,"*",0b101,zmul(0b1001,0b101),"1000100")
test(0b1000101,"/",0b101,zdiv(0b1000101,0b101),"1001 rem 1")

test(0b10,"+",0b10,zadd(0b10,0b10),"101")
test(0b101,"+",0b10,zadd(0b101,0b10),"1001")
test(0b1001,"+",0b1001,zadd(0b1001,0b1001),"10101")
test(0b10101,"+",0b1000,zadd(0b10101,0b1000),"100101")
test(0b100101,"+",0b10101,zadd(0b100101,0b10101),"1010000")
test(0b1000,"-",0b101,zsub(0b1000,0b101),"1")
test(0b10101010,"-",0b1010101,zsub(0b10101010,0b1010101),"1000000")
test(0b1001,"*",0b101,zmul(0b1001,0b101),"1000100")
test(0b101010,"+",0b101,zadd(0b101010,0b101),"1000100")

test(0b10100,"+",0b1010,zadd(0b10100,0b1010),"101000")
test(0b101000,"-",0b1010,zsub(0b101000,0b1010),"10100")

test(0b100010,"*",0b100101,zmul(0b100010,0b100101),"100001000001")
test(0b100001000001,"/",0b100,zdiv(0b100001000001,0b100),"101010001 rem 0")
test(0b101000101,"*",0b101001,zmul(0b101000101,0b101001),"101010000010101")
test(0b101010000010101,"/",0b100,zdiv(0b101010000010101,0b100),"1001010001001 rem 10")

test(0b10100010010100,"+",0b1001000001,zadd(0b10100010010100,0b1001000001),"100000000010101")
test(0b10100010010100,"-",0b1001000001,zsub(0b10100010010100,0b1001000001),"10010001000010")
test(0b10000,"*",0b1001000001,zmul(0b10000,0b1001000001),"10100010010100")
test(0b1010001010000001001,"/",0b100000000100000,zdiv(0b1010001010000001001,0b100000000100000),"10001 rem 10100001010101")

test(0b10100,"+",0b1010,zadd(0b10100,0b1010),"101000")
test(0b10100,"-",0b1010,zsub(0b10100,0b1010),"101")
test(0b10100,"*",0b1010,zmul(0b10100,0b1010),"101000001")
test(0b10100,"/",0b1010,zdiv(0b10100,0b1010),"1 rem 101")
integer m = zmul(0b10100,0b1010)
test(m,"/",0b1010,zdiv(m,0b1010),"10100 rem 0")
```
Output:
``` 0:       0 (0)
1:       1 (1)
2:      10 (2)
3:     100 (3)
4:     101 (4)
5:    1000 (5)
6:    1001 (6)
7:    1010 (7)
8:   10000 (8)
9:   10001 (9)
10:   10010 (10)
11:   10100 (11)
12:   10101 (12)
13:  100000 (13)
14:  100001 (14)
15:  100010 (15)
16:  100100 (16)
17:  100101 (17)
18:  101000 (18)
19:  101001 (19)
20:  101010 (20)
0 + 0 = 0, aka 0 + 0 = 0
101 + 101 = 10000, aka 4 + 4 = 8
10100 - 1000 = 1001, aka 11 - 5 = 6
100100 - 1000 = 10100, aka 16 - 5 = 11
1001 * 101 = 1000100, aka 6 * 4 = 24
1000101 / 101 = 1001 rem 1, aka 25 / 4 = 6 rem 1
10 + 10 = 101, aka 2 + 2 = 4
101 + 10 = 1001, aka 4 + 2 = 6
1001 + 1001 = 10101, aka 6 + 6 = 12
10101 + 1000 = 100101, aka 12 + 5 = 17
100101 + 10101 = 1010000, aka 17 + 12 = 29
1000 - 101 = 1, aka 5 - 4 = 1
10101010 - 1010101 = 1000000, aka 54 - 33 = 21
1001 * 101 = 1000100, aka 6 * 4 = 24
101010 + 101 = 1000100, aka 20 + 4 = 24
10100 + 1010 = 101000, aka 11 + 7 = 18
101000 - 1010 = 10100, aka 18 - 7 = 11
100010 * 100101 = 100001000001, aka 15 * 17 = 255
100001000001 / 100 = 101010001 rem 0, aka 255 / 3 = 85 rem 0
101000101 * 101001 = 101010000010101, aka 80 * 19 = 1520
101010000010101 / 100 = 1001010001001 rem 10, aka 1520 / 3 = 506 rem 2
10100010010100 + 1001000001 = 100000000010101, aka 888 + 111 = 999
10100010010100 - 1001000001 = 10010001000010, aka 888 - 111 = 777
10000 * 1001000001 = 10100010010100, aka 8 * 111 = 888
1010001010000001001 / 100000000100000 = 10001 rem 10100001010101, aka 9876 / 1000 = 9 rem 876
10100 + 1010 = 101000, aka 11 + 7 = 18
10100 - 1010 = 101, aka 11 - 7 = 4
10100 * 1010 = 101000001, aka 11 * 7 = 77
10100 / 1010 = 1 rem 101, aka 11 / 7 = 1 rem 4
101000001 / 1010 = 10100 rem 0, aka 77 / 7 = 11 rem 0
```

## PicoLisp

```(seed (in "/dev/urandom" (rd 8)))

(de unpad (Lst)
(while (=0 (car Lst))
(pop 'Lst) )
Lst )

(de numz (N)
(let Fibs (1 1)
(while (>= N (+ (car Fibs) (cadr Fibs)))
(push 'Fibs (+ (car Fibs) (cadr Fibs))) )
(make
(for I (uniq Fibs)
(if (> I N)
(link 0)
(link 1)
(dec 'N I) ) ) ) ) )

(de znum (Lst)
(let Fibs (1 1)
(do (dec (length Lst))
(push 'Fibs (+ (car Fibs) (cadr Fibs))) )
(sum
'((X Y) (unless (=0 X) Y))
Lst
(uniq Fibs) ) ) )

(de incz (Lst)
(addz Lst (1)) )

(de decz (Lst)
(subz Lst (1)) )

(de addz (Lst1 Lst2)
(let Max (max (length Lst1) (length Lst2))
(reorg
(mapcar + (need Max Lst1 0) (need Max Lst2 0)) ) ) )

(de subz (Lst1 Lst2)
(use (@A @B)
(let
(Max (max (length Lst1) (length Lst2))
Lst (mapcar - (need Max Lst1 0) (need Max Lst2 0)) )
(loop
(while (match '(@A 1 0 0 @B) Lst)
(setq Lst (append @A (0 1 1) @B)) )
(while (match '(@A 1 -1 0 @B) Lst)
(setq Lst (append @A (0 0 1) @B)) )
(while (match '(@A 1 -1 1 @B) Lst)
(setq Lst (append @A (0 0 2) @B)) )
(while (match '(@A 1 0 -1 @B) Lst)
(setq Lst (append @A (0 1 0) @B)) )
(while (match '(@A 2 0 0 @B) Lst)
(setq Lst (append @A (1 1 1) @B)) )
(while (match '(@A 2 -1 0 @B) Lst)
(setq Lst (append @A (1 0 1) @B)) )
(while (match '(@A 2 -1 1 @B) Lst)
(setq Lst (append @A (1 0 2) @B)) )
(while (match '(@A 2 0 -1 @B) Lst)
(setq Lst (append @A (1 1 0) @B)) )
(while (match '(@A 1 -1) Lst)
(setq Lst (append @A (0 1))) )
(while (match '(@A 2 -1) Lst)
(setq Lst (append @A (1 1))) )
(NIL (match '(@A -1 @B) Lst)) )
(reorg (unpad Lst)) ) ) )

(de mulz (Lst1 Lst2)
(let (Sums (list Lst1) Mulz (0))
(mapc
'((X)
(when (= 1 (car X))
(setq Mulz (addz (cdr X) Mulz)) )
Mulz )
(mapcar
'((X)
(cons
X
(push 'Sums (addz (car Sums) (cadr Sums))) ) )
(reverse Lst2) ) ) ) )

(de divz (Lst1 Lst2)
(let Q 0
(while (lez Lst2 Lst1)
(setq Lst1 (subz Lst1 Lst2))
(setq Q (incz Q)) )
(list Q (or Lst1 (0))) ) )

(de reorg (Lst)
(use (@A @B)
(let Lst (reverse Lst)
(loop
(while (match '(@A 1 1 @B) Lst)
(if @B
(inc (nth @B 1))
(setq @B (1)) )
(setq Lst (append @A (0 0) @B) ) )
(while (match '(@A 2 @B) Lst)
(inc
(if (cdr @A)
(tail 2 @A)
@A ) )
(if @B
(inc (nth @B 1))
(setq @B (1)) )
(setq Lst (append @A (0) @B)) )
(NIL
(or
(match '(@A 1 1 @B) Lst)
(match '(@A 2 @B) Lst) ) ) )
(reverse Lst) ) ) )

(de lez (Lst1 Lst2)
(let Max (max (length Lst1) (length Lst2))
(<= (need Max Lst1 0) (need Max Lst2 0)) ) )

(let (X 0 Y 0)
(do 1024
(setq X (rand 1 1024))
(setq Y (rand 1 1024))
(test (numz (+ X Y)) (addz (numz X) (numz Y)))
(test (numz (* X Y)) (mulz (numz X) (numz Y)))
(test (numz (+ X 1)) (incz (numz X))) )

(do 1024
(setq X (rand 129 1024))
(setq Y (rand 1 128))
(test (numz (- X Y)) (subz (numz X) (numz Y)))
(test (numz (/ X Y)) (car (divz (numz X) (numz Y))))
(test (numz (% X Y)) (cadr (divz (numz X) (numz Y))))
(test (numz (- X 1)) (decz (numz X))) ) )

(bye)
```

## Python

```import copy

class Zeckendorf:
def __init__(self, x='0'):
q = 1
i = len(x) - 1
self.dLen = int(i / 2)
self.dVal = 0
while i >= 0:
self.dVal = self.dVal + (ord(x[i]) - ord('0')) * q
q = q * 2
i = i -1

def a(self, n):
i = n
while True:
if self.dLen < i:
self.dLen = i
j = (self.dVal >> (i * 2)) & 3
if j == 0 or j == 1:
return
if j == 2:
if (self.dVal >> ((i + 1) * 2) & 1) != 1:
return
self.dVal = self.dVal + (1 << (i * 2 + 1))
return
if j == 3:
temp = 3 << (i * 2)
temp = temp ^ -1
self.dVal = self.dVal & temp
self.b((i + 1) * 2)
i = i + 1

def b(self, pos):
if pos == 0:
self.inc()
return
if (self.dVal >> pos) & 1 == 0:
self.dVal = self.dVal + (1 << pos)
self.a(int(pos / 2))
if pos > 1:
self.a(int(pos / 2) - 1)
else:
temp = 1 << pos
temp = temp ^ -1
self.dVal = self.dVal & temp
self.b(pos + 1)
self.b(pos - (2 if pos > 1 else 1))

def c(self, pos):
if (self.dVal >> pos) & 1 == 1:
temp = 1 << pos
temp = temp ^ -1
self.dVal = self.dVal & temp
return
self.c(pos + 1)
if pos > 0:
self.b(pos - 1)
else:
self.inc()

def inc(self):
self.dVal = self.dVal + 1
self.a(0)

def __add__(self, rhs):
copy = self
rhs_dVal = rhs.dVal
limit = (rhs.dLen + 1) * 2
for gn in range(0, limit):
if ((rhs_dVal >> gn) & 1) == 1:
copy.b(gn)
return copy

def __sub__(self, rhs):
copy = self
rhs_dVal = rhs.dVal
limit = (rhs.dLen + 1) * 2
for gn in range(0, limit):
if (rhs_dVal >> gn) & 1 == 1:
copy.c(gn)
while (((copy.dVal >> ((copy.dLen * 2) & 31)) & 3) == 0) or (copy.dLen == 0):
copy.dLen = copy.dLen - 1
return copy

def __mul__(self, rhs):
na = copy.deepcopy(rhs)
nb = copy.deepcopy(rhs)
nr = Zeckendorf()
dVal = self.dVal
for i in range(0, (self.dLen + 1) * 2):
if ((dVal >> i) & 1) > 0:
nr = nr + nb
nt = copy.deepcopy(nb)
nb = nb + na
na = copy.deepcopy(nt)
return nr

def __str__(self):
dig = ["00", "01", "10"]
dig1 = ["", "1", "10"]

if self.dVal == 0:
return '0'
idx = (self.dVal >> ((self.dLen * 2) & 31)) & 3
sb = dig1[idx]
i = self.dLen - 1
while i >= 0:
idx = (self.dVal >> (i * 2)) & 3
sb = sb + dig[idx]
i = i - 1
return sb

# main
print "Addition:"
g = Zeckendorf("10")
g = g + Zeckendorf("10")
print g
g = g + Zeckendorf("10")
print g
g = g + Zeckendorf("1001")
print g
g = g + Zeckendorf("1000")
print g
g = g + Zeckendorf("10101")
print g
print

print "Subtraction:"
g = Zeckendorf("1000")
g = g - Zeckendorf("101")
print g
g = Zeckendorf("10101010")
g = g - Zeckendorf("1010101")
print g
print

print "Multiplication:"
g = Zeckendorf("1001")
g = g * Zeckendorf("101")
print g
g = Zeckendorf("101010")
g = g + Zeckendorf("101")
print g
```
Output:
```Addition:
101
1001
10101
100101
1010000

Subtraction:
1
1000000

Multiplication:
1000100
1000100```

## Quackery

Unsigned (non-negative) Zeckendorf arithmetic.

Implements the required functions; addition, subtraction, multiplication and division, the optional decrement, increment and comparative functions, and additionally double and modulus, since they come for free with addition and division respectively.

The algorithms are of my own devising, without reference to the description in the task or existing research, so are potentially novel, but probably not.

I really should have taken notes as I was going along, so here is the hand-wavy explanation:

Mostly Zeckendorf numbers are represented bitwise to benefit from the inherent parallelism of bitwise logic, but occasionally as nests (the Quackery name for dynamic arrays) of 0s and 1s for ease of coding.

The word `canonise` puts a Zecekendorf number in canonical form; no two adjacent bits are set to 1, runs of 0s are allowed. The converse operation, `defrock` puts a number as far from canonical form as possible; no two adjacent bits are set to 0, runs of 1s are allowed. Despite their similarities they are coded quite differently as there was a long gap between coding one and then the other and as noted above, I didn't take notes.

Addition works by isolating the bits in both arguments that are set to 1, removing them from both and then bitwise xoring them together and canonising. After this the isolated bits are doubled and these two numbers (the xored number and the isolated bits number) are added. This is repeated until the xored number is 0. Doubling is achieved by shifting the number an appropriate distance left and right and adding the left shifted and right shifted numbers. `zadd` and `zdouble` are mutually recursive.

`2blit` separates the lowest two bits from a Zeckedorf number so that `zdouble` can treat them as a special case.

Multiplication is basically the Russian Peasant algorithm with the twist that instead of doubling we start with two instances of one of the multiplicands and repeatedly add them Fibonacci style.

Subtraction is implemented as difference (i.e. abs(a-b) as this is an unsigned implementation.) The process is to reduce both numbers in value until the smaller one equals zero. Continuing the naming theme established by `canonise` and `defrock`, the word that removes the bits that are set to 1 in both arguments is called `exorcise`. The appropriate sequence of exorcisms and defrockings will reduce the smaller argument to zero much of the time.

However, numbers which alternate 1s and 0s (e.g. ...01010101...) are immune to both canonisation and defrocking. When this occurs we add the smaller number to the larger number and double the smaller number and repeat the exorisms and defrockings. Extensive testing leads me to believe with a very high degree of confidence this is sufficient, but I have not proved it in a mathematical sense.

Division is basic binary long division with a twist; instead of multiplying the divisor by 2 until it's large enough, use it to make a fibonacci style sequence, except starting with a couple of copies of the divisor rather than 1s.

The while-again loop computes a nest of all the fibonacci multiples up to the dividend, the witheach loop tries subracting each one from largest to smallest and builds up the result accordingly. The remainder (modulus) comes for free as what is left at the end of all the subtractions.

To demonstrate that these words correctly implement Zeckendorf arithmetic, I have used them to implement Euclid's algorithm for Greatest Common Denominator, and used that to implement Largest Common Multiple. We repeatedly give `zlcm` two random numbers up to one quintillion (converted to Zeckendorf notation) and print the result (converted back to decimal), next to the same computation made using the conventional representation. `zgcd` and `zlcm` exercise the multiplication, division and modulus routines repeatedly, and those exercise the addition, subtraction and comparison routines.

`bin` is an extension to the Quackery compiler to allow it to understand numbers in binary notation (and hence also Zeckendorf notation).

`gcd` and `lcm` are defined at Least common multiple#Quackery, and `n->z` and `z->n` are defined at Zeckendorf number representation#Quackery.

```  [ nextword
dup \$ "" = if
[ \$ '"bin" needs to be followed by a string.'
message put bail ]
dup
2 base put
\$->n
base release
not if
[ drop
\$ " is not a binary number."
join message put
bail ]
nip
swap dip join ]         builds bin       ( [ \$ --> [ \$ )

[ ^ not ]                     is zeq       ( z z --> b   )

[ zeq not ]                   is zne       ( z z --> b   )

[ false unrot
[ 2dup zne while
rot drop
dup 1 & unrot
1 >> dip [ 1 >> ]
again ]
2drop ]                     is zlt       ( z z --> b   )

[ swap zlt ]                  is zgt       ( z z --> b   )

[ zlt not ]                   is zge       ( z z --> b   )

[ zgt not ]                   is zle       ( z z --> b   )

[ dup 1 << & 0 zeq ]          is canonical (   z --> b   )

[ [] swap
[ dup 1 & rot join
swap 1 >>
dup not until ]
drop ]                      is bits      (   z --> [   )

[ dup canonical if done
0 0 rot bits
witheach
[ |
[ table
[ 1 << 0 ]
[ 1 << 1 | bin 10 ]
[ 1 << 0 ]
[ 1 >> 1 |
bin 10 << 0 ] ]
do ]
drop again ]                is canonise  (   z --> z   )

[ dup bin -100
& swap bin 11 & ]           is 2blit     (   z --> z z )

[ 2blit bit | canonise ]      is zinc      (   z --> z   )

[ dup 0 zeq if
[ \$ "Cannot zdec zero."
fail ]
1
[ 2dup & if done
1 << again ]
tuck ^
swap 1 <<
[ bin 10 >>
tuck | swap
dup 0 zeq until ]
drop ]                      is zdec      (   z --> z   )

forward is zadd      (   z --> z   )

[ dup 2blit
[ table
0 bin 10 bin 101 ]
unrot bin 10 >>
swap 1 <<
rot | zadd ]                is zdouble   (   z --> z   )

[ 2dup ^ canonise
unrot &
dup 0 zeq iff
drop done
zdouble again ]       resolves zadd      ( z z --> z   )

[ tuck take zadd swap put ]   is ztally    ( z s -->     )

[ 0 temp put
dip dup
[ dup while
dup 1 & if
[ over temp ztally ]
dip [ tuck zadd ]
1 >> again ]
drop 2drop temp take ]      is zmult     ( z z --> z   )

[ 2dup & ~ tuck & dip & ]     is exorcise  ( z z --> z z )

[ dup
[ 0 ' [ 0 0 0 ] rot 1
[ 2dup > while
1 << again ]
1 <<
[ dup while
2swap 2over & 0 !=
dip
[ dup
' [ 1 0 0 ]
= if
[ drop
' [ 0 1 1 ] ] ]
join
behead
rot 1 << | swap
2swap 1 >> again ]
2drop
witheach
[ dip [ 1 << ] | ]
dup bin 111 &
bin 100 zeq if
[ bin -1000 &
bin 11 | ] ]
2dup zeq iff drop done
nip again ]               is defrock   (   z --> z   )

[ 2dup zlt if swap
dup 0 zeq iff drop done
[ exorcise dup while
dip defrock
exorcise dup while
dup dip zadd
zdouble
again ]
drop canonise ]             is zdiff     ( z z --> z   )

[ dup 0 zeq if
[ \$ "Z-division by zero."
fail ]
0 unrot swap
temp put
dup nested
[ dup 0 peek
tuck dip rot zadd
temp share
over zge while
swap join
again ]
drop nip
temp take
swap witheach
[ rot 1 << unrot
2dup zge iff
[ zdiff
dip [ 1 | ] ]
else drop ] ]           is zdivmod   ( z z --> z z )

[ zdivmod drop ]              is zdiv      ( z z --> z   )

[ zdivmod nip ]               is zmod      ( z z --> z   )

[ [ dup while
tuck zmod again ]
drop ]                      is zgcd      ( z z --> z   )

[ 2dup and iff
[ 2dup zgcd
zdiv zmult ]
else and ]                  is zlcm      ( z z --> z    )

10 times
[ 10 15 ** random
10 15 ** random
2dup lcm echo cr
n->z dip n->z
zlcm z->n echo cr cr ]```
Output:
```25624571429859946191396654570
25624571429859946191396654570

24702413608219494319878326100
24702413608219494319878326100

177592191573881063687998734000
177592191573881063687998734000

28221788451919578670971892845
28221788451919578670971892845

99008448632249766843573255321
99008448632249766843573255321

312648960463735816244223692220
312648960463735816244223692220

146093274904252809568841733264
146093274904252809568841733264

169485448104022309641359784180
169485448104022309641359784180

593337022246602746222083444716
593337022246602746222083444716

50904418052185753625716614402
50904418052185753625716614402
```

## Racket

This implementation only handles natural (non-negative numbers). The algorithms for addition and subtraction use the techniques explained in the paper "Efficient algorithms for Zeckendorf arithmetic" (http://arxiv.org/pdf/1207.4497.pdf).

```#lang racket (require math)

(define sqrt5 (sqrt 5))
(define phi (* 0.5 (+ 1 sqrt5)))

;; What is the nth fibonnaci number, shifted by 2 so that
;; F(0) = 1, F(1) = 2, ...?
;;
(define (F n)
(fibonacci (+ n 2)))

;; What is the largest n such that F(n) <= m?
;;
(define (F* m)
(let ([n (- (inexact->exact (round (/ (log (* m sqrt5)) (log phi)))) 2)])
(if (<= (F n) m) n (sub1 n))))

(define (zeck->natural z)
(for/sum ([i (reverse z)]
[j (in-naturals)])
(* i (F j))))

(define (natural->zeck n)
(if (zero? n)
null
(for/list ([i (in-range (F* n) -1 -1)])
(let ([f (F i)])
(cond [(>= n f) (set! n (- n f))
1]
[else 0])))))

; Extend list to the right to a length of len with repeated padding elements
;
(define (pad lst len [padding 0])
(append lst (make-list (- len (length lst)) padding)))

; Strip padding elements from the left of the list
;
(define (unpad lst [padding 0])
(cond [(null? lst) lst]
[(equal? (first lst) padding) (unpad (rest lst) padding)]
[else lst]))

;; Run a filter function across a window in a list from left to right
;;
(define (left->right width fn)
(λ (lst)
(let F ([a lst])
(if (< (length a) width)
a
(let ([f (fn (take a width))])
(cons (first f) (F (append (rest f) (drop a width)))))))))

;; Run a function fn across a window in a list from right to left
;;
(define (right->left width fn)
(λ (lst)
(let F ([a lst])
(if (< (length a) width)
a
(let ([f (fn (take-right a width))])
(append (F (append (drop-right a width) (drop-right f 1)))
(list (last f))))))))

;; (a0 a1 a2 ... an) -> (a0 a1 a2 ... (fn ... an))
;;
(define (replace-tail width fn)
(λ (lst)
(append (drop-right lst width) (fn (take-right lst width)))))

(define (rule-a lst)
(match lst
[(list 0 2 0 x) (list 1 0 0 (add1 x))]
[(list 0 3 0 x) (list 1 1 0 (add1 x))]
[(list 0 2 1 x) (list 1 1 0 x)]
[(list 0 1 2 x) (list 1 0 1 x)]
[else lst]))

(define (rule-a-tail lst)
(match lst
[(list x 0 3 0) (list x 1 1 1)]
[(list x 0 2 0) (list x 1 0 1)]
[(list 0 1 2 0) (list 1 0 1 0)]
[(list x y 0 3) (list x y 1 1)]
[(list x y 0 2) (list x y 1 0)]
[(list x 0 1 2) (list x 1 0 0)]
[else lst]))

(define (rule-b lst)
(match lst
[(list 0 1 1) (list 1 0 0)]
[else lst]))

(define (rule-c lst)
(match lst
[(list 1 0 0) (list 0 1 1)]
[(list 1 -1 0) (list 0 0 1)]
[(list 1 -1 1) (list 0 0 2)]
[(list 1 0 -1) (list 0 1 0)]
[(list 2 0 0) (list 1 1 1)]
[(list 2 -1 0) (list 1 0 1)]
[(list 2 -1 1) (list 1 0 2)]
[(list 2 0 -1) (list 1 1 0)]
[else lst]))

(define (zeck-combine op y z [f identity])
(let* ([bits (max (add1 (length y)) (add1 (length z)) 4)]
[f0 (λ (x) (pad (reverse x) bits))]
[f1 (left->right 4 rule-a)]
[f2 (replace-tail 4 rule-a-tail)]
[f3 (right->left 3 rule-b)]
[f4 (left->right 3 rule-b)])
((compose1 unpad f4 f3 f2 f1 f reverse) (map op (f0 y) (f0 z)))))

(define (zeck+ y z)
(zeck-combine + y z))

(define (zeck- y z)
(when (zeck< y z) (error (format "~a" `(zeck-: cannot subtract since ,y < ,z))))
(zeck-combine - y z (left->right 3 rule-c)))

(define (zeck* y z)
(define (M ry Zn Zn_1 [acc null])
(if (null? ry)
acc
(M (rest ry) (zeck+ Zn Zn_1) Zn
(if (zero? (first ry)) acc (zeck+ acc Zn)))))
(cond [(zeck< z y) (zeck* z y)]
[(null? y) null]               ; 0 * z -> 0
[else (M (reverse y) z z)]))

(define (zeck-quotient/remainder y z)
(define (M Zn acc)
(if (zeck< y Zn)
(drop-right acc 1)
(M (zeck+ Zn (first acc)) (cons Zn acc))))
(define (D x m [acc null])
(if (null? m)
(values (reverse acc) x)
(let* ([v (first m)]
[smaller (zeck< v x)]
[bit (if smaller 1 0)]
[x_ (if smaller (zeck- x v) x)])
(D x_ (rest m) (cons bit acc)))))
(D y (M z (list z))))

(define (zeck-quotient y z)
(let-values ([(quotient _) (zeck-quotient/remainder y z)])
quotient))

(define (zeck-remainder y z)
(let-values ([(_ remainder) (zeck-quotient/remainder y z)])
remainder))

(define (zeck-add1 z)
(zeck+ z '(1)))

(define (zeck= y z)
(equal? (unpad y) (unpad z)))

(define (zeck< y z)
; Compare equal-length unpadded zecks
(define (LT a b)
(if (null? a)
#f
(let ([a0 (first a)] [b0 (first b)])
(if (= a0 b0)
(LT (rest a) (rest b))
(= a0 0)))))

(let* ([a (unpad y)] [len-a (length a)]
[b (unpad z)] [len-b (length b)])
(cond [(< len-a len-b) #t]
[(> len-a len-b) #f]
[else (LT a b)])))

(define (zeck> y z)
(not (or (zeck= y z) (zeck< y z))))

;; Examples
;;
(define (example op-name op a b)
(let* ([y (natural->zeck a)]
[z (natural->zeck b)]
[x (op y z)]
[c (zeck->natural x)])
(printf "~a ~a ~a = ~a ~a ~a = ~a = ~a\n"
a op-name b y op-name z x c)))

(example '+ zeck+ 888 111)
(example '- zeck- 888 111)
(example '* zeck* 8 111)
(example '/ zeck-quotient 9876 1000)
(example '% zeck-remainder 9876 1000)
```
Output:
```888 + 111 = (1 0 1 0 0 0 1 0 0 1 0 1 0 0) + (1 0 0 1 0 0 0 0 0 1) = (1 0 0 0 0 0 0 0 0 0 1 0 1 0 1) = 999
888 - 111 = (1 0 1 0 0 0 1 0 0 1 0 1 0 0) - (1 0 0 1 0 0 0 0 0 1) = (1 0 0 1 0 0 0 1 0 0 0 0 1 0) = 777
8 * 111 = (1 0 0 0 0) * (1 0 0 1 0 0 0 0 0 1) = (1 0 1 0 0 0 1 0 0 1 0 1 0 0) = 888
9876 / 1000 = (1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0 1) / (1 0 0 0 0 0 0 0 0 1 0 0 0 0 0) = (1 0 0 0 1) = 9
9876 % 1000 = (1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0 1) % (1 0 0 0 0 0 0 0 0 1 0 0 0 0 0) = (1 0 1 0 0 0 0 1 0 1 0 1 0 1) = 876
```

## Raku

(formerly Perl 6)

This is a somewhat limited implementation of Zeckendorf arithmetic operators. They only handle positive integer values. There are no actual calculations, everything is done with string manipulations, so it doesn't matter what glyphs you use for 1 and 0.

Implemented arithmetic operators:

```  +z addition
-z subtraction
×z multiplication
/z division (more of a divmod really)
++z post increment
--z post decrement
```

Comparison operators:

``` eqz equal
nez not equal
gtz greater than
ltz less than
```
```my \$z1 = '1'; # glyph to use for a '1'
my \$z0 = '0'; # glyph to use for a '0'

sub zorder(\$a) { (\$z0 lt \$z1) ?? \$a !! \$a.trans([\$z0, \$z1] => [\$z1, \$z0]) }

######## Zeckendorf comparison operators #########

# less than
sub infix:<ltz>(\$a, \$b) { \$a.&zorder lt \$b.&zorder }

# greater than
sub infix:<gtz>(\$a, \$b) { \$a.&zorder gt \$b.&zorder }

# equal
sub infix:<eqz>(\$a, \$b) { \$a eq \$b }

# not equal
sub infix:<nez>(\$a, \$b) { \$a ne \$b }

######## Operators for Zeckendorf arithmetic ########

# post increment
sub postfix:<++z>(\$a is rw) {
\$a = ("\$z0\$z0"~\$a).subst(/("\$z0\$z0")(\$z1+ %% \$z0)?\$/,
-> \$/ { "\$z0\$z1" ~ (\$1 ?? \$z0 x \$1.chars !! '') });
\$a ~~ s/^\$z0+//;
\$a
}

# post decrement
sub postfix:<--z>(\$a is rw) {
\$a.=subst(/\$z1(\$z0*)\$/,
-> \$/ {\$z0 ~ "\$z1\$z0" x \$0.chars div 2 ~ \$z1 x \$0.chars mod 2});
\$a ~~ s/^\$z0+(.+)\$/\$0/;
\$a
}

# addition
sub infix:<+z>(\$a is copy, \$b is copy) { \$a++z; \$a++z while \$b--z nez \$z0; \$a }

# subtraction
sub infix:<-z>(\$a is copy, \$b is copy) { \$a--z; \$a--z while \$b--z nez \$z0; \$a }

# multiplication
sub infix:<×z>(\$a, \$b) {
return \$z0 if \$a eqz \$z0 or \$b eqz \$z0;
return \$a if \$b eqz \$z1;
return \$b if \$a eqz \$z1;
my \$c = \$a;
my \$d = \$z1;
repeat {
my \$e = \$z0;
repeat { \$c++z; \$e++z } until \$e eqz \$a;
\$d++z;
} until \$d eqz \$b;
\$c
}

# division  (really more of a div mod)
sub infix:</z>(\$a is copy, \$b is copy) {
fail "Divide by zero" if \$b eqz \$z0;
return \$a if \$a eqz \$z0 or \$b eqz \$z1;
my \$c = \$z0;
repeat {
my \$d = \$b +z (\$z1 ~ \$z0);
\$c++z;
\$a++z;
\$a--z while \$d--z nez \$z0
} until \$a ltz \$b;
\$c ~= " remainder \$a" if \$a nez \$z0;
\$c
}

###################### Testing ######################

# helper sub to translate constants into the particular glyphs you used
sub z(\$a) { \$a.trans([<1 0>] => [\$z1, \$z0]) }

say "Using the glyph '\$z1' for 1 and '\$z0' for 0\n";

my \$fmt = "%-22s = %15s  %s\n";

my \$zeck = \$z1;

printf( \$fmt, "\$zeck++z", \$zeck++z, '# increment' ) for 1 .. 10;

printf \$fmt, "\$zeck +z {z('1010')}", \$zeck +z= z('1010'), '# addition';

printf \$fmt, "\$zeck -z {z('100')}", \$zeck -z= z('100'), '# subtraction';

printf \$fmt, "\$zeck ×z {z('100101')}", \$zeck ×z= z('100101'), '# multiplication';

printf \$fmt, "\$zeck /z {z('100')}", \$zeck /z= z('100'), '# division';

printf( \$fmt, "\$zeck--z", \$zeck--z, '# decrement' ) for 1 .. 5;

printf \$fmt, "\$zeck ×z {z('101001')}", \$zeck ×z= z('101001'), '# multiplication';

printf \$fmt, "\$zeck /z {z('100')}", \$zeck /z= z('100'), '# division';
```

Testing Output

```Using the glyph '1' for 1 and '0' for 0

1++z                   =              10  # increment
10++z                  =             100  # increment
100++z                 =             101  # increment
101++z                 =            1000  # increment
1000++z                =            1001  # increment
1001++z                =            1010  # increment
1010++z                =           10000  # increment
10000++z               =           10001  # increment
10001++z               =           10010  # increment
10010++z               =           10100  # increment
10100 +z 1010          =          101000  # addition
101000 -z 100          =          100010  # subtraction
100010 ×z 100101       =    100001000001  # multiplication
100001000001 /z 100    =       101010001  # division
101010001--z           =       101010000  # decrement
101010000--z           =       101001010  # decrement
101001010--z           =       101001001  # decrement
101001001--z           =       101001000  # decrement
101001000--z           =       101000101  # decrement
101000101 ×z 101001    = 101010000010101  # multiplication
101010000010101 /z 100 = 1001010001001 remainder 10  # division```

Using alternate glyphs:

```Using the glyph 'X' for 1 and 'O' for 0

X++z                   =              XO  # increment
XO++z                  =             XOO  # increment
XOO++z                 =             XOX  # increment
XOX++z                 =            XOOO  # increment
XOOO++z                =            XOOX  # increment
XOOX++z                =            XOXO  # increment
XOXO++z                =           XOOOO  # increment
XOOOO++z               =           XOOOX  # increment
XOOOX++z               =           XOOXO  # increment
XOOXO++z               =           XOXOO  # increment
XOXOO +z XOXO          =          XOXOOO  # addition
XOXOOO -z XOO          =          XOOOXO  # subtraction
XOOOXO ×z XOOXOX       =    XOOOOXOOOOOX  # multiplication
XOOOOXOOOOOX /z XOO    =       XOXOXOOOX  # division
XOXOXOOOX--z           =       XOXOXOOOO  # decrement
XOXOXOOOO--z           =       XOXOOXOXO  # decrement
XOXOOXOXO--z           =       XOXOOXOOX  # decrement
XOXOOXOOX--z           =       XOXOOXOOO  # decrement
XOXOOXOOO--z           =       XOXOOOXOX  # decrement
XOXOOOXOX ×z XOXOOX    = XOXOXOOOOOXOXOX  # multiplication
XOXOXOOOOOXOXOX /z XOO = XOOXOXOOOXOOX remainder XO  # division```

## Scala

Works with: Scala version 2.13.1

The addition is an implementation of an algorithm suggested in http[:]//arxiv.org/pdf/1207.4497.pdf: Efficient Algorithms for Zeckendorf Arithmetic.

```import scala.collection.mutable.ListBuffer

object ZeckendorfArithmetic extends App {

val elapsed: (=> Unit) => Long = f => {
val s = System.currentTimeMillis

f

(System.currentTimeMillis - s) / 1000
}
val add: (Z, Z) => Z = (z1, z2) => z1 + z2
val subtract: (Z, Z) => Z = (z1, z2) => z1 - z2
val multiply: (Z, Z) => Z = (z1, z2) => z1 * z2
val divide: (Z, Z) => Option[Z] = (z1, z2) => z1 / z2
val modulo: (Z, Z) => Option[Z] = (z1, z2) => z1 % z2
val ops = Map(("+", add), ("-", subtract), ("*", multiply), ("/", divide), ("%", modulo))
val calcs = List(
(Z("101"), "+", Z("10100"))
, (Z("101"), "-", Z("10100"))
, (Z("101"), "*", Z("10100"))
, (Z("101"), "/", Z("10100"))
, (Z("-1010101"), "+", Z("10100"))
, (Z("-1010101"), "-", Z("10100"))
, (Z("-1010101"), "*", Z("10100"))
, (Z("-1010101"), "/", Z("10100"))
, (Z("1000101010"), "+", Z("10101010"))
, (Z("1000101010"), "-", Z("10101010"))
, (Z("1000101010"), "*", Z("10101010"))
, (Z("1000101010"), "/", Z("10101010"))
, (Z("10100"), "+", Z("1010"))
, (Z("100101"), "-", Z("100"))
, (Z("1010101010101010101"), "+", Z("-1010101010101"))
, (Z("1010101010101010101"), "-", Z("-1010101010101"))
, (Z("1010101010101010101"), "*", Z("-1010101010101"))
, (Z("1010101010101010101"), "/", Z("-1010101010101"))
, (Z("1010101010101010101"), "%", Z("-1010101010101"))
, (Z("1010101010101010101"), "+", Z("101010101010101"))
, (Z("1010101010101010101"), "-", Z("101010101010101"))
, (Z("1010101010101010101"), "*", Z("101010101010101"))
, (Z("1010101010101010101"), "/", Z("101010101010101"))
, (Z("1010101010101010101"), "%", Z("101010101010101"))
, (Z("10101010101010101010"), "+", Z("1010101010101010"))
, (Z("10101010101010101010"), "-", Z("1010101010101010"))
, (Z("10101010101010101010"), "*", Z("1010101010101010"))
, (Z("10101010101010101010"), "/", Z("1010101010101010"))
, (Z("10101010101010101010"), "%", Z("1010101010101010"))
, (Z("1010"), "%", Z("10"))
, (Z("1010"), "%", Z("-10"))
, (Z("-1010"), "%", Z("10"))
, (Z("-1010"), "%", Z("-10"))
, (Z("100"), "/", Z("0"))
, (Z("100"), "%", Z("0"))
)
val iadd: (BigInt, BigInt) => BigInt = (a, b) => a + b
val isub: (BigInt, BigInt) => BigInt = (a, b) => a - b

// just for result checking:

import Z._

val imul: (BigInt, BigInt) => BigInt = (a, b) => a * b
val idiv: (BigInt, BigInt) => Option[BigInt] = (a, b) => if (b == 0) None else Some(a / b)
val imod: (BigInt, BigInt) => Option[BigInt] = (a, b) => if (b == 0) None else Some(a % b)
val iops = Map(("+", iadd), ("-", isub), ("*", imul), ("/", idiv), ("%", imod))

case class Z(var zs: String) {

import Z._

require((zs.toSet -- Set('-', '0', '1') == Set()) && (!zs.contains("11")))

//--- fa(summand1.z,summand2.z) --------------------------
val fa: (BigInt, BigInt) => BigInt = (z1, z2) => {
val v = z1.toString.toCharArray.map(_.asDigit).reverse.padTo(5, 0).zipAll(z2.toString.toCharArray.map(_.asDigit).reverse, 0, 0)
val arr1 = (v.map(p => p._1 + p._2) :+ 0).reverse
(0 to arr1.length - 4) foreach { i => //stage1
val a = arr1.slice(i, i + 4).toList
val b = a.foldRight("")("" + _ + _) dropRight 1
val a1 = b match {
case "020" => List(1, 0, 0, a(3) + 1)
case "030" => List(1, 1, 0, a(3) + 1)
case "021" => List(1, 1, 0, a(3))
case "012" => List(1, 0, 1, a(3))
case _ => a
}
0 to 3 foreach { j => arr1(j + i) = a1(j) }
}
val arr2 = arr1.foldRight("")("" + _ + _)
.replace("0120", "1010").replace("030", "111").replace("003", "100").replace("020", "101")
.replace("003", "100").replace("012", "101").replace("021", "110")
.replace("02", "10").replace("03", "11")
.reverse.toArray
(0 to arr2.length - 3) foreach { i => //stage2, step1
val a = arr2.slice(i, i + 3).toList
val b = a.foldRight("")("" + _ + _)
val a1 = b match {
case "110" => List('0', '0', '1')
case _ => a
}
0 to 2 foreach { j => arr2(j + i) = a1(j) }
}
val arr3 = arr2.foldRight("")("" + _ + _).concat("0").reverse.toArray
(0 to arr3.length - 3) foreach { i => //stage2, step2
val a = arr3.slice(i, i + 3).toList
val b = a.foldRight("")("" + _ + _)
val a1 = b match {
case "011" => List('1', '0', '0')
case _ => a
}
0 to 2 foreach { j => arr3(j + i) = a1(j) }
}
BigInt(arr3.foldRight("")("" + _ + _))
}
//--- fs(minuend.z,subtrahend.z) -------------------------
val fs: (BigInt, BigInt) => BigInt = (min, sub) => {
val zmvr = min.toString.toCharArray.map(_.asDigit).reverse
val zsvr = sub.toString.toCharArray.map(_.asDigit).reverse.padTo(zmvr.length, 0)
val v = zmvr.zipAll(zsvr, 0, 0).reverse
val last = v.length - 1
val zma = zmvr.reverse.toArray

val zsa = zsvr.reverse.toArray
for (i <- (0 to last).reverse) {
val e = zma(i) - zsa(i)
if (e < 0) {
zma(i - 1) = zma(i - 1) - 1
zma(i) = 0
val part = Z(((i to last).map(zma(_))).foldRight("")("" + _ + _))
val carry = Z("1".padTo(last - i, "0").foldRight("")("" + _ + _))
val sum = part + carry

val sums = sum.z.toString
(1 to sum.size) foreach { j => zma(last - sum.size + j) = sums(j - 1).asDigit }
if (zma(i - 1) < 0) {
for (j <- (0 until i).reverse) {
if (zma(j) < 0) {
zma(j - 1) = zma(j - 1) - 1
zma(j) = 0
val part = Z(((j to last).map(zma(_))).foldRight("")("" + _ + _))
val carry = Z("1".padTo(last - j, "0").foldRight("")("" + _ + _))
val sum = part + carry

val sums = sum.z.toString
(1 to sum.size) foreach { k => zma(last - sum.size + k) = sums(k - 1).asDigit }
}
}
}
}
else zma(i) = e
zsa(i) = 0
}
BigInt(zma.foldRight("")("" + _ + _))
}
//--- fm(multiplicand.z,multplier.z) ---------------------
val fm: (BigInt, BigInt) => BigInt = (mc, mp) => {
val mct = mt(Z(mc.toString))
val mpxi = mp.toString.reverse.toCharArray.map(_.asDigit).zipWithIndex.filter(_._1 != 0).map(_._2)
mpxi.foldRight(Z("0"))((fi, sum) => sum + mct(fi)).z
}
//--- fd(dividend.z,divisor.z) ---------------------------
val fd: (BigInt, BigInt) => BigInt = (dd, ds) => {
val dst = dt(Z(dd.toString), Z(ds.toString)).reverse
var diff = Z(dd.toString)
val zd = ListBuffer[String]()
0 until dst.length foreach { i =>
if (dst(i) > diff) zd += "0" else {
diff = diff - dst(i)

zd += "1"
}
}
BigInt(zd.mkString)
}
val fasig: (Z, Z) => Int = (z1, z2) => if (z1.z.abs > z2.z.abs) z1.z.signum else z2.z.signum
val fssig: (Z, Z) => Int = (z1, z2) =>
if ((z1.z.abs > z2.z.abs && z1.z.signum > 0) || (z1.z.abs < z2.z.abs && z1.z.signum < 0)) 1 else -1
var z: BigInt = BigInt(zs)

override def toString: String = "" + z + "Z(i:" + z2i(this) + ")"

def size: Int = z.abs.toString.length

def ++ : Z = {
val za = this + Z("1")

this.zs = za.zs

this.z = za.z

this
}

def +(that: Z): Z =
if (this == Z("0")) that
else if (that == Z("0")) this
else if (this.z.signum == that.z.signum) Z((fa(this.z.abs.max(that.z.abs), this.z.abs.min(that.z.abs)) * this.z.signum).toString)
else if (this.z.abs == that.z.abs) Z("0")
else Z((fs(this.z.abs.max(that.z.abs), this.z.abs.min(that.z.abs)) * fasig(this, that)).toString)

def -- : Z = {
val zs = this - Z("1")

this.zs = zs.zs

this.z = zs.z

this
}

def -(that: Z): Z =
if (this == Z("0")) Z((that.z * (-1)).toString)
else if (that == Z("0")) this
else if (this.z.signum != that.z.signum) Z((fa(this.z.abs.max(that.z.abs), this.z.abs.min(that.z.abs)) * this.z.signum).toString)
else if (this.z.abs == that.z.abs) Z("0")
else Z((fs(this.z.abs.max(that.z.abs), this.z.abs.min(that.z.abs)) * fssig(this, that)).toString)

def %(that: Z): Option[Z] =
if (that == Z("0")) None
else if (this == Z("0")) Some(Z("0"))
else if (that == Z("1")) Some(Z("0"))
else if (this.z.abs < that.z.abs) Some(this)
else if (this.z == that.z) Some(Z("0"))
else this / that match {
case None => None

case Some(z) => Some(this - z * that)
}

def *(that: Z): Z =
if (this == Z("0") || that == Z("0")) Z("0")
else if (this == Z("1")) that
else if (that == Z("1")) this
else Z((fm(this.z.abs.max(that.z.abs), this.z.abs.min(that.z.abs)) * this.z.signum * that.z.signum).toString)

def /(that: Z): Option[Z] =
if (that == Z("0")) None
else if (this == Z("0")) Some(Z("0"))
else if (that == Z("1")) Some(Z("1"))
else if (this.z.abs < that.z.abs) Some(Z("0"))
else if (this.z == that.z) Some(Z("1"))
else Some(Z((fd(this.z.abs.max(that.z.abs), this.z.abs.min(that.z.abs)) * this.z.signum * that.z.signum).toString))

def <(that: Z): Boolean = this.z < that.z

def <=(that: Z): Boolean = this.z <= that.z

def >(that: Z): Boolean = this.z > that.z

def >=(that: Z): Boolean = this.z >= that.z

}

object Z {
// only for comfort and result checking:
val fibs: LazyList[BigInt] = {
def series(i: BigInt, j: BigInt): LazyList[BigInt] = i #:: series(j, i + j)

series(1, 0).tail.tail.tail
}
val z2i: Z => BigInt = z => z.z.abs.toString.toCharArray.map(_.asDigit).reverse.zipWithIndex.map { case (v, i) => v * fibs(i) }.foldRight(BigInt(0))(_ + _) * z.z.signum

var fmts: Map[Z, List[Z]] = Map(Z("0") -> List[Z](Z("0"))) //map of Fibonacci multiples table of divisors

// get division table (division weight vector)
def dt(dd: Z, ds: Z): List[Z] = {
val wv = new ListBuffer[Z]
wv ++= mt(ds)
var zs = ds.z.abs.toString
val upper = dd.z.abs.toString
while ((zs.length < upper.length)) {
wv += (wv.toList.last + wv.toList.reverse.tail.head)

zs = "1" + zs
}
wv.toList
}

// get multiply table from fmts
def mt(z: Z): List[Z] = {
fmts.getOrElse(z, Nil) match {
case Nil =>
val e = mwv(z)
fmts = fmts + (z -> e)
e
case l => l
}
}

// multiply weight vector
def mwv(z: Z): List[Z] = {
val wv = new ListBuffer[Z]
wv += z
wv += (z + z)
var zs = "11"
val upper = z.z.abs.toString
while ((zs.length < upper.length)) {
wv += (wv.toList.last + wv.toList.reverse.tail.head)

zs = "1" + zs
}
wv.toList
}
}

println("elapsed time: " + elapsed {
calcs foreach { case (op1, op, op2) => println("" + op1 + " " + op + " " + op2 + " = "
+ {
(ops(op)) (op1, op2) match {
case None => None

case Some(z) => z

case z => z
}
}
.ensuring { x =>
(iops(op)) (z2i(op1), z2i(op2)) match {
case None => None == x

case Some(i) => i == z2i(x.asInstanceOf[Z])

case i => i == z2i(x.asInstanceOf[Z])
}
})
}
} + " sec"
)

}
```

Output:

```101Z(i:4) + 10100Z(i:11) = 100010Z(i:15)
101Z(i:4) - 10100Z(i:11) = -1010Z(i:-7)
101Z(i:4) * 10100Z(i:11) = 10010010Z(i:44)
101Z(i:4) / 10100Z(i:11) = 0Z(i:0)
-1010101Z(i:-33) + 10100Z(i:11) = -1000001Z(i:-22)
-1010101Z(i:-33) - 10100Z(i:11) = -10010010Z(i:-44)
-1010101Z(i:-33) * 10100Z(i:11) = -101010001010Z(i:-363)
-1010101Z(i:-33) / 10100Z(i:11) = -100Z(i:-3)
1000101010Z(i:109) + 10101010Z(i:54) = 10000101001Z(i:163)
1000101010Z(i:109) - 10101010Z(i:54) = 100000000Z(i:55)
1000101010Z(i:109) * 10101010Z(i:54) = 101000001000101001Z(i:5886)
1000101010Z(i:109) / 10101010Z(i:54) = 10Z(i:2)
10100Z(i:11) + 1010Z(i:7) = 101000Z(i:18)
100101Z(i:17) - 100Z(i:3) = 100001Z(i:14)
1010101010101010101Z(i:10945) + -1010101010101Z(i:-609) = 1010100000000000000Z(i:10336)
1010101010101010101Z(i:10945) - -1010101010101Z(i:-609) = 10000001010101010100Z(i:11554)
1010101010101010101Z(i:10945) * -1010101010101Z(i:-609) = -100010001000001001010010100100001Z(i:-6665505)
1010101010101010101Z(i:10945) / -1010101010101Z(i:-609) = -100101Z(i:-17)
1010101010101010101Z(i:10945) % -1010101010101Z(i:-609) = 1010100100100Z(i:592)
1010101010101010101Z(i:10945) + 101010101010101Z(i:1596) = 10000101010101010100Z(i:12541)
1010101010101010101Z(i:10945) - 101010101010101Z(i:1596) = 1010000000000000000Z(i:9349)
1010101010101010101Z(i:10945) * 101010101010101Z(i:1596) = 10001000100001010001001010001001001Z(i:17468220)
1010101010101010101Z(i:10945) / 101010101010101Z(i:1596) = 1001Z(i:6)
1010101010101010101Z(i:10945) % 101010101010101Z(i:1596) = 101000000001000Z(i:1369)
10101010101010101010Z(i:17710) + 1010101010101010Z(i:2583) = 100001010101010101001Z(i:20293)
10101010101010101010Z(i:17710) - 1010101010101010Z(i:2583) = 10100000000000000000Z(i:15127)
10101010101010101010Z(i:17710) * 1010101010101010Z(i:2583) = 1000100010001000000000001000100010001Z(i:45744930)
10101010101010101010Z(i:17710) / 1010101010101010Z(i:2583) = 1001Z(i:6)
10101010101010101010Z(i:17710) % 1010101010101010Z(i:2583) = 1010000000001000Z(i:2212)
1010Z(i:7) % 10Z(i:2) = 1Z(i:1)
1010Z(i:7) % -10Z(i:-2) = 1Z(i:1)
-1010Z(i:-7) % 10Z(i:2) = -1Z(i:-1)
-1010Z(i:-7) % -10Z(i:-2) = -1Z(i:-1)
100Z(i:3) / 0Z(i:0) = None
100Z(i:3) % 0Z(i:0) = None
elapsed time: 1 sec```

## Tcl

Translation of: Raku
```namespace eval zeckendorf {
# Want to use alternate symbols? Change these
variable zero "0"
variable one "1"

# Base operations: increment and decrement
proc zincr var {
upvar 1 \$var a
namespace upvar [namespace current] zero 0 one 1
if {![regsub "\$0\$" \$a \$1\$0 a]} {append a \$1}
while {[regsub "\$0\$1\$1" \$a "\$1\$0\$0" a]
|| [regsub "^\$1\$1" \$a "\$1\$0\$0" a]} {}
regsub ".\$" \$a "" a
return \$a
}
proc zdecr var {
upvar 1 \$var a
namespace upvar [namespace current] zero 0 one 1
regsub "^\$0+(.+)\$" [subst [regsub "\${1}(\$0*)\$" \$a "\$0\[
string repeat {\$1\$0} \[regsub -all .. {\\1} {} x]]\[
string repeat {\$1} \[expr {\\$x ne {}}]]"]
] {\1} a
return \$a
}

# Exported operations
proc eq {a b} {
expr {\$a eq \$b}
}
proc add {a b} {
variable zero
while {![eq \$b \$zero]} {
zincr a
zdecr b
}
return \$a
}
proc sub {a b} {
variable zero
while {![eq \$b \$zero]} {
zdecr a
zdecr b
}
return \$a
}
proc mul {a b} {
variable zero
variable one
if {[eq \$a \$zero] || [eq \$b \$zero]} {return \$zero}
if {[eq \$a \$one]} {return \$b}
if {[eq \$b \$one]} {return \$a}
set c \$a
while {![eq [zdecr b] \$zero]} {
set c [add \$c \$a]
}
return \$c
}
proc div {a b} {
variable zero
variable one
if {[eq \$b \$zero]} {error "div zero"}
if {[eq \$a \$zero] || [eq \$b \$one]} {return \$a}
set r \$zero
while {![eq \$a \$zero]} {
if {![eq \$a [add [set a [sub \$a \$b]] \$b]]} break
zincr r
}
return \$r
}
# Note that there aren't any ordering operations in this version

# Assemble into a coherent API
namespace export \[a-y\]*
namespace ensemble create
}
```

Demonstrating:

```puts [zeckendorf add "10100" "1010"]
puts [zeckendorf sub "10100" "1010"]
puts [zeckendorf mul "10100" "1010"]
puts [zeckendorf div "10100" "1010"]
puts [zeckendorf div [zeckendorf mul "10100" "1010"] "1010"]
```
Output:
```101000
101
101000001
1
10100
```

## Visual Basic .NET

Translation of: C#
```Imports System.Text

Module Module1

Class Zeckendorf
Implements IComparable(Of Zeckendorf)

Private Shared ReadOnly dig As String() = {"00", "01", "10"}
Private Shared ReadOnly dig1 As String() = {"", "1", "10"}

Private dVal As Integer = 0
Private dLen As Integer = 0

Public Sub New(Optional x As String = "0")
Dim q = 1
Dim i = x.Length - 1
dLen = i \ 2

Dim z = Asc("0")
While i >= 0
Dim a = Asc(x(i))
dVal += (a - z) * q
q *= 2
i -= 1
End While
End Sub

Private Sub A(n As Integer)
Dim i = n
While True
If dLen < i Then
dLen = i
End If
Dim j = (dVal >> (i * 2)) And 3
If j = 0 OrElse j = 1 Then
Return
ElseIf j = 2 Then
If ((dVal >> ((i + 1) * 2)) And 1) <> 1 Then
Return
End If
dVal += 1 << (i * 2 + 1)
Return
ElseIf j = 3 Then
Dim temp = 3 << (i * 2)
temp = temp Xor -1
dVal = dVal And temp
B((i + 1) * 2)
End If
i += 1
End While
End Sub

Private Sub B(pos As Integer)
If pos = 0 Then
Inc()
Return
End If
If ((dVal >> pos) And 1) = 0 Then
dVal += 1 << pos
A(pos \ 2)
If pos > 1 Then
A(pos \ 2 - 1)
End If
Else
Dim temp = 1 << pos
temp = temp Xor -1
dVal = dVal And temp
B(pos + 1)
B(pos - If(pos > 1, 2, 1))
End If
End Sub

Private Sub C(pos As Integer)
If ((dVal >> pos) And 1) = 1 Then
Dim temp = 1 << pos
temp = temp Xor -1
dVal = dVal And temp
Return
End If
C(pos + 1)
If pos > 0 Then
B(pos - 1)
Else
Inc()
End If
End Sub

Public Function Inc() As Zeckendorf
dVal += 1
A(0)
Return Me
End Function

Public Function Copy() As Zeckendorf
Dim z As New Zeckendorf With {
.dVal = dVal,
.dLen = dLen
}
Return z
End Function

Public Sub PlusAssign(other As Zeckendorf)
Dim gn = 0
While gn < (other.dLen + 1) * 2
If ((other.dVal >> gn) And 1) = 1 Then
B(gn)
End If
gn += 1
End While
End Sub

Public Sub MinusAssign(other As Zeckendorf)
Dim gn = 0
While gn < (other.dLen + 1) * 2
If ((other.dVal >> gn) And 1) = 1 Then
C(gn)
End If
gn += 1
End While
While (((dVal >> dLen * 2) And 3) = 0) OrElse dLen = 0
dLen -= 1
End While
End Sub

Public Sub TimesAssign(other As Zeckendorf)
Dim na = other.Copy
Dim nb = other.Copy
Dim nt As Zeckendorf
Dim nr As New Zeckendorf
Dim i = 0
While i < (dLen + 1) * 2
If ((dVal >> i) And 1) > 0 Then
nr.PlusAssign(nb)
End If
nt = nb.Copy
nb.PlusAssign(na)
na = nt.Copy
i += 1
End While
dVal = nr.dVal
dLen = nr.dLen
End Sub

Public Function CompareTo(other As Zeckendorf) As Integer Implements IComparable(Of Zeckendorf).CompareTo
Return dVal.CompareTo(other.dVal)
End Function

Public Overrides Function ToString() As String
If dVal = 0 Then
Return "0"
End If

Dim idx = (dVal >> (dLen * 2)) And 3
Dim sb As New StringBuilder(dig1(idx))
Dim i = dLen - 1
While i >= 0
idx = (dVal >> (i * 2)) And 3
sb.Append(dig(idx))
i -= 1
End While
Return sb.ToString
End Function
End Class

Sub Main()
Console.WriteLine("Addition:")
Dim g As New Zeckendorf("10")
g.PlusAssign(New Zeckendorf("10"))
Console.WriteLine(g)
g.PlusAssign(New Zeckendorf("10"))
Console.WriteLine(g)
g.PlusAssign(New Zeckendorf("1001"))
Console.WriteLine(g)
g.PlusAssign(New Zeckendorf("1000"))
Console.WriteLine(g)
g.PlusAssign(New Zeckendorf("10101"))
Console.WriteLine(g)
Console.WriteLine()

Console.WriteLine("Subtraction:")
g = New Zeckendorf("1000")
g.MinusAssign(New Zeckendorf("101"))
Console.WriteLine(g)
g = New Zeckendorf("10101010")
g.MinusAssign(New Zeckendorf("1010101"))
Console.WriteLine(g)
Console.WriteLine()

Console.WriteLine("Multiplication:")
g = New Zeckendorf("1001")
g.TimesAssign(New Zeckendorf("101"))
Console.WriteLine(g)
g = New Zeckendorf("101010")
g.PlusAssign(New Zeckendorf("101"))
Console.WriteLine(g)
End Sub

End Module
```
Output:
```Addition:
101
1001
10101
100101
1010000

Subtraction:
1
1000000

Multiplication:
1000100
1000100```

## V (Vlang)

Translation of: Go
```import strings
const (
dig  = ["00", "01", "10"]
dig1 = ["", "1", "10"]
)

struct Zeckendorf {
mut:
d_val int
d_len int
}

fn new_zeck(xx string) Zeckendorf {
mut z := Zeckendorf{}
mut x := xx
if x == "" {
x = "0"
}
mut q := 1
mut i := x.len - 1
z.d_len = i / 2
for ; i >= 0; i-- {
z.d_val += int(x[i]-'0'[0]) * q
q *= 2
}
return z
}

fn (mut z Zeckendorf) a(ii int) {
mut i:=ii
for ; ; i++ {
if z.d_len < i {
z.d_len = i
}
j := (z.d_val >> u32(i*2)) & 3
if j in [0, 1] {
return
} else if j==2 {
if ((z.d_val >> (u32(i+1) * 2)) & 1) != 1 {
return
}
z.d_val += 1 << u32(i*2+1)
return
} else {// 3
z.d_val &= ~(3 << u32(i*2))
z.b((i + 1) * 2)
}
}
}

fn (mut z Zeckendorf) b(p int) {
mut pos := p
if pos == 0 {
z.inc()
return
}
if ((z.d_val >> u32(pos)) & 1) == 0 {
z.d_val += 1 << u32(pos)
z.a(pos / 2)
if pos > 1 {
z.a(pos/2 - 1)
}
} else {
z.d_val &= ~(1 << u32(pos))
z.b(pos + 1)
mut temp := 1
if pos > 1 {
temp = 2
}
z.b(pos - temp)
}
}

fn (mut z Zeckendorf) c(p int) {
mut pos := p
if ((z.d_val >> u32(pos)) & 1) == 1 {
z.d_val &= ~(1 << u32(pos))
return
}
z.c(pos + 1)
if pos > 0 {
z.b(pos - 1)
} else {
z.inc()
}
}

fn (mut z Zeckendorf) inc() {
z.d_val++
z.a(0)
}

fn (mut z1 Zeckendorf) plus_assign(z2 Zeckendorf) {
for gn := 0; gn < (z2.d_len+1)*2; gn++ {
if ((z2.d_val >> u32(gn)) & 1) == 1 {
z1.b(gn)
}
}
}

fn (mut z1 Zeckendorf) minus_assign(z2 Zeckendorf) {
for gn := 0; gn < (z2.d_len+1)*2; gn++ {
if ((z2.d_val >> u32(gn)) & 1) == 1 {
z1.c(gn)
}
}

for z1.d_len > 0 && ((z1.d_val>>u32(z1.d_len*2))&3) == 0 {
z1.d_len--
}
}

fn (mut z1 Zeckendorf) times_assign(z2 Zeckendorf) {
mut na := z2.copy()
mut nb := z2.copy()
mut nr := Zeckendorf{}
for i := 0; i <= (z1.d_len+1)*2; i++ {
if ((z1.d_val >> u32(i)) & 1) > 0 {
nr.plus_assign(nb)
}
nt := nb.copy()
nb.plus_assign(na)
na = nt.copy()
}
z1.d_val = nr.d_val
z1.d_len = nr.d_len
}

fn (z Zeckendorf) copy() Zeckendorf {
return Zeckendorf{z.d_val, z.d_len}
}

fn (z1 Zeckendorf) compare(z2 Zeckendorf) int {
if z1.d_val < z2.d_val {
return -1
} else if z1.d_val > z2.d_val {
return 1
} else {
return 0
}
}

fn (z Zeckendorf) str() string {
if z.d_val == 0 {
return "0"
}
mut sb := strings.new_builder(128)
sb.write_string(dig1[(z.d_val>>u32(z.d_len*2))&3])
for i := z.d_len - 1; i >= 0; i-- {
sb.write_string(dig[(z.d_val>>u32(i*2))&3])
}
return sb.str()
}

fn main() {
println("Addition:")
mut g := new_zeck("10")
g.plus_assign(new_zeck("10"))
println(g)
g.plus_assign(new_zeck("10"))
println(g)
g.plus_assign(new_zeck("1001"))
println(g)
g.plus_assign(new_zeck("1000"))
println(g)
g.plus_assign(new_zeck("10101"))
println(g)

println("\nSubtraction:")
g = new_zeck("1000")
g.minus_assign(new_zeck("101"))
println(g)
g = new_zeck("10101010")
g.minus_assign(new_zeck("1010101"))
println(g)

println("\nMultiplication:")
g = new_zeck("1001")
g.times_assign(new_zeck("101"))
println(g)
g = new_zeck("101010")
g.plus_assign(new_zeck("101"))
println(g)
}```
Output:
```Addition:
101
1001
10101
100101
1010000

Subtraction:
1
1000000

Multiplication:
1000100
1000100
```

## Wren

Translation of: Kotlin
Library: Wren-trait
```import "/trait" for Comparable

class Zeckendorf is Comparable {
static dig  { ["00", "01", "10"] }
static dig1 { ["", "1", "10"] }

construct new(x) {
var q = 1
var i = x.count - 1
_dLen = (i / 2).floor
_dVal = 0
while (i >= 0) {
_dVal = _dVal + (x[i].bytes[0] - 48) * q
q = q * 2
i = i - 1
}
}

dLen { _dLen }
dVal { _dVal }

dLen=(v) { _dLen = v }
dVal=(v) { _dVal = v }

a(n) {
var i = n
while (true) {
if (_dLen < i) _dLen = i
var j = (_dVal >> (i * 2)) & 3
if (j == 0 || j == 1) return
if (j == 2) {
if (((_dVal >> ((i + 1) * 2)) & 1) != 1) return
_dVal = _dVal + (1 << (i * 2 + 1))
return
}
if (j == 3) {
_dVal = _dVal & ~(3 << (i * 2))
b((i + 1) * 2)
}
i = i + 1
}
}

b(pos) {
if (pos == 0) {
var thiz = this
thiz.inc
return
}
if (((_dVal >> pos) & 1) == 0) {
_dVal = _dVal + (1 << pos)
a((pos / 2).floor)
if (pos > 1) a((pos / 2).floor - 1)
} else {
_dVal = _dVal & ~(1 << pos)
b(pos + 1)
b(pos - ((pos > 1) ? 2 : 1))
}
}

c(pos) {
if (((_dVal >> pos) & 1) == 1) {
_dVal = _dVal & ~(1 << pos)
return
}
c(pos + 1)
if (pos > 0) {
b(pos - 1)
} else {
var thiz = this
thiz.inc
}
}

inc {
_dVal = _dVal + 1
a(0)
return this
}

plusAssign(other) {
for (gn in 0...(other.dLen + 1) * 2) {
if (((other.dVal >> gn) & 1) == 1) b(gn)
}
}

minusAssign(other) {
for (gn in 0...(other.dLen + 1) * 2) {
if (((other.dVal >> gn) & 1) == 1) c(gn)
}
while ((((_dVal >> _dLen * 2) & 3) == 0) || (_dLen == 0)) _dLen = _dLen - 1
}

timesAssign(other) {
var na = other.copy()
var nb = other.copy()
var nr = Zeckendorf.new("0")
for (i in 0..(_dLen + 1) * 2) {
if (((_dVal >> i) & 1) > 0) nr.plusAssign(nb)
var nt = nb.copy()
nb.plusAssign(na)
na = nt.copy()
}
_dVal = nr.dVal
_dLen = nr.dLen
}

compare(other) { (_dVal - other.dVal).sign }

toString {
if (_dVal == 0) return "0"
var sb = Zeckendorf.dig1[(_dVal >> (_dLen * 2)) & 3]
if (_dLen > 0) {
for (i in _dLen - 1..0) {
sb = sb + Zeckendorf.dig[(_dVal >> (i * 2)) & 3]
}
}
return sb
}

copy() {
var z = Zeckendorf.new("0")
z.dVal = _dVal
z.dLen = _dLen
return z
}
}

var Z = Zeckendorf // type alias
System.print("Addition:")
var g = Z.new("10")
g.plusAssign(Z.new("10"))
System.print(g)
g.plusAssign(Z.new("10"))
System.print(g)
g.plusAssign(Z.new("1001"))
System.print(g)
g.plusAssign(Z.new("1000"))
System.print(g)
g.plusAssign(Z.new("10101"))
System.print(g)
System.print("\nSubtraction:")
g = Z.new("1000")
g.minusAssign(Z.new("101"))
System.print(g)
g = Z.new("10101010")
g.minusAssign(Z.new("1010101"))
System.print(g)
System.print("\nMultiplication:")
g = Z.new("1001")
g.timesAssign(Z.new("101"))
System.print(g)
g = Z.new("101010")
g.plusAssign(Z.new("101"))
System.print(g)```
Output:
```Addition:
101
1001
10101
100101
1010000

Subtraction:
1
1000000

Multiplication:
1000100
1000100
```