# Arithmetic/Rational

(Redirected from Rational Arithmetic)
Arithmetic/Rational
You are encouraged to solve this task according to the task description, using any language you may know.

Create a reasonably complete implementation of rational arithmetic in the particular language using the idioms of the language.

Example

Define a new type called frac with binary operator "//" of two integers that returns a structure made up of the numerator and the denominator (as per a rational number).

Further define the appropriate rational unary operators abs and '-', with the binary operators for addition '+', subtraction '-', multiplication '×', division '/', integer division '÷', modulo division, the comparison operators (e.g. '<', '≤', '>', & '≥') and equality operators (e.g. '=' & '≠').

Define standard coercion operators for casting int to frac etc.

If space allows, define standard increment and decrement operators (e.g. '+:=' & '-:=' etc.).

Finally test the operators: Use the new type frac to find all perfect numbers less than 219 by summing the reciprocal of the factors.

## Action!

Calculations on a real Atari 8-bit computer take quite long time. It is recommended to use an emulator capable with increasing speed of Atari CPU.

```INCLUDE "D2:REAL.ACT" ;from the Action! Tool Kit

TYPE Frac=[INT num,den]

REAL half

PROC PrintFrac(Frac POINTER x)
PrintI(x.num) Put('/) PrintI(x.den)
RETURN

INT FUNC Gcd(INT a,b)
INT tmp

IF a<b THEN
tmp=a a=b b=tmp
FI

WHILE b#0
DO
tmp=a MOD b
a=b
b=tmp
OD
RETURN (a)

PROC Init(INT n,d Frac POINTER res)
IF d>0 THEN
res.num=n res.den=d
ELSEIF d<0 THEN
res.num=-n res.den=-d
ELSE
Print("Denominator cannot be zero!")
Break()
FI
RETURN

PROC Assign(Frac POINTER x,res)
Init(x.num,x.den,res)
RETURN

PROC Neg(Frac POINTER x,res)
Init(-x.num,x.den,res)
RETURN

PROC Inverse(Frac POINTER x,res)
Init(x.den,x.num)
RETURN

PROC Abs(Frac POINTER x,res)
IF x.num<0 THEN
Neg(x,res)
ELSE
Assign(x,res)
FI
RETURN

INT common,xDen,yDen

common=Gcd(x.den,y.den)
xDen=x.den/common
yDen=y.den/common
Init(x.num*yDen+y.num*xDen,xDen*y.den,res)
RETURN

PROC Sub(Frac POINTER x,y,res)
Frac n

RETURN

PROC Mult(Frac POINTER x,y,res)
Init(x.num*y.num,x.den*y.den,res)
RETURN

PROC Div(Frac POINTER x,y,res)
Frac i

Inverse(y,i) Mult(x,i,res)
RETURN

BYTE FUNC Greater(Frac POINTER x,y)
Frac diff

Sub(x,y,diff)
IF diff.num>0 THEN
RETURN (1)
FI
RETURN (0)

BYTE FUNC Less(Frac POINTER x,y)
RETURN (Greater(y,x))

BYTE FUNC GreaterEqual(Frac POINTER x,y)
Frac diff

Sub(x,y,diff)
IF diff.num>=0 THEN
RETURN (1)
FI
RETURN (0)

BYTE FUNC LessEqual(Frac POINTER x,y)
RETURN (GreaterEqual(y,x))

BYTE FUNC Equal(Frac POINTER x,y)
Frac diff

Sub(x,y,diff)
IF diff.num=0 THEN
RETURN (1)
FI
RETURN (0)

BYTE FUNC NotEqual(Frac POINTER x,y)
IF Equal(x,y) THEN
RETURN (0)
FI
RETURN (1)

INT FUNC Sqrt(INT x)
REAL r1,r2

IF x=0 THEN
RETURN (0)
FI
IntToReal(x,r1)
Power(r1,half,r2)
RETURN (RealToInt(r2))

PROC Main()
DEFINE MAXINT="32767"
INT i,f,max2
Frac sum,tmp1,tmp2,tmp3,one

Put(125) PutE() ;clear screen
ValR("0.5",half)
Init(1,1,one)
FOR i=2 TO MAXINT
DO
Init(1,i,sum) ;sum=1/i
max2=Sqrt(i)
FOR f=2 TO max2
DO
IF i MOD f=0 THEN
Init(1,f,tmp1)     ;tmp1=1/f
Init(f,i,tmp3)     ;tmp3=f/i
FI
OD

IF Equal(sum,one) THEN
PrintF("%I is perfect%E",i)
FI
OD
RETURN```
Output:
```6 is perfect
28 is perfect
496 is perfect
8128 is perfect
```

[This section is included from a subpage and should be edited there, not here.]

The generic package specification:

```generic
type Number is range <>;
package Generic_Rational is
type Rational is private;

function "abs"   (A : Rational) return Rational;
function "+"     (A : Rational) return Rational;
function "-"     (A : Rational) return Rational;
function Inverse (A : Rational) return Rational;

function "+" (A : Rational; B : Rational) return Rational;
function "+" (A : Rational; B : Number  ) return Rational;
function "+" (A : Number;   B : Rational) return Rational;

function "-" (A : Rational; B : Rational) return Rational;
function "-" (A : Rational; B : Number  ) return Rational;
function "-" (A : Number;   B : Rational) return Rational;

function "*" (A : Rational; B : Rational) return Rational;
function "*" (A : Rational; B : Number  ) return Rational;
function "*" (A : Number;   B : Rational) return Rational;

function "/" (A : Rational; B : Rational) return Rational;
function "/" (A : Rational; B : Number  ) return Rational;
function "/" (A : Number;   B : Rational) return Rational;
function "/" (A : Number;   B : Number)   return Rational;

function ">"  (A : Rational; B : Rational) return Boolean;
function ">"  (A : Number;   B : Rational) return Boolean;
function ">"  (A : Rational; B : Number)   return Boolean;

function "<"  (A : Rational; B : Rational) return Boolean;
function "<"  (A : Number;   B : Rational) return Boolean;
function "<"  (A : Rational; B : Number)   return Boolean;

function ">=" (A : Rational; B : Rational) return Boolean;
function ">=" (A : Number;   B : Rational) return Boolean;
function ">=" (A : Rational; B : Number)   return Boolean;

function "<=" (A : Rational; B : Rational) return Boolean;
function "<=" (A : Number;   B : Rational) return Boolean;
function "<=" (A : Rational; B : Number)   return Boolean;

function "="  (A : Number;   B : Rational) return Boolean;
function "="  (A : Rational; B : Number)   return Boolean;

function Numerator   (A : Rational) return Number;
function Denominator (A : Rational) return Number;

Zero : constant Rational;
One  : constant Rational;
private
type Rational is record
Numerator   : Number;
Denominator : Number;
end record;

Zero : constant Rational := (0, 1);
One  : constant Rational := (1, 1);
end Generic_Rational;
```

The package can be instantiated with any integer type. It provides rational numbers represented by a numerator and denominator cleaned from the common divisors. Mixed arithmetic of the base integer type and the rational type is supported. Division to zero raises Constraint_Error. The implementation of the specification above is as follows:

```package body Generic_Rational is

function GCD (A, B : Number) return Number is
begin
if A = 0 then
return B;
end if;
if B = 0 then
return A;
end if;
if A > B then
return GCD (B, A mod B);
else
return GCD (A, B mod A);
end if;
end GCD;

function Inverse (A : Rational) return Rational is
begin
if A.Numerator > 0 then
return (A.Denominator, A.Numerator);
elsif A.Numerator < 0 then
return (-A.Denominator, -A.Numerator);
else
raise Constraint_Error;
end if;
end Inverse;

function "abs" (A : Rational) return Rational is
begin
return (abs A.Numerator, A.Denominator);
end "abs";

function "+" (A : Rational) return Rational is
begin
return A;
end "+";

function "-" (A : Rational) return Rational is
begin
return (-A.Numerator, A.Denominator);
end "-";

function "+" (A : Rational; B : Rational) return Rational is
Common        : constant Number := GCD (A.Denominator, B.Denominator);
A_Denominator : constant Number := A.Denominator / Common;
B_Denominator : constant Number := B.Denominator / Common;
begin
return (A.Numerator * B_Denominator + B.Numerator * A_Denominator) /
(A_Denominator * B.Denominator);
end "+";

function "+" (A : Rational; B : Number) return Rational is
begin
return (A.Numerator + B * A.Denominator) / A.Denominator;
end "+";

function "+" (A : Number; B : Rational) return Rational is
begin
return B + A;
end "+";

function "-" (A : Rational; B : Rational) return Rational is
begin
return A + (-B);
end "-";

function "-" (A : Rational; B : Number) return Rational is
begin
return A + (-B);
end "-";

function "-" (A : Number; B : Rational) return Rational is
begin
return A + (-B);
end "-";

function "*" (A : Rational; B : Rational) return Rational is
begin
return (A.Numerator * B.Numerator) / (A.Denominator * B.Denominator);
end "*";

function "*" (A : Rational; B : Number) return Rational is
Common : constant Number := GCD (A.Denominator, abs B);
begin
return (A.Numerator * B / Common, A.Denominator / Common);
end "*";

function "*" (A : Number; B : Rational) return Rational is
begin
return B * A;
end "*";

function "/" (A : Rational; B : Rational) return Rational is
begin
return A * Inverse (B);
end "/";

function "/" (A : Rational; B : Number) return Rational is
Common : constant Number := GCD (abs A.Numerator, abs B);
begin
if B > 0 then
return (A.Numerator / Common, A.Denominator * (B / Common));
else
return ((-A.Numerator) / Common, A.Denominator * ((-B) / Common));
end if;
end "/";

function "/" (A : Number; B : Rational) return Rational is
begin
return Inverse (B) * A;
end "/";

function "/" (A : Number; B : Number) return Rational is
Common : constant Number := GCD (abs A, abs B);
begin
if B = 0 then
raise Constraint_Error;
elsif A = 0 then
return (0, 1);
elsif A > 0 xor B > 0 then
return (-(abs A / Common), abs B / Common);
else
return (abs A / Common, abs B / Common);
end if;
end "/";

function ">" (A, B : Rational) return Boolean is
Diff : constant Rational := A - B;
begin
return Diff.Numerator > 0;
end ">";

function ">" (A : Number; B : Rational) return Boolean is
Diff : constant Rational := A - B;
begin
return Diff.Numerator > 0;
end ">";

function ">" (A : Rational; B : Number) return Boolean is
Diff : constant Rational := A - B;
begin
return Diff.Numerator > 0;
end ">";

function "<" (A, B : Rational) return Boolean is
Diff : constant Rational := A - B;
begin
return Diff.Numerator < 0;
end "<";

function "<" (A : Number; B : Rational) return Boolean is
Diff : constant Rational := A - B;
begin
return Diff.Numerator < 0;
end "<";

function "<" (A : Rational; B : Number) return Boolean is
Diff : constant Rational := A - B;
begin
return Diff.Numerator < 0;
end "<";

function ">=" (A, B : Rational) return Boolean is
Diff : constant Rational := A - B;
begin
return Diff.Numerator >= 0;
end ">=";

function ">=" (A : Number; B : Rational) return Boolean is
Diff : constant Rational := A - B;
begin
return Diff.Numerator >= 0;
end ">=";

function ">=" (A : Rational; B : Number) return Boolean is
Diff : constant Rational := A - B;
begin
return Diff.Numerator >= 0;
end ">=";

function "<=" (A, B : Rational) return Boolean is
Diff : constant Rational := A - B;
begin
return Diff.Numerator <= 0;
end "<=";

function "<=" (A : Number; B : Rational) return Boolean is
Diff : constant Rational := A - B;
begin
return Diff.Numerator <= 0;
end "<=";

function "<=" (A : Rational; B : Number) return Boolean is
Diff : constant Rational := A - B;
begin
return Diff.Numerator <= 0;
end "<=";

function "=" (A : Number; B : Rational) return Boolean is
Diff : constant Rational := A - B;
begin
return Diff.Numerator = 0;
end "=";

function "=" (A : Rational; B : Number) return Boolean is
Diff : constant Rational := A - B;
begin
return Diff.Numerator = 0;
end "=";

function Numerator (A : Rational) return Number is
begin
return A.Numerator;
end Numerator;

function Denominator (A : Rational) return Number is
begin
return A.Denominator;
end Denominator;

end Generic_Rational;
```

The implementation uses solution of the greatest common divisor task. Here is the implementation of the test:

```with Ada.Numerics.Elementary_Functions;  use Ada.Numerics.Elementary_Functions;
with Generic_Rational;

procedure Test_Rational is
package Integer_Rational is new Generic_Rational (Integer);
use Integer_Rational;
begin
for Candidate in 2..2**15 loop
declare
Sum  : Rational := 1 / Candidate;
begin
for Divisor in 2..Integer (Sqrt (Float (Candidate))) loop
if Candidate mod Divisor = 0 then -- Factor is a divisor of Candidate
Sum := Sum + One / Divisor + Rational'(Divisor / Candidate);
end if;
end loop;
if Sum = 1 then
Put_Line (Integer'Image (Candidate) & " is perfect");
end if;
end;
end loop;
end Test_Rational;
```

The perfect numbers are searched by summing of the reciprocal of each of the divisors of a candidate except 1. This sum must be 1 for a perfect number.

Output:
``` 6 is perfect
28 is perfect
496 is perfect
8128 is perfect
```

## ALGOL 68

Works with: ALGOL 68 version Standard - no extensions to language used
Works with: ALGOL 68G version Any - tested with release mk15-0.8b.fc9.i386
``` MODE FRAC = STRUCT( INT num #erator#,  den #ominator#);
FORMAT frac repr = \$g(-0)"//"g(-0)\$;

PROC gcd = (INT a, b) INT: # greatest common divisor #
(a = 0 | b |: b = 0 | a |: ABS a > ABS b  | gcd(b, a MOD b) | gcd(a, b MOD a));

PROC lcm = (INT a, b)INT: # least common multiple #
a OVER gcd(a, b) * b;

PROC raise not implemented error = ([]STRING args)VOID: (
put(stand error, ("Not implemented error: ",args, newline));
stop
);

PRIO // = 9; # higher then the ** operator #
OP // = (INT num, den)FRAC: ( # initialise and normalise #
INT common = gcd(num, den);
IF den < 0 THEN
( -num OVER common, -den OVER common)
ELSE
( num OVER common, den OVER common)
FI
);

OP + = (FRAC a, b)FRAC: (
INT common = lcm(den OF a, den OF b);
FRAC result := ( common OVER den OF a * num OF a + common OVER den OF b * num OF b, common );
num OF result//den OF result
);

OP - = (FRAC a, b)FRAC: a + -b,
* = (FRAC a, b)FRAC: (
INT num = num OF a * num OF b,
den = den OF a * den OF b;
INT common = gcd(num, den);
(num OVER common) // (den OVER common)
);

OP /  = (FRAC a, b)FRAC: a * FRAC(den OF b, num OF b),# real division #
%  = (FRAC a, b)INT: ENTIER (a / b),               # integer divison #
%* = (FRAC a, b)FRAC: a/b - FRACINIT ENTIER (a/b), # modulo division #
** = (FRAC a, INT exponent)FRAC:
IF exponent >= 0 THEN
(num OF a ** exponent, den OF a ** exponent )
ELSE
(den OF a ** exponent, num OF a ** exponent )
FI;

OP REALINIT = (FRAC frac)REAL: num OF frac / den OF frac,
FRACINIT = (INT num)FRAC: num // 1,
FRACINIT = (REAL num)FRAC: (
# express real number as a fraction # # a future execise! #
raise not implemented error(("Convert a REAL to a FRAC","!"));
SKIP
);

OP <  = (FRAC a, b)BOOL: num OF (a - b) <  0,
>  = (FRAC a, b)BOOL: num OF (a - b) >  0,
<= = (FRAC a, b)BOOL: NOT ( a > b ),
>= = (FRAC a, b)BOOL: NOT ( a < b ),
=  = (FRAC a, b)BOOL: (num OF a, den OF a) = (num OF b, den OF b),
/= = (FRAC a, b)BOOL: (num OF a, den OF a) /= (num OF b, den OF b);

# Unary operators #
OP - = (FRAC frac)FRAC: (-num OF frac, den OF frac),
ABS = (FRAC frac)FRAC: (ABS num OF frac, ABS den OF frac),
ENTIER = (FRAC frac)INT: (num OF frac OVER den OF frac) * den OF frac;

COMMENT Operators for extended characters set, and increment/decrement:
OP +:= = (REF FRAC a, FRAC b)REF FRAC: ( a := a + b ),
+=: = (FRAC a, REF FRAC b)REF FRAC: ( b := a + b ),
-:= = (REF FRAC a, FRAC b)REF FRAC: ( a := a - b ),
*:= = (REF FRAC a, FRAC b)REF FRAC: ( a := a * b ),
/:= = (REF FRAC a, FRAC b)REF FRAC: ( a := a / b ),
%:= = (REF FRAC a, FRAC b)REF FRAC: ( a := FRACINIT (a % b) ),
%*:= = (REF FRAC a, FRAC b)REF FRAC: ( a := a %* b );

# OP aliases for extended character sets (eg: Unicode, APL, ALCOR and GOST 10859) #
OP ×  = (FRAC a, b)FRAC: a * b,
÷  = (FRAC a, b)INT: a OVER b,
÷× = (FRAC a, b)FRAC: a MOD b,
÷* = (FRAC a, b)FRAC: a MOD b,
%× = (FRAC a, b)FRAC: a MOD b,
≤  = (FRAC a, b)FRAC: a <= b,
≥  = (FRAC a, b)FRAC: a >= b,
≠  = (FRAC a, b)BOOL: a /= b,
↑  = (FRAC frac, INT exponent)FRAC: frac ** exponent,

÷×:= = (REF FRAC a, FRAC b)REF FRAC: ( a := a MOD b ),
%×:= = (REF FRAC a, FRAC b)REF FRAC: ( a := a MOD b ),
÷*:= = (REF FRAC a, FRAC b)REF FRAC: ( a := a MOD b );

# BOLD aliases for CPU that only support uppercase for 6-bit bytes  - wrist watches #
OP OVER = (FRAC a, b)INT: a % b,
MOD = (FRAC a, b)FRAC: a %*b,
LT = (FRAC a, b)BOOL: a <  b,
GT = (FRAC a, b)BOOL: a >  b,
LE = (FRAC a, b)BOOL: a <= b,
GE = (FRAC a, b)BOOL: a >= b,
EQ = (FRAC a, b)BOOL: a =  b,
NE = (FRAC a, b)BOOL: a /= b,
UP = (FRAC frac, INT exponent)FRAC: frac**exponent;

# the required standard assignment operators #
OP PLUSAB  = (REF FRAC a, FRAC b)REF FRAC: ( a +:= b ), # PLUS #
PLUSTO  = (FRAC a, REF FRAC b)REF FRAC: ( a +=: b ), # PRUS #
MINUSAB = (REF FRAC a, FRAC b)REF FRAC: ( a *:= b ),
DIVAB   = (REF FRAC a, FRAC b)REF FRAC: ( a /:= b ),
OVERAB  = (REF FRAC a, FRAC b)REF FRAC: ( a %:= b ),
MODAB   = (REF FRAC a, FRAC b)REF FRAC: ( a %*:= b );

END COMMENT
Example: searching for Perfect Numbers.
FRAC sum:= FRACINIT 0;
FORMAT perfect = \$b(" perfect!","")\$;

FOR i FROM 2 TO 2**19 DO
INT candidate := i;
FRAC sum := 1 // candidate;
REAL real sum := 1 / candidate;
FOR factor FROM 2 TO ENTIER sqrt(candidate) DO
IF candidate MOD factor = 0 THEN
sum :=  sum + 1 // factor + 1 // ( candidate OVER factor);
real sum +:= 1 / factor + 1 / ( candidate OVER factor)
FI
OD;
IF den OF sum  = 1 THEN
printf((\$"Sum of reciprocal factors of "g(-0)" = "g(-0)" exactly, about "g(0,real width) f(perfect)l\$,
candidate, ENTIER sum, real sum, ENTIER sum = 1))
FI
OD```
Output:
```Sum of reciprocal factors of 6 = 1 exactly, about 1.0000000000000000000000000001 perfect!
Sum of reciprocal factors of 28 = 1 exactly, about 1.0000000000000000000000000001 perfect!
Sum of reciprocal factors of 120 = 2 exactly, about 2.0000000000000000000000000002
Sum of reciprocal factors of 496 = 1 exactly, about 1.0000000000000000000000000001 perfect!
Sum of reciprocal factors of 672 = 2 exactly, about 2.0000000000000000000000000001
Sum of reciprocal factors of 8128 = 1 exactly, about 1.0000000000000000000000000001 perfect!
Sum of reciprocal factors of 30240 = 3 exactly, about 3.0000000000000000000000000002
Sum of reciprocal factors of 32760 = 3 exactly, about 3.0000000000000000000000000003
Sum of reciprocal factors of 523776 = 2 exactly, about 2.0000000000000000000000000005
```

## Arturo

Arturo comes with built-in support for rational numbers.

```a: to :rational [1 2]
b: to :rational [3 4]

print ["a:" a]
print ["b:" b]

print ["a + b  :" a + b]
print ["a - b  :" a - b]
print ["a * b  :" a * b]
print ["a / b  :" a / b]
print ["a // b :" a // b]
print ["a % b  :" a % b]

print ["reciprocal b:" reciprocal b]
print ["neg a:" neg a]

print ["pi ~=" to :rational 3.14]
```
Output:
```a: 1/2
b: 3/4
a + b  : 5/4
a - b  : -1/4
a * b  : 3/8
a / b  : 2/3
a // b : 2/3
a % b  : 1/2
reciprocal b: 4/3
neg a: -1/2
pi ~= 157/50```

## BBC BASIC

```      *FLOAT64
DIM frac{num, den}
DIM Sum{} = frac{}, Kf{} = frac{}, One{} = frac{}
One.num = 1 : One.den = 1

FOR n% = 2 TO 2^19-1
Sum.num = 1 : Sum.den = n%
FOR k% = 2 TO SQR(n%)
IF (n% MOD k%) = 0 THEN
Kf.num = 1 : Kf.den = k%
PROCnormalise(Sum{})
Kf.den = n% DIV k%
PROCnormalise(Sum{})
ENDIF
NEXT
IF FNeq(Sum{}, One{}) PRINT n% " is perfect"
NEXT n%
END

DEF PROCabs(a{}) : a.num = ABS(a.num) : ENDPROC
DEF PROCneg(a{}) : a.num = -a.num : ENDPROC

LOCAL t : t = a.den * b.den
a.num = a.num * b.den + b.num * a.den
a.den = t
ENDPROC

DEF PROCsub(a{}, b{})
LOCAL t : t = a.den * b.den
a.num = a.num * b.den - b.num * a.den
a.den = t
ENDPROC

DEF PROCmul(a{}, b{})
a.num *= b.num : a.den *= b.den
ENDPROC

DEF PROCdiv(a{}, b{})
a.num *= b.den : a.den *= b.num
ENDPROC

DEF FNeq(a{}, b{}) = a.num * b.den = b.num * a.den
DEF FNlt(a{}, b{}) = a.num * b.den < b.num * a.den
DEF FNgt(a{}, b{}) = a.num * b.den > b.num * a.den
DEF FNne(a{}, b{}) = a.num * b.den <> b.num * a.den
DEF FNle(a{}, b{}) = a.num * b.den <= b.num * a.den
DEF FNge(a{}, b{}) = a.num * b.den >= b.num * a.den

DEF PROCnormalise(a{})
LOCAL a, b, t
a = a.num : b = a.den
WHILE b <> 0
t = a
a = b
b = t - b * INT(t / b)
ENDWHILE
a.num /= a : a.den /= a
IF a.den < 0 a.num *= -1 : a.den *= -1
ENDPROC
```

Output:

```         6 is perfect
28 is perfect
496 is perfect
8128 is perfect
```

## C

C does not have overloadable operators. The following implementation does not define all operations so as to keep the example short. Note that the code passes around struct values instead of pointers to keep it simple, a practice normally avoided for efficiency reasons.

```#include <stdio.h>
#include <stdlib.h>
#define FMT "%lld"
typedef long long int fr_int_t;
typedef struct { fr_int_t num, den; } frac;

fr_int_t gcd(fr_int_t m, fr_int_t n)
{
fr_int_t t;
while (n) { t = n; n = m % n; m = t; }
return m;
}

frac frac_new(fr_int_t num, fr_int_t den)
{
frac a;
if (!den) {
printf("divide by zero: "FMT"/"FMT"\n", num, den);
abort();
}

int g = gcd(num, den);

if (g)	{ num /= g; den /= g; }
else	{ num = 0; den = 1;   }

if (den < 0) {
den = -den;
num = -num;
}
a.num = num; a.den = den;
return a;
}

#define BINOP(op, n, d) frac frac_##op(frac a, frac b) { return frac_new(n,d); }
BINOP(add, a.num * b.den + b.num * a.den, a.den * b.den);
BINOP(sub, a.num * b.den - b.num + a.den, a.den * b.den);
BINOP(mul, a.num * b.num, a.den * b.den);
BINOP(div, a.num * b.den, a.den * b.num);

int frac_cmp(frac a, frac b) {
int l = a.num * b.den, r = a.den * b.num;
return l < r ? -1 : l > r;
}
#define frac_cmp_int(a, b) frac_cmp(a, frac_new(b, 1))
int frtoi(frac a) { return a.den / a.num; }
double frtod(frac a) { return (double)a.den / a.num; }

int main()
{
int n, k;
frac sum, kf;

for (n = 2; n < 1<<19; n++) {
sum = frac_new(1, n);

for (k = 2; k * k < n; k++) {
if (n % k) continue;
kf = frac_new(1, k);

kf = frac_new(1, n / k);
}
if (frac_cmp_int(sum, 1) == 0) printf("%d\n", n);
}

return 0;
}
```

## C#

[This section is included from a subpage and should be edited there, not here.]
```using System;

struct Fraction : IEquatable<Fraction>, IComparable<Fraction>
{

public Fraction(long num, long denom)
{
if (num == 0)
{
denom = 1;
}
else if (denom == 0)
{
throw new ArgumentException("Denominator may not be zero", "denom");
}
else if (denom < 0)
{
num = -num;
denom = -denom;
}

long d = GCD(num, denom);
this.Num = num / d;
this.Denom = denom / d;
}

private static long GCD(long x, long y)
{
return y == 0 ? x : GCD(y, x % y);
}

private static long LCM(long x, long y)
{
return x / GCD(x, y) * y;
}

public Fraction Abs()
{
return new Fraction(Math.Abs(Num), Denom);
}

public Fraction Reciprocal()
{
return new Fraction(Denom, Num);
}

#region Conversion Operators

public static implicit operator Fraction(long i)
{
return new Fraction(i, 1);
}

public static explicit operator double(Fraction f)
{
return f.Num == 0 ? 0 : (double)f.Num / f.Denom;
}

#endregion

#region Arithmetic Operators

public static Fraction operator -(Fraction f)
{
return new Fraction(-f.Num, f.Denom);
}

public static Fraction operator +(Fraction a, Fraction b)
{
long m = LCM(a.Denom, b.Denom);
long na = a.Num * m / a.Denom;
long nb = b.Num * m / b.Denom;
return new Fraction(na + nb, m);
}

public static Fraction operator -(Fraction a, Fraction b)
{
return a + (-b);
}

public static Fraction operator *(Fraction a, Fraction b)
{
return new Fraction(a.Num * b.Num, a.Denom * b.Denom);
}

public static Fraction operator /(Fraction a, Fraction b)
{
return a * b.Reciprocal();
}

public static Fraction operator %(Fraction a, Fraction b)
{
long l = a.Num * b.Denom, r = a.Denom * b.Num;
long n = l / r;
return new Fraction(l - n * r, a.Denom * b.Denom);
}

#endregion

#region Comparison Operators

public static bool operator ==(Fraction a, Fraction b)
{
return a.Num == b.Num && a.Denom == b.Denom;
}

public static bool operator !=(Fraction a, Fraction b)
{
return a.Num != b.Num || a.Denom != b.Denom;
}

public static bool operator <(Fraction a, Fraction b)
{
return (a.Num * b.Denom) < (a.Denom * b.Num);
}

public static bool operator >(Fraction a, Fraction b)
{
return (a.Num * b.Denom) > (a.Denom * b.Num);
}

public static bool operator <=(Fraction a, Fraction b)
{
return !(a > b);
}

public static bool operator >=(Fraction a, Fraction b)
{
return !(a < b);
}

#endregion

#region Object Members

public override bool Equals(object obj)
{
if (obj is Fraction)
return ((Fraction)obj) == this;
else
return false;
}

public override int GetHashCode()
{
return Num.GetHashCode() ^ Denom.GetHashCode();
}

public override string ToString()
{
return Num.ToString() + "/" + Denom.ToString();
}

#endregion

#region IEquatable<Fraction> Members

public bool Equals(Fraction other)
{
return other == this;
}

#endregion

#region IComparable<Fraction> Members

public int CompareTo(Fraction other)
{
return (this.Num * other.Denom).CompareTo(this.Denom * other.Num);
}

#endregion
}
```

Test program:

```using System;

static class Program
{
static void Main(string[] args)
{
int max = 1 << 19;
for (int candidate = 2; candidate < max; candidate++)
{
Fraction sum = new Fraction(1, candidate);
int max2 = (int)Math.Sqrt(candidate);
for (int factor = 2; factor <= max2; factor++)
{
if (candidate % factor == 0)
{
sum += new Fraction(1, factor);
sum += new Fraction(1, candidate / factor);
}
}

if (sum == 1)
Console.WriteLine("{0} is perfect", candidate);
}
}
}
```
Output:
```6 is perfect
28 is perfect
496 is perfect
8128 is perfect```

## C++

Library: Boost

Boost provides a rational number template.

```#include <iostream>
#include "math.h"
#include "boost/rational.hpp"

typedef  boost::rational<int> frac;

bool is_perfect(int c)
{
frac sum(1, c);
for (int f = 2;f < sqrt(static_cast<float>(c)); ++f){

if (c % f == 0) sum += frac(1,f) + frac(1, c/f);
}
if (sum.denominator() == 1){
return (sum == 1);
}
return false;
}

int main()
{
for (int candidate = 2; candidate < 0x80000; ++candidate){
if (is_perfect(candidate))
std::cout << candidate << " is perfect" << std::endl;
}
return 0;
}
```

### Without using external libraries

```#include <cmath>
#include <cstdint>
#include <iostream>
#include <numeric>
#include <stdexcept>

class Rational {
public:
/// Constructors ///
Rational() : numer(0), denom(1) {}

Rational(const int64_t number) : numer(number), denom(1) {}

Rational(const int64_t& numerator, const int64_t& denominator) : numer(numerator), denom(denominator) {
if ( numer == 0 ) {
denom = 1;
} else if ( denom == 0 ) {
throw std::invalid_argument("Denominator cannot be zero: " + denom);
} else if ( denom < 0 ) {
numer = -numer;
denom = -denom;
}

int64_t divisor = std::gcd(numerator, denom);
numer = numer / divisor;
denom = denom / divisor;
}

Rational(const Rational& other) : numer(other.numer), denom(other.denom) {}

/// Operators ///
Rational& operator=(const Rational& other) {
if ( *this != other ) { numer = other.numer; denom = other.denom; }
return *this;
}

bool operator!=(const Rational& other) const { return ! ( *this == other ); }

bool operator==(const Rational& other) const {
if ( numer == other.numer && denom == other.denom ) { return true; }
return false;
}

Rational& operator+=(const Rational& other) {
*this = Rational(numer* other.denom + other.numer * denom, denom * other.denom);
return *this;
}

Rational operator+(const Rational& other) const { return Rational(*this) += other; }

Rational& operator-=(const Rational& other) {
Rational temp(other);
temp.numer = -temp.numer;
return *this += temp;
}
Rational operator-(const Rational& other) const { return Rational(*this) -= other; }

Rational& operator*=(const Rational& other) {
*this = Rational(numer * other.numer, denom * other.denom);
return *this;
}

Rational operator*(const Rational& other) const { return Rational(*this) *= other; };

Rational& operator/=(const Rational other) {
Rational temp(other.denom, other.numer);
*this *= temp;
return *this;
}

Rational operator/(const Rational& other) const { return Rational(*this) /= other; };

bool operator<(const Rational& other) const { return numer * other.denom < denom * other.numer; }

bool operator<=(const Rational& other) const { return ! ( other < *this ); }

bool operator>(const Rational& other) const { return other < *this; }

bool operator>=(const Rational& other) const { return ! ( *this < other ); }

Rational operator-() const { return Rational(-numer, denom); }

Rational& operator++() { numer += denom; return *this; }

Rational operator++(int) { Rational temp = *this; ++*this; return temp; }

Rational& operator--() { numer -= denom; return *this; }

Rational operator--(int) { Rational temp = *this; --*this; return temp; }

friend std::ostream& operator<<(std::ostream& outStream, const Rational& other) {
outStream << other.numer << "/" << other.denom;
return outStream;
}

/// Methods ///
Rational reciprocal() const { return Rational(denom, numer); }

Rational positive() const { return Rational(abs(numer), denom); }

int64_t to_integer() const { return numer / denom; }

double to_double() const { return (double) numer / denom; }

int64_t hash() const { return std::hash<int64_t>{}(numer) ^ std::hash<int64_t>{}(denom); }

private:
int64_t numer;
int64_t denom;
};

int main() {
std::cout << "Perfect numbers less than 2^19:" << std::endl;
const int32_t limit = 1 << 19;
for ( int32_t candidate = 2; candidate < limit; ++candidate ) {
Rational sum = Rational(1, candidate);
int32_t square_root = (int32_t) sqrt(candidate);
for ( int32_t factor = 2; factor <= square_root; ++factor ) {
if ( candidate % factor == 0 ) {
sum += Rational(1, factor);
sum += Rational(1, candidate / factor);
}
}

if ( sum == Rational(1) ) {
std::cout << candidate << std::endl;
}
}
}
```
Output:
```Perfect numbers less than 2^19:
6
28
496
8128
```

## Clojure

Ratios are built in to Clojure and support math operations already. They automatically reduce and become Integers if possible.

```user> 22/7
22/7
user> 34/2
17
user> (+ 37/5 42/9)
181/15
```

## Common Lisp

Common Lisp has rational numbers built-in and integrated with all other number types. Common Lisp's number system is not extensible so reimplementing rational arithmetic would require all-new operator names.

```(loop for candidate from 2 below (expt 2 19)
for sum = (+ (/ candidate)
(loop for factor from 2 to (isqrt candidate)
when (zerop (mod candidate factor))
sum (+ (/ factor) (/ (floor candidate factor)))))
when (= sum 1)
collect candidate)
```

## D

```import std.bigint, std.traits, std.conv;

// std.numeric.gcd doesn't work with BigInt.
T gcd(T)(in T a, in T b) pure nothrow {
return (b != 0) ? gcd(b, a % b) : (a < 0) ? -a : a;
}

T lcm(T)(in T a, in T b) pure nothrow {
return a / gcd(a, b) * b;
}

struct RationalT(T) if (!isUnsigned!T) {
private T num, den; // Numerator & denominator.

private enum Type { NegINF = -2,
NegDEN = -1,
NaRAT  =  0,
NORMAL =  1,
PosINF =  2 };

this(U : RationalT)(U n) pure nothrow {
num = n.num;
den = n.den;
}

this(U)(in U n) pure nothrow if (isIntegral!U) {
num = toT(n);
den = 1UL;
}

this(U, V)(in U n, in V d) pure nothrow {
num = toT(n);
den = toT(d);
const common = gcd(num, den);
if (common != 0) {
num /= common;
den /= common;
} else { // infinite or NOT a Number
num = (num == 0) ? 0 : (num < 0) ? -1 : 1;
den = 0;
}
if (den < 0) { // Assure den is non-negative.
num = -num;
den = -den;
}
}

static T toT(U)(in ref U n) pure nothrow if (is(U == T)) {
return n;
}

static T toT(U)(in ref U n) pure nothrow if (!is(U == T)) {
T result = n;
return result;
}

T numerator() const pure nothrow @property {
return num;
}

T denominator() const pure nothrow @property {
return den;
}

string toString() const /*pure nothrow*/ {
if (den != 0)
return num.text ~ (den == 1 ? "" : "/" ~ den.text);
if (num == 0)
return "NaRat";
else
return ((num < 0) ? "-" : "+") ~ "infRat";
}

real toReal() pure const nothrow {
static if (is(T == BigInt))
return num.toLong / real(den.toLong);
else
return num / real(den);
}

RationalT opBinary(string op)(in RationalT r)
const pure nothrow if (op == "+" || op == "-") {
T common = lcm(den, r.den);
T n = mixin("common / den * num" ~ op ~
"common / r.den * r.num" );
return RationalT(n, common);
}

RationalT opBinary(string op)(in RationalT r)
const pure nothrow if (op == "*") {
return RationalT(num * r.num, den * r.den);
}

RationalT opBinary(string op)(in RationalT r)
const pure nothrow if (op == "/") {
return RationalT(num * r.den, den * r.num);
}

RationalT opBinary(string op, U)(in U r)
const pure nothrow if (isIntegral!U && (op == "+" ||
op == "-" || op == "*" || op == "/")) {
return opBinary!op(RationalT(r));
}

RationalT opBinary(string op)(in size_t p)
const pure nothrow if (op == "^^") {
return RationalT(num ^^ p, den ^^ p);
}

RationalT opBinaryRight(string op, U)(in U l)
const pure nothrow if (isIntegral!U) {
return RationalT(l).opBinary!op(RationalT(num, den));
}

RationalT opOpAssign(string op, U)(in U l) pure /*nothrow*/ {
mixin("this = this " ~ op ~ "l;");
return this;
}

RationalT opUnary(string op)()
const pure nothrow if (op == "+" || op == "-") {
return RationalT(mixin(op ~ "num"), den);
}

bool opCast(U)() const if (is(U == bool)) {
return num != 0;
}

bool opEquals(U)(in U r) const pure nothrow {
RationalT rhs = RationalT(r);
if (type() == Type.NaRAT || rhs.type() == Type.NaRAT)
return false;
return num == rhs.num && den == rhs.den;
}

int opCmp(U)(in U r) const pure nothrow {
auto rhs = RationalT(r);
if (type() == Type.NaRAT || rhs.type() == Type.NaRAT)
throw new Error("Compare involve a NaRAT.");
if (type() != Type.NORMAL ||
rhs.type() != Type.NORMAL) // for infinite
return (type() == rhs.type()) ? 0 :
((type() < rhs.type()) ? -1 : 1);
auto diff = num * rhs.den - den * rhs.num;
return (diff == 0) ? 0 : ((diff < 0) ? -1 : 1);
}

Type type() const pure nothrow {
if (den > 0) return Type.NORMAL;
if (den < 0) return Type.NegDEN;
if (num > 0) return Type.PosINF;
if (num < 0) return Type.NegINF;
return Type.NaRAT;
}
}

RationalT!U rational(U)(in U n) pure nothrow {
return typeof(return)(n);
}

RationalT!(CommonType!(U1, U2))
rational(U1, U2)(in U1 n, in U2 d) pure nothrow {
return typeof(return)(n, d);
}

alias Rational = RationalT!BigInt;

version (arithmetic_rational_main) { // Test.
void main() {
import std.stdio, std.math;
alias RatL = RationalT!long;

foreach (immutable p; 2 .. 2 ^^ 19) {
auto sum = RatL(1, p);
immutable limit = 1 + cast(uint)real(p).sqrt;
foreach (immutable factor; 2 .. limit)
if (p % factor == 0)
sum += RatL(1, factor) + RatL(factor, p);
if (sum.denominator == 1)
writefln("Sum of recipr. factors of %6s = %s exactly%s",
p, sum, (sum == 1) ? ", perfect." : ".");
}
}
}
```

Use the `-version=rational_arithmetic_main` compiler switch to run the test code.

Output:
```Sum of recipr. factors of      6 = 1 exactly, perfect.
Sum of recipr. factors of     28 = 1 exactly, perfect.
Sum of recipr. factors of    120 = 2 exactly.
Sum of recipr. factors of    496 = 1 exactly, perfect.
Sum of recipr. factors of    672 = 2 exactly.
Sum of recipr. factors of   8128 = 1 exactly, perfect.
Sum of recipr. factors of  30240 = 3 exactly.
Sum of recipr. factors of  32760 = 3 exactly.
Sum of recipr. factors of 523776 = 2 exactly.```

Currently RationalT!BigInt is not fast.

## Delphi

[[1]]

Translation of: C#
```program Arithmetic_Rational;

{\$APPTYPE CONSOLE}

uses
System.SysUtils,
Boost.Rational;

var
sum: TFraction;
max: Integer = 1 shl 19;
candidate, max2, factor: Integer;

begin
for candidate := 2 to max - 1 do
begin
sum := Fraction(1, candidate);
max2 := Trunc(Sqrt(candidate));
for factor := 2 to max2 do
begin
if (candidate mod factor) = 0 then
begin
sum := sum + Fraction(1, factor);
sum := sum + Fraction(1, candidate div factor);
end;
end;
if sum = Fraction(1) then
Writeln(candidate, ' is perfect');
end;
end.
```
Output:
```6 is perfect
28 is perfect
496 is perfect
8128 is perfect```

## EchoLisp

EchoLisp supports rational numbers as native type. "Big" rational i.e bigint/bigint are not supported.

```;; Finding perfect numbers
(define (sum/inv n) ;; look for div's in [2..sqrt(n)] and add 1/n
(for/fold (acc (/ n)) [(i (in-range 2 (sqrt n)))]
#:break (> acc 1) ; no hope
(when (zero? (modulo n i ))
(set! acc (+ acc (/ i) (/ i n))))))
```
Output:
```;; rational operations
(+ 1/42 1/666) → 59/2331
42/666 → 7/111
(expt 3/4 7) → 2187/16384 ; 3/4 ^7
(/ 6 8) → 3/4   ;; / operator → rational
(// 6 8) → 0.75 ;; // operator → float
(* 6/7 14/12) → 1

;; even perfect numbers (up to 100000)
(for [(i (in-range 4 100000 2))] ;; 8 seconds
(when (= (sum/inv i) 1)
(printf "🍏 🍒 🍓 %d is perfect." i)))

🍏 🍒 🍓 6 is perfect.
🍏 🍒 🍓 28 is perfect.
🍏 🍒 🍓 496 is perfect.
🍏 🍒 🍓 8128 is perfect.
```

## Elisa

```component RationalNumbers;
type Rational;
Rational(Numerator = integer, Denominater = integer) -> Rational;

Rational + Rational -> Rational;
Rational - Rational -> Rational;
Rational * Rational -> Rational;
Rational / Rational -> Rational;

Rational == Rational -> boolean;
Rational <> Rational -> boolean;
Rational >= Rational -> boolean;
Rational <= Rational -> boolean;
Rational >  Rational -> boolean;
Rational <  Rational -> boolean;

+ Rational -> Rational;
- Rational -> Rational;
abs(Rational) -> Rational;

Rational(integer) -> Rational;
Numerator(Rational) -> integer;
Denominator(Rational) -> integer;
begin
Rational(A,B) = Rational:[A;B];

R1 + R2 = Normalize( R1.A * R2.B + R1.B * R2.A, R1.B * R2.B);
R1 - R2 = Normalize( R1.A * R2.B - R1.B * R2.A, R1.B * R2.B);
R1 * R2 = Normalize( R1.A * R2.A, R1.B * R2.B);
R1 / R2 = Normalize( R1.A * R2.B, R1.B * R2.A);

R1 == R2 = [ R = (R1 - R2); R.A == 0];
R1 <> R2 = [ R = (R1 - R2); R.A <> 0];
R1 >= R2 = [ R = (R1 - R2); R.A >= 0];
R1 <= R2 = [ R = (R1 - R2); R.A <= 0];
R1 > R2  = [ R = (R1 - R2); R.A > 0];
R1 < R2  = [ R = (R1 - R2); R.A < 0];

+ R = R;
- R = Rational(-R.A, R.B);

abs(R) = Rational(abs(R.A), abs(R.B));
Rational(I) = Rational (I, 1);
Numerator(R) = R.A;
Denominator(R) = R.B;

<< internal definitions >>

Normalize (A = integer, B = integer) -> Rational;
Normalize (A, B) = [ exception( B == 0, "Illegal Rational Number");
Common = GCD(abs(A), abs(B));
if B < 0 then Rational(-A / Common, -B / Common)
else Rational( A / Common,  B / Common) ];

GCD (A = integer, B = integer) -> integer;
GCD (A, B) = [ if A == 0 then return(B);
if B == 0 then return(A);
if A > B  then GCD (B, mod(A,B))
else GCD (A, mod(B,A)) ];

end component RationalNumbers;```

Tests

```use RationalNumbers;

PerfectNumbers( Limit = integer) -> multi(integer);
PerfectNumbers( Limit) =
[ Candidate = 2 .. Limit;
Sum:= Rational(1,Candidate);
[ Divisor = 2 .. integer(sqrt(real(Candidate)));
if mod(Candidate, Divisor) == 0 then
Sum := Sum + Rational(1, Divisor) + Rational(Divisor, Candidate);
];
if Sum == Rational(1,1) then Candidate
];

PerfectNumbers(10000)?```
Output:
```6
28
496
8128
```

## Elixir

```defmodule Rational do
import Kernel, except: [div: 2]

defstruct numerator: 0, denominator: 1

def new(numerator), do: %Rational{numerator: numerator, denominator: 1}

def new(numerator, denominator) do
sign = if numerator * denominator < 0, do: -1, else: 1
{numerator, denominator} = {abs(numerator), abs(denominator)}
gcd = gcd(numerator, denominator)
%Rational{numerator: sign * Kernel.div(numerator, gcd),
denominator: Kernel.div(denominator, gcd)}
end

{a, b} = convert(a, b)
new(a.numerator * b.denominator + b.numerator * a.denominator,
a.denominator * b.denominator)
end

def sub(a, b) do
{a, b} = convert(a, b)
new(a.numerator * b.denominator - b.numerator * a.denominator,
a.denominator * b.denominator)
end

def mult(a, b) do
{a, b} = convert(a, b)
new(a.numerator * b.numerator, a.denominator * b.denominator)
end

def div(a, b) do
{a, b} = convert(a, b)
new(a.numerator * b.denominator, a.denominator * b.numerator)
end

defp convert(a), do: if is_integer(a), do: new(a), else: a

defp convert(a, b), do: {convert(a), convert(b)}

defp gcd(a, 0), do: a
defp gcd(a, b), do: gcd(b, rem(a, b))
end

defimpl Inspect, for: Rational do
def inspect(r, _opts) do
"%Rational<#{r.numerator}/#{r.denominator}>"
end
end

Enum.each(2..trunc(:math.pow(2,19)), fn candidate ->
sum = 2 .. round(:math.sqrt(candidate))
|> Enum.reduce(Rational.new(1, candidate), fn factor,sum ->
if rem(candidate, factor) == 0 do
else
sum
end
end)
if sum.denominator == 1 do
:io.format "Sum of recipr. factors of ~6w = ~w exactly ~s~n",
[candidate, sum.numerator, (if sum.numerator == 1, do: "perfect!", else: "")]
end
end)
```
Output:
```Sum of recipr. factors of      6 = 1 exactly perfect!
Sum of recipr. factors of     28 = 1 exactly perfect!
Sum of recipr. factors of    120 = 2 exactly
Sum of recipr. factors of    496 = 1 exactly perfect!
Sum of recipr. factors of    672 = 2 exactly
Sum of recipr. factors of   8128 = 1 exactly perfect!
Sum of recipr. factors of  30240 = 3 exactly
Sum of recipr. factors of  32760 = 3 exactly
Sum of recipr. factors of 523776 = 2 exactly
```

## ERRE

```PROGRAM RATIONAL_ARITH

!
! for rosettacode.org
!

TYPE RATIONAL=(NUM,DEN)

DIM SUM:RATIONAL,ONE:RATIONAL,KF:RATIONAL

DIM A:RATIONAL,B:RATIONAL
PROCEDURE ABS(A.->A.)
A.NUM=ABS(A.NUM)
END PROCEDURE

PROCEDURE NEG(A.->A.)
A.NUM=-A.NUM
END PROCEDURE

LOCAL T
T=A.DEN*B.DEN
A.NUM=A.NUM*B.DEN+B.NUM*A.DEN
A.DEN=T
END PROCEDURE

PROCEDURE SUB(A.,B.->A.)
LOCAL T
T=A.DEN*B.DEN
A.NUM=A.NUM*B.DEN-B.NUM*A.DEN
A.DEN=T
END PROCEDURE

PROCEDURE MULT(A.,B.->A.)
A.NUM*=B.NUM  A.DEN*=B.DEN
END PROCEDURE

PROCEDURE DIVIDE(A.,B.->A.)
A.NUM*=B.DEN
A.DEN*=B.NUM
END PROCEDURE

PROCEDURE EQ(A.,B.->RES%)
RES%=A.NUM*B.DEN=B.NUM*A.DEN
END PROCEDURE

PROCEDURE LT(A.,B.->RES%)
RES%=A.NUM*B.DEN<B.NUM*A.DEN
END PROCEDURE

PROCEDURE GT(A.,B.->RES%)
RES%=A.NUM*B.DEN>B.NUM*A.DEN
END PROCEDURE

PROCEDURE NE(A.,B.->RES%)
RES%=A.NUM*B.DEN<>B.NUM*A.DEN
END PROCEDURE

PROCEDURE LE(A.,B.->RES%)
RES%=A.NUM*B.DEN<=B.NUM*A.DEN
END PROCEDURE

PROCEDURE GE(A.,B.->RES%)
RES%=A.NUM*B.DEN>=B.NUM*A.DEN
END PROCEDURE

PROCEDURE NORMALIZE(A.->A.)
LOCAL A,B,T
A=A.NUM   B=A.DEN
WHILE B<>0 DO
T=A
A=B
B=T-B*INT(T/B)
END WHILE
A.NUM/=A  A.DEN/=A
IF A.DEN<0 THEN A.NUM*=-1 A.DEN*=-1 END IF
END PROCEDURE

BEGIN
ONE.NUM=1 ONE.DEN=1
FOR N=2 TO 2^19-1 DO
SUM.NUM=1 SUM.DEN=N
FOR K=2 TO SQR(N) DO
IF N=K*INT(N/K) THEN
KF.NUM=1 KF.DEN=K
NORMALIZE(SUM.->SUM.)
KF.DEN=INT(N/K)
NORMALIZE(SUM.->SUM.)
END IF
END FOR
EQ(SUM.,ONE.->RES%)
IF RES% THEN PRINT(N;" is perfect") END IF
END FOR
END PROGRAM```
Output:
``` 6  is perfect
28  is perfect
496  is perfect
8128  is perfect
```

## F#

The F# Powerpack library defines the BigRational data type.

```type frac = Microsoft.FSharp.Math.BigRational

let perf n = 1N = List.fold (+) 0N (List.map (fun i -> if n % i = 0 then 1N/frac.FromInt(i) else 0N) [2..n])

for i in 1..(1<<<19) do if (perf i) then printfn "%i is perfect" i
```

## Factor

`ratio` is a built-in numeric type.

```USING: generalizations io kernel math math.functions
math.primes.factors math.ranges prettyprint sequences ;
IN: rosetta-code.arithmetic-rational

2/5              ! literal syntax 2/5
2/4              ! automatically simplifies to 1/2
5/1              ! automatically coerced to 5
26/5             ! mixed fraction 5+1/5
13/178 >fraction ! get the numerator and denominator 13 178
8 recip          ! get the reciprocal 1/8

! ratios can be any size
12417829731289312/61237812937138912735712
8 ndrop ! clear the stack
! arithmetic works the same as any other number.

: perfect? ( n -- ? )
divisors rest [ recip ] map-sum 1 = ;

"Perfect numbers <= 2^19: " print
2 19 ^ [1,b] [ perfect? ] filter .
```
Output:
```Perfect numbers <= 2^19:
V{ 6 28 496 8128 }
```

## Fermat

Fermat supports rational aritmetic natively.

```for n=2 to 2^19 by 2 do
s:=3/n;
m:=1;
while m<=n/3 do
if Divides(m,n) then s:=s+1/m; fi;
m:=m+1;
od;
if s=2 then !!n fi;
od;```
Output:
```6
28
496
8128
```

## Forth

```\ Rationals can use any double cell operations:  2!, 2@, 2dup, 2swap, etc.
\ Uses the stack convention of the built-in "*/" for int * frac -> int

: numerator  drop ;
: denominator nip ;

: s>rat      1 ;		\ integer to rational  (n/1)
: rat>s      / ;		\ integer
: rat>frac   mod ;		\ fractional part
: rat>float  swap s>f s>f f/ ;

: rat.  swap 1 .r [char] / emit . ;

: gcd ( a b -- gcd ) begin ?dup while tuck mod repeat ;
: rat-normalize ( rat -- rat ) 2dup gcd tuck / >r / r> ;

: rat-abs     swap abs    swap ;
: rat-negate  swap negate swap ;
: 1/rat       over 0< if negate swap negate else swap then ;

: rat+ ( a b c d -- ad+bc bd )
rot 2dup * >r
rot * >r * r> +
r> rat-normalize ;
: rat-  rat-negate rat+ ;

: rat* ( a b c d -- ac bd )
rot * >r * r> rat-normalize ;
: rat/  swap rat* ;

: rat-equal  d= ;
: rat-less ( a b c d -- ad<bc )
-rot * >r * r> < ;
: rat-more  2swap rat-less ;

: rat-inc  tuck + swap ;
: rat-dec  tuck - swap ;
```

## Fortran

Works with: Fortran version 90 and later
```module module_rational

implicit none
private
public :: rational
public :: rational_simplify
public :: assignment (=)
public :: operator (//)
public :: operator (+)
public :: operator (-)
public :: operator (*)
public :: operator (/)
public :: operator (<)
public :: operator (<=)
public :: operator (>)
public :: operator (>=)
public :: operator (==)
public :: operator (/=)
public :: abs
public :: int
public :: modulo
type rational
integer :: numerator
integer :: denominator
end type rational
interface assignment (=)
module procedure assign_rational_int, assign_rational_real
end interface
interface operator (//)
module procedure make_rational
end interface
interface operator (+)
end interface
interface operator (-)
module procedure rational_minus, rational_subtract
end interface
interface operator (*)
module procedure rational_multiply
end interface
interface operator (/)
module procedure rational_divide
end interface
interface operator (<)
module procedure rational_lt
end interface
interface operator (<=)
module procedure rational_le
end interface
interface operator (>)
module procedure rational_gt
end interface
interface operator (>=)
module procedure rational_ge
end interface
interface operator (==)
module procedure rational_eq
end interface
interface operator (/=)
module procedure rational_ne
end interface
interface abs
module procedure rational_abs
end interface
interface int
module procedure rational_int
end interface
interface modulo
module procedure rational_modulo
end interface

contains

recursive function gcd (i, j) result (res)
integer, intent (in) :: i
integer, intent (in) :: j
integer :: res
if (j == 0) then
res = i
else
res = gcd (j, modulo (i, j))
end if
end function gcd

function rational_simplify (r) result (res)
type (rational), intent (in) :: r
type (rational) :: res
integer :: g
g = gcd (r % numerator, r % denominator)
res = r % numerator / g // r % denominator / g
end function rational_simplify

function make_rational (numerator, denominator) result (res)
integer, intent (in) :: numerator
integer, intent (in) :: denominator
type (rational) :: res
res = rational (numerator, denominator)
end function make_rational

subroutine assign_rational_int (res, i)
type (rational), intent (out), volatile :: res
integer, intent (in) :: i
res = i // 1
end subroutine assign_rational_int

subroutine assign_rational_real (res, x)
type (rational), intent(out), volatile :: res
real, intent (in) :: x
integer :: x_floor
real :: x_frac
x_floor = floor (x)
x_frac = x - x_floor
if (x_frac == 0) then
res = x_floor // 1
else
res = (x_floor // 1) + (1 // floor (1 / x_frac))
end if
end subroutine assign_rational_real

function rational_add (r, s) result (res)
type (rational), intent (in) :: r
type (rational), intent (in) :: s
type (rational) :: res
res = r % numerator * s % denominator + r % denominator * s % numerator // &
& r % denominator * s % denominator

function rational_minus (r) result (res)
type (rational), intent (in) :: r
type (rational) :: res
res = - r % numerator // r % denominator
end function rational_minus

function rational_subtract (r, s) result (res)
type (rational), intent (in) :: r
type (rational), intent (in) :: s
type (rational) :: res
res = r % numerator * s % denominator - r % denominator * s % numerator // &
& r % denominator * s % denominator
end function rational_subtract

function rational_multiply (r, s) result (res)
type (rational), intent (in) :: r
type (rational), intent (in) :: s
type (rational) :: res
res = r % numerator * s % numerator // r % denominator * s % denominator
end function rational_multiply

function rational_divide (r, s) result (res)
type (rational), intent (in) :: r
type (rational), intent (in) :: s
type (rational) :: res
res = r % numerator * s % denominator // r % denominator * s % numerator
end function rational_divide

function rational_lt (r, s) result (res)
type (rational), intent (in) :: r
type (rational), intent (in) :: s
type (rational) :: r_simple
type (rational) :: s_simple
logical :: res
r_simple = rational_simplify (r)
s_simple = rational_simplify (s)
res = r_simple % numerator * s_simple % denominator < &
& s_simple % numerator * r_simple % denominator
end function rational_lt

function rational_le (r, s) result (res)
type (rational), intent (in) :: r
type (rational), intent (in) :: s
type (rational) :: r_simple
type (rational) :: s_simple
logical :: res
r_simple = rational_simplify (r)
s_simple = rational_simplify (s)
res = r_simple % numerator * s_simple % denominator <= &
& s_simple % numerator * r_simple % denominator
end function rational_le

function rational_gt (r, s) result (res)
type (rational), intent (in) :: r
type (rational), intent (in) :: s
type (rational) :: r_simple
type (rational) :: s_simple
logical :: res
r_simple = rational_simplify (r)
s_simple = rational_simplify (s)
res = r_simple % numerator * s_simple % denominator > &
& s_simple % numerator * r_simple % denominator
end function rational_gt

function rational_ge (r, s) result (res)
type (rational), intent (in) :: r
type (rational), intent (in) :: s
type (rational) :: r_simple
type (rational) :: s_simple
logical :: res
r_simple = rational_simplify (r)
s_simple = rational_simplify (s)
res = r_simple % numerator * s_simple % denominator >= &
& s_simple % numerator * r_simple % denominator
end function rational_ge

function rational_eq (r, s) result (res)
type (rational), intent (in) :: r
type (rational), intent (in) :: s
logical :: res
res = r % numerator * s % denominator == s % numerator * r % denominator
end function rational_eq

function rational_ne (r, s) result (res)
type (rational), intent (in) :: r
type (rational), intent (in) :: s
logical :: res
res = r % numerator * s % denominator /= s % numerator * r % denominator
end function rational_ne

function rational_abs (r) result (res)
type (rational), intent (in) :: r
type (rational) :: res
res = sign (r % numerator, r % denominator) // r % denominator
end function rational_abs

function rational_int (r) result (res)
type (rational), intent (in) :: r
integer :: res
res = r % numerator / r % denominator
end function rational_int

function rational_modulo (r) result (res)
type (rational), intent (in) :: r
integer :: res
res = modulo (r % numerator, r % denominator)
end function rational_modulo

end module module_rational
```

Example:

```program perfect_numbers

use module_rational
implicit none
integer, parameter :: n_min = 2
integer, parameter :: n_max = 2 ** 19 - 1
integer :: n
integer :: factor
type (rational) :: sum

do n = n_min, n_max
sum = 1 // n
factor = 2
do
if (factor * factor >= n) then
exit
end if
if (modulo (n, factor) == 0) then
sum = rational_simplify (sum + (1 // factor) + (factor // n))
end if
factor = factor + 1
end do
if (sum % numerator == 1 .and. sum % denominator == 1) then
write (*, '(i0)') n
end if
end do

end program perfect_numbers
```
Output:
```6
28
496
8128```

## Frink

Rational numbers are built into Frink and the numerator and denominator can be arbitrarily-sized. They are automatically simplified and collapsed into integers if necessary. All functions in the language can work with rational numbers. Rational numbers are treated as exact. Rational numbers can exist in complex numbers or intervals.

```1/2 + 2/3
// 7/6 (approx. 1.1666666666666667)

1/2 + 1/2
// 1

// 600001/200000000000000000000 (exactly 3.000005e-15)

8^(1/3)
// 2    (note the exact integer result.)```

## GAP

Rational numbers are built-in.

```2/3 in Rationals;
# true
2/3 + 3/4;
# 17/12
```

## Go

Go does not have user defined operators. Go does however have a rational number type in the `math/big` package of the standard library. The big.Rat type supports the operations of the task, although typically with methods rather than operators:

• Rat.Abs
• Rat.Neg
• Rat.Sub
• Rat.Mul
• Rat.Quo
• Rat.Cmp
• Rat.SetInt

Code here implements the perfect number test described in the task using the standard library.

```package main

import (
"fmt"
"math"
"math/big"
)

func main() {
var recip big.Rat
max := int64(1 << 19)
for candidate := int64(2); candidate < max; candidate++ {
sum := big.NewRat(1, candidate)
max2 := int64(math.Sqrt(float64(candidate)))
for factor := int64(2); factor <= max2; factor++ {
if candidate%factor == 0 {
if f2 := candidate / factor; f2 != factor {
}
}
}
if sum.Denom().Int64() == 1 {
perfectstring := ""
if sum.Num().Int64() == 1 {
perfectstring = "perfect!"
}
fmt.Printf("Sum of recipr. factors of %d = %d exactly %s\n",
candidate, sum.Num().Int64(), perfectstring)
}
}
}
```
Output:
```Sum of recipr. factors of 6 = 1 exactly perfect!
Sum of recipr. factors of 28 = 1 exactly perfect!
Sum of recipr. factors of 120 = 2 exactly
Sum of recipr. factors of 496 = 1 exactly perfect!
Sum of recipr. factors of 672 = 2 exactly
Sum of recipr. factors of 8128 = 1 exactly perfect!
Sum of recipr. factors of 30240 = 3 exactly
Sum of recipr. factors of 32760 = 3 exactly
Sum of recipr. factors of 523776 = 2 exactly
```

## Groovy

Groovy does not provide any built-in facility for rational arithmetic. However, it does support arithmetic operator overloading. Thus it is not too hard to build a fairly robust, complete, and intuitive rational number class, such as the following:

```class Rational extends Number implements Comparable {
final BigInteger num, denom

static final Rational ONE = new Rational(1)
static final Rational ZERO = new Rational(0)

Rational(BigDecimal decimal) {
this(
decimal.scale() < 0 ? decimal.unscaledValue() * 10 ** -decimal.scale() : decimal.unscaledValue(),
decimal.scale() < 0 ? 1                                                : 10 ** decimal.scale()
)
}

Rational(BigInteger n, BigInteger d = 1) {
if (!d || n == null) { n/d }
(num, denom) = reduce(n, d)
}

private List reduce(BigInteger n, BigInteger d) {
BigInteger sign = ((n < 0) ^ (d < 0)) ? -1 : 1
(n, d) = [n.abs(), d.abs()]
BigInteger commonFactor = gcd(n, d)

[n.intdiv(commonFactor) * sign, d.intdiv(commonFactor)]
}

Rational toLeastTerms() { reduce(num, denom) as Rational }

private BigInteger gcd(BigInteger n, BigInteger m) {
n == 0 ? m : { while(m%n != 0) { (n, m) = [m%n, n] }; n }()
}

Rational plus(Rational r) { [num*r.denom + r.num*denom, denom*r.denom] }
Rational plus(BigInteger n) { [num + n*denom, denom] }
Rational plus(Number n) { this + ([n] as Rational) }

Rational next() { [num + denom, denom] }

Rational minus(Rational r) { [num*r.denom - r.num*denom, denom*r.denom] }
Rational minus(BigInteger n) { [num - n*denom, denom] }
Rational minus(Number n) { this - ([n] as Rational) }

Rational previous() { [num - denom, denom] }

Rational multiply(Rational r) { [num*r.num, denom*r.denom] }
Rational multiply(BigInteger n) { [num*n, denom] }
Rational multiply(Number n) { this * ([n] as Rational) }

Rational div(Rational r) { new Rational(num*r.denom, denom*r.num) }
Rational div(BigInteger n) { new Rational(num, denom*n) }
Rational div(Number n) { this / ([n] as Rational) }

BigInteger intdiv(BigInteger n) { num.intdiv(denom*n) }

Rational negative() { [-num, denom] }

Rational abs() { [num.abs(), denom] }

Rational reciprocal() { new Rational(denom, num) }

Rational power(BigInteger n) {
def (nu, de) = (n < 0 ? [denom, num] : [num, denom])*.power(n.abs())
new Rational (nu, de)
}

boolean asBoolean() { num != 0 }

BigDecimal toBigDecimal() { (num as BigDecimal)/(denom as BigDecimal) }

BigInteger toBigInteger() { num.intdiv(denom) }

Double toDouble() { toBigDecimal().toDouble() }
double doubleValue() { toDouble() as double }

Float toFloat() { toBigDecimal().toFloat() }
float floatValue() { toFloat() as float }

Integer toInteger() { toBigInteger().toInteger() }
int intValue() { toInteger() as int }

Long toLong() { toBigInteger().toLong() }
long longValue() { toLong() as long }

Object asType(Class type) {
switch (type) {
case this.class:              return this
case [Boolean, Boolean.TYPE]: return asBoolean()
default: throw new ClassCastException("Cannot convert from type Rational to type " + type)
}
}

boolean equals(o) { compareTo(o) == 0 }

int compareTo(o) {
o instanceof Rational
? compareTo(o as Rational)
: o instanceof Number
? compareTo(o as Number)
: (Double.NaN as int)
}
int compareTo(Rational r) { num*r.denom <=> denom*r.num }
int compareTo(Number n) { num <=> denom*(n as BigInteger) }

int hashCode() { [num, denom].hashCode() }

String toString() {
"\${num}//\${denom}"
}
}
```

The following RationalCategory class allows for modification of regular Number behavior when interacting with Rational.

```import org.codehaus.groovy.runtime.DefaultGroovyMethods

class RationalCategory {
static Rational plus (Number a, Rational b) { ([a] as Rational) + b }
static Rational minus (Number a, Rational b) { ([a] as Rational) - b }
static Rational multiply (Number a, Rational b) { ([a] as Rational) * b }
static Rational div (Number a, Rational b) { ([a] as Rational) / b  }

static <T> T asType (Number a, Class<T> type) {
type == Rational \
? [a] as Rational
: DefaultGroovyMethods.asType(a, type)
}
}
```

Test Program (mixes the RationalCategory methods into the Number class):

```Number.metaClass.mixin RationalCategory

def x = [5, 20] as Rational
def y = [9, 12] as Rational
def z = [0, 10000] as Rational

println x
println y
println z
println (x <=> y)
println (x.compareTo(y))
assert x < y
assert x*3 == y
assert x*5.5 == 5.5*x
assert (z + 1) <= y*4
assert x + 1.3 == 1.3 + x
assert 24 - y == -(y - 24)
assert 3 / y == (y / 3).reciprocal()
assert x != y

println "x + y == \${x} + \${y} == \${x + y}"
println "x + z == \${x} + \${z} == \${x + z}"
println "x - y == \${x} - \${y} == \${x - y}"
println "x - z == \${x} - \${z} == \${x - z}"
println "x * y == \${x} * \${y} == \${x * y}"
println "y ** 3 == \${y} ** 3 == \${y ** 3}"
println "y ** -3 == \${y} ** -3 == \${y ** -3}"
println "x * z == \${x} * \${z} == \${x * z}"
println "x / y == \${x} / \${y} == \${x / y}"
try { print "x / z == \${x} / \${z} == "; println "\${x / z}" }
catch (Throwable t) { println t.message }

println "-x == -\${x} == \${-x}"
println "-y == -\${y} == \${-y}"
println "-z == -\${z} == \${-z}"

print "x as int == \${x} as int == "; println x.intValue()
print "x as double == \${x} as double == "; println x.doubleValue()
print "1 / x as int == 1 / \${x} as int == "; println x.reciprocal().intValue()
print "1.0 / x == 1.0 / \${x} == "; println x.reciprocal().doubleValue()
print "y as int == \${y} as int == "; println y.intValue()
print "y as double == \${y} as double == "; println y.doubleValue()
print "1 / y as int == 1 / \${y} as int == "; println y.reciprocal().intValue()
print "1.0 / y == 1.0 / \${y} == "; println y.reciprocal().doubleValue()
print "z as int == \${z} as int == "; println z.intValue()
print "z as double == \${z} as double == "; println z.doubleValue()
try { print "1 / z as int == 1 / \${z} as int == "; println z.reciprocal().intValue() }
catch (Throwable t) { println t.message }
try { print "1.0 / z == 1.0 / \${z} == "; println z.reciprocal().doubleValue() }
catch (Throwable t) { println t.message }

println "++x == ++ \${x} == \${++x}"
println "++y == ++ \${y} == \${++y}"
println "++z == ++ \${z} == \${++z}"
println "-- --x == -- -- \${x} == \${-- (--x)}"
println "-- --y == -- -- \${y} == \${-- (--y)}"
println "-- --z == -- -- \${z} == \${-- (--z)}"
println x
println y
println z

println (x <=> y)
assert x*3 == y
assert (z + 1) <= y*4
assert (x < y)

println 25 as Rational
println 25.0 as Rational
println 0.25 as Rational

def ε = 0.000000001  // tolerance (epsilon): acceptable "wrongness" to account for rounding error

def π = Math.PI
def α = π as Rational
assert (π - (α as BigDecimal)).abs() < ε
println π
println α
println (α.toBigDecimal())
println (α as BigDecimal)
println (α as Double)
println (α as double)
println (α as boolean)
println (z as boolean)
try { println (α as Date) }
catch (Throwable t) { println t.message }
try { println (α as char) }
catch (Throwable t) { println t.message }
```
Output:
```1//4
3//4
0//1
-1
-1
x + y == 1//4 + 3//4 == 1//1
x + z == 1//4 + 0//1 == 1//4
x - y == 1//4 - 3//4 == -1//2
x - z == 1//4 - 0//1 == 1//4
x * y == 1//4 * 3//4 == 3//16
y ** 3 == 3//4 ** 3 == 27//64
y ** -3 == 3//4 ** -3 == 64//27
x * z == 1//4 * 0//1 == 0//1
x / y == 1//4 / 3//4 == 1//3
x / z == 1//4 / 0//1 == Division by zero
-x == -1//4 == -1//4
-y == -3//4 == -3//4
-z == -0//1 == 0//1
x as int == 1//4 as int == 0
x as double == 1//4 as double == 0.25
1 / x as int == 1 / 1//4 as int == 4
1.0 / x == 1.0 / 1//4 == 4.0
y as int == 3//4 as int == 0
y as double == 3//4 as double == 0.75
1 / y as int == 1 / 3//4 as int == 1
1.0 / y == 1.0 / 3//4 == 1.3333333333
z as int == 0//1 as int == 0
z as double == 0//1 as double == 0.0
1 / z as int == 1 / 0//1 as int == Division by zero
1.0 / z == 1.0 / 0//1 == Division by zero
++x == ++ 1//4 == 5//4
++y == ++ 3//4 == 7//4
++z == ++ 0//1 == 1//1
-- --x == -- -- 5//4 == -3//4
-- --y == -- -- 7//4 == -1//4
-- --z == -- -- 1//1 == -1//1
1//4
3//4
0//1
-1
25//1
25//1
1//4
3.141592653589793
884279719003555//281474976710656
3.141592653589793115997963468544185161590576171875
3.141592653589793115997963468544185161590576171875
3.141592653589793
3.141592653589793
true
false
Cannot convert from type Rational to type class java.util.Date
Cannot convert from type Rational to type char
```

The following uses the Rational class, with RationalCategory mixed into Number, to find all perfect numbers less than 219:

```Number.metaClass.mixin RationalCategory

def factorize = { target ->
assert target > 0
if (target == 1L) { return [1L] }
if ([2L, 3L].contains(target)) { return [1L, target] }
def targetSqrt = Math.sqrt(target)
def lowFactors = (2L..targetSqrt).findAll { (target % it) == 0 }

if (!lowFactors) { return [1L, target] }
def highFactors = lowFactors[-1..0].findResults { target.intdiv(it) } - lowFactors[-1]

return [1L] + lowFactors + highFactors + [target]
}

def perfect = {
def factors = factorize(it)
2 as Rational == factors.sum{ factor -> new Rational(1, factor) } \
? [perfect: it, factors: factors]
: null
}

def trackProgress = { if ((it % (100*1000)) == 0) { println it } else if ((it % 1000) == 0) { print "." } }

(1..(2**19)).findResults { trackProgress(it); perfect(it) }.each { println(); print it }
```
Output:
```...................................................................................................100000
...................................................................................................200000
...................................................................................................300000
...................................................................................................400000
...................................................................................................500000
........................
[perfect:6, factors:[1, 2, 3, 6]]
[perfect:28, factors:[1, 2, 4, 7, 14, 28]]
[perfect:496, factors:[1, 2, 4, 8, 16, 31, 62, 124, 248, 496]]
[perfect:8128, factors:[1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064, 8128]]```

Haskell provides a `Rational` type, which is really an alias for `Ratio Integer` (`Ratio` being a polymorphic type implementing rational numbers for any `Integral` type of numerators and denominators). The fraction is constructed using the `%` operator.

```import Data.Ratio ((%))

-- Prints the first N perfect numbers.
main = do
let n = 4
mapM_ print \$
take
n
[ candidate
| candidate <- [2 .. 2 ^ 19]
, getSum candidate == 1 ]
where
getSum candidate =
1 % candidate +
sum
[ 1 % factor + 1 % (candidate `div` factor)
| factor <- [2 .. floor (sqrt (fromIntegral candidate))]
, candidate `mod` factor == 0 ]
```

For a sample implementation of `Ratio`, see the Haskell 98 Report.

## Icon and Unicon

The IPL provides support for rational arithmetic

• The data type is called 'rational' not 'frac'.
• Use the record constructor 'rational' to create a rational. Sign must be 1 or -1.
• Neither Icon nor Unicon supports operator overloading. Augmented assignments make little sense w/o this.
• Procedures include 'negrat' (unary -), 'addrat' (+), 'subrat' (-), 'mpyrat' (*), 'divrat' (modulo /).

• 'makerat' (make), 'absrat' (abs), 'eqrat' (=), 'nerat' (~=), 'ltrat' (<), 'lerat' (<=), 'gerat' (>=), 'gtrat' (>)
```procedure main()
limit := 2^19

write("Perfect numbers up to ",limit," (using rational arithmetic):")
every write(is_perfect(c := 2 to limit))
write("End of perfect numbers")

# verify the rest of the implementation

zero := makerat(0)          # from integer
half := makerat(0.5)        # from real
qtr  := makerat("1/4")      # from strings ...
one  := makerat("1")
mone := makerat("-1")

verifyrat("eqrat",zero,zero)
verifyrat("ltrat",zero,half)
verifyrat("ltrat",half,zero)
verifyrat("gtrat",zero,half)
verifyrat("gtrat",half,zero)
verifyrat("nerat",zero,half)
verifyrat("nerat",zero,zero)
verifyrat("absrat",mone,)

end

procedure is_perfect(c)       #: test for perfect numbers using rational arithmetic
rsum := rational(1, c, 1)
every f := 2 to sqrt(c) do
if 0 = c % f then
if rsum.numer = rsum.denom = 1 then
return c
end
```
Output:
```Perfect numbers up to 524288 (using rational arithmetic):
6
28
496
8128
End of perfect numbers
Testing eqrat( (0/1), (0/1) ) ==> returned (0/1)
Testing ltrat( (0/1), (1/2) ) ==> returned (1/2)
Testing ltrat( (1/2), (0/1) ) ==> failed
Testing gtrat( (0/1), (1/2) ) ==> failed
Testing gtrat( (1/2), (0/1) ) ==> returned (0/1)
Testing nerat( (0/1), (1/2) ) ==> returned (1/2)
Testing nerat( (0/1), (0/1) ) ==> failed
Testing absrat( (-1/1),  ) ==> returned (1/1)```

The following task functions are missing from the IPL:

```procedure verifyrat(p,r1,r2)  #: verification tests for rational procedures
return write("Testing ",p,"( ",rat2str(r1),", ",rat2str(\r2) | &null," ) ==> ","returned " || rat2str(p(r1,r2)) | "failed")
end

procedure makerat(x)          #: make rational (from integer, real, or strings)
local n,d
static c
initial c := &digits++'+-'

return case type(x) of {
"real"    : real2rat(x)
"integer" : ratred(rational(x,1,1))
"string"  : if x ? ( n := integer(tab(many(c))), ="/", d := integer(tab(many(c))), pos(0)) then
ratred(rational(n,d,1))
else
makerat(numeric(x))
}
end

procedure absrat(r1)          #: abs(rational)
r1 := ratred(r1)
r1.sign := 1
return r1
end

invocable all                 #  for string invocation

procedure xoprat(op,r1,r2)    #: support procedure for binary operations that cross denominators
local numer, denom, div

r1 := ratred(r1)
r2 := ratred(r2)

return if op(r1.numer * r2.denom,r2.numer * r1.denom) then r2   # return right argument on success
end

procedure eqrat(r1,r2)        #: rational r1 = r2
return xoprat("=",r1,r2)
end

procedure nerat(r1,r2)        #: rational r1 ~= r2
return xoprat("~=",r1,r2)
end

procedure ltrat(r1,r2)        #: rational r1 < r2
return xoprat("<",r1,r2)
end

procedure lerat(r1,r2)        #: rational r1 <= r2
return xoprat("<=",r1,r2)
end

procedure gerat(r1,r2)        #: rational r1 >= r2
return xoprat(">=",r1,r2)
end

procedure gtrat(r1,r2)        #: rational r1 > r2
return xoprat(">",r1,r2)
end

```

The

provides rational and gcd in numbers. Record definition and usage is shown below:

```   record rational(numer, denom, sign)        # rational type

divrat(r1,r2) # Divide rational numbers r1 and r2.
medrat(r1,r2) # Form mediant of r1 and r2.
mpyrat(r1,r2) # Multiply rational numbers r1 and r2.
negrat(r)     # Produce negative of rational number r.
rat2real(r)   # Produce floating-point approximation of r
rat2str(r)    # Convert the rational number r to its string representation.
real2rat(v,p) # Convert real to rational with precision p (default 1e-10). Warning: excessive p gives ugly fractions
reciprat(r)   # Produce the reciprocal of rational number r.
str2rat(s)    # Convert the string representation (such as "3/2") to a rational number
subrat(r1,r2) # Subtract rational numbers r1 and r2.

gcd(i, j)     # returns greatest common divisor of i and j
```

## J

Rational numbers in J may be formed from fixed precision integers by first upgrading them to arbitrary precision integers and then dividing them:

```  (x: 3) % (x: -4)
_3r4
3 %&x: -4
_3r4
```

Note that the syntax is analogous to the syntax for floating point numbers, but uses `r` to separate the numerator and denominator instead of `e` to separate the mantissa and exponent. Thus:

```   | _3r4             NB. absolute value
3r4
-2r5               NB. negation
_2r5
23r20
3r4-2r5            NB. subtraction
7r20
3r4*2r5            NB. multiplication
3r10
3r4%2r5            NB. division
15r8
3r4 <.@% 2r5       NB. integer division
1
3r4 (-~ <.)@% 2r5  NB. remainder
_7r8
3r4 < 2r5          NB. less than
0
3r4 <: 2r5         NB. less than or equal
0
3r4 > 2r5          NB. greater than
1
3r4 >: 2r5         NB. greater than or equal
1
3r4 = 2r5          NB. equal
0
3r4 ~: 2r5         NB. not equal
1
```

You can also coerce numbers directly to rational using x: (or to integer or floating point as appropriate using its inverse)

```   x: 3%4
3r4
x:inv 3%4
0.75
```

Increment and decrement are also included in the language, but you could just as easily add or subtract 1:

```   >: 3r4
7r4
<: 3r4
_1r4
```

J does not encourage the use of specialized mutators, but those could also be defined:

```mutadd=:adverb define
(m)=: (".m)+y
)

(m)=: (".m)-y
)
```

Note that the name whose association is being modified in this fashion needs to be quoted (or you can use an expression to provide the name):

```   n=: 3r4
7r4
'n' mutsub 1
3r4
'n' mutsub 1
_1r4
```

(Bare words to the immediate left of the assignment operator are implicitly quoted - but this is just syntactic sugar because that is such an overwhelmingly common case.)

That said, note that J's floating point numbers work just fine for the stated problem:

```   is_perfect_rational=: 2 = (1 + i.) +/@:%@([ #~ 0 = |) ]
```

Faster version (but the problem, as stated, is still tremendously inefficient):

```factors=: */&>@{@((^ i.@>:)&.>/)@q:~&__
is_perfect_rational=: 2= +/@:%@,@factors
```

Exhaustive testing would take forever:

```   I.is_perfect_rational@"0 i.2^19
6 28 496 8128
I.is_perfect_rational@x:@"0 i.2^19x
6 28 496 8128
```

More limited testing takes reasonable amounts of time:

```   (#~ is_perfect_rational"0) (* <:@+:) 2^i.10x
6 28 496 8128
```

## Java

Uses BigRational class: Arithmetic/Rational/Java

```public class BigRationalFindPerfectNumbers {
public static void main(String[] args) {
int MAX_NUM = 1 << 19;
System.out.println("Searching for perfect numbers in the range [1, " + (MAX_NUM - 1) + "]");

BigRational TWO = BigRational.valueOf(2);
for (int i = 1; i < MAX_NUM; i++) {
BigRational reciprocalSum = BigRational.ONE;
if (i > 1)
int maxDivisor = (int) Math.sqrt(i);
if (maxDivisor >= i)
maxDivisor--;

for (int divisor = 2; divisor <= maxDivisor; divisor++) {
if (i % divisor == 0) {
int dividend = i / divisor;
if (divisor != dividend)
}
}
if (reciprocalSum.equals(TWO))
System.out.println(String.valueOf(i) + " is a perfect number");
}
}
}
```
Output:
```Searching for perfect numbers in the range [1, 524287]
6 is a perfect number
28 is a perfect number
496 is a perfect number
8128 is a perfect number```

## JavaScript

[This section is included from a subpage and should be edited there, not here.]
The core of the Rational class

<lang javascript>// the constructor function Rational(numerator, denominator) {

```   if (denominator === undefined)
denominator = 1;
else if (denominator == 0)
throw "divide by zero";
```
```   this.numer = numerator;
if (this.numer == 0)
this.denom = 1;
else
this.denom = denominator;
```
```   this.normalize();
```

}

// getter methods Rational.prototype.numerator = function() {return this.numer}; Rational.prototype.denominator = function() {return this.denom};

// clone a rational Rational.prototype.dup = function() {

```   return new Rational(this.numerator(), this.denominator());
```

};

// conversion methods Rational.prototype.toString = function() {

```   if (this.denominator() == 1) {
return this.numerator().toString();
} else {
// implicit conversion of numbers to strings
return this.numerator() + '/' + this.denominator()
}
```

}; Rational.prototype.toFloat = function() {return eval(this.toString())} Rational.prototype.toInt = function() {return Math.floor(this.toFloat())};

// reduce Rational.prototype.normalize = function() {

```   // greatest common divisor
var a=Math.abs(this.numerator()), b=Math.abs(this.denominator())
while (b != 0) {
var tmp = a;
a = b;
b = tmp % b;
}
// a is the gcd
```
```   this.numer /= a;
this.denom /= a;
if (this.denom < 0) {
this.numer *= -1;
this.denom *= -1;
}
return this;
```

}

// absolute value // returns a new rational Rational.prototype.abs = function() {

```   return new Rational(Math.abs(this.numerator()), this.denominator());
```

};

// inverse // returns a new rational Rational.prototype.inv = function() {

```   return new Rational(this.denominator(), this.numerator());
```

};

// // arithmetic methods

```   for (var i = 0; i < arguments.length; i++) {
this.numer = this.numer * arguments[i].denominator() + this.denom * arguments[i].numerator();
this.denom = this.denom * arguments[i].denominator();
}
return this.normalize();
```

};

```   for (var i = 0; i < arguments.length; i++) {
this.numer = this.numer * arguments[i].denominator() - this.denom * arguments[i].numerator();
this.denom = this.denom * arguments[i].denominator();
}
return this.normalize();
```

};

// unary "-" operator // returns a new rational Rational.prototype.neg = function() {

```   return (new Rational(0)).subtract(this);
```

};

```   for (var i = 0; i < arguments.length; i++) {
this.numer *= arguments[i].numerator();
this.denom *= arguments[i].denominator();
}
return this.normalize();
```

};

// modifies receiver Rational.prototype.divide = function(rat) {

```   return this.multiply(rat.inv());
```

}

// increment // modifies receiver Rational.prototype.inc = function() {

```   this.numer += this.denominator();
return this.normalize();
```

}

// decrement // modifies receiver Rational.prototype.dec = function() {

```   this.numer -= this.denominator();
return this.normalize();
```

}

// // comparison methods

Rational.prototype.isZero = function() {

```   return (this.numerator() == 0);
```

} Rational.prototype.isPositive = function() {

```   return (this.numerator() > 0);
```

} Rational.prototype.isNegative = function() {

```   return (this.numerator() < 0);
```

}

Rational.prototype.eq = function(rat) {

```   return this.dup().subtract(rat).isZero();
```

} Rational.prototype.ne = function(rat) {

```   return !(this.eq(rat));
```

} Rational.prototype.lt = function(rat) {

```   return this.dup().subtract(rat).isNegative();
```

} Rational.prototype.gt = function(rat) {

```   return this.dup().subtract(rat).isPositive();
```

} Rational.prototype.le = function(rat) {

```   return !(this.gt(rat));
```

} Rational.prototype.ge = function(rat) {

```   return !(this.lt(rat));
```

}</lang>

Testing

<lang javascript>function assert(cond, msg) { if (!cond) throw msg; }

print('testing') var a, b, c, d, e, f;

//test creation a = new Rational(0); assert(a.toString() == "0", "Rational(0).toString() == '0'") a = new Rational(2); assert(a.toString() == "2", "Rational(2).toString() == '2'") a = new Rational(1,2); assert(a.toString() == "1/2", "Rational(1,2).toString() == '1/2'") b = new Rational(2,-12); assert(b.toString() == "-1/6", "Rational(1,6).toString() == '1/6'") f = new Rational(0,9)

a = new Rational(1,3) b = new Rational(1,2) c = new Rational(1,3)

assert(!(a.eq(b)), "1/3 == 1/2") assert(a.eq(c), "1/3 == 1/3") assert(a.ne(b), "1/3 != 1/2") assert(!(a.ne(c)), "1/3 != 1/3") assert(a.lt(b), "1/3 < 1/2") assert(!(b.lt(a)), "1/2 < 1/3") assert(!(a.lt(c)), "1/3 < 1/3") assert(!(a.gt(b)), "1/3 > 1/2") assert(b.gt(a), "1/2 > 1/3") assert(!(a.gt(c)), "1/3 > 1/3")

assert(a.le(b), "1/3 <= 1/2") assert(!(b.le(a)), "1/2 <= 1/3") assert(a.le(c), "1/3 <= 1/3") assert(!(a.ge(b)), "1/3 >= 1/2") assert(b.ge(a), "1/2 >= 1/3") assert(a.ge(c), "1/3 >= 1/3")

a = new Rational(1,2) b = new Rational(1,6) a.add(b); assert(a.eq(new Rational(2,3)), "1/2 + 1/6 == 2/3") c = a.neg(); assert(a.eq(new Rational(2,3)), "neg(1/2) == -1/2")

```            assert(c.eq(new Rational(2,-3)), "neg(1/2) == -1/2")
```

d = c.abs(); assert(c.eq(new Rational(-2,3)), "abs(neg(1/2)) == 1/2")

```            assert(d.eq(new Rational(2,3)), "abs(neg(1/2)) == 1/2")
```

b.subtract(a); assert(b.eq(new Rational(-1,2)), "1/6 - 1/2 == -1/3")

c = a.neg().abs(); assert(c.eq(a), "abs(neg(1/2)) == 1/2") c = (new Rational(-1,3)).inv(); assert(c.toString() == '-3', "inv(1/6 - 1/2) == -3") try {

```   e = f.inv();
throw "should have been an error: " +f + '.inv() = ' + e
```

} catch (e) {

```   assert(e == "divide by zero", "0.inv() === error")
```

}

b = new Rational(1,6) b.add(new Rational(2,3), new Rational(4,2)); assert(b.toString() == "17/6", "1/6+2/3+4/2 == 17/6");

a = new Rational(1,3); b = new Rational(1,6) c = new Rational(5,6); d = new Rational(1/5); e = new Rational(2); f = new Rational(0,9);

assert(c.dup().multiply(d).eq(b), "5/6 * 1/5 = 1/6") assert(c.dup().multiply(d,e).eq(a), "5/6 * 1/5 *2 = 1/3") assert(c.dup().multiply(d,e,f).eq(f), "5/6 * 1/5 *2*0 = 0")

c.divide(new Rational(5)); assert(c.eq(b), "5/6 / 5 = 1/6b")

try {

```   e = c.divide(f)
throw "should have been an error: " + c + "/" + f + '= ' + e
```

} catch (e) {

```   assert(e == "divide by zero", "0.inv() === error")
```

}

print('all tests passed');</lang>

Finding perfect numbers

<lang javascript>function factors(num) {

```   var factors = new Array();
var sqrt = Math.floor(Math.sqrt(num));
for (var i = 1; i <= sqrt; i++) {
if (num % i == 0) {
factors.push(i);
if (num / i != i)
factors.push(num / i);
}
}
factors.sort(function(a,b){return a-b});  // numeric sort
return factors;
```

}

function isPerfect(n) {

```   var sum = new Rational(0);
var fctrs = factors(n);
for (var i = 0; i < fctrs.length; i++)
```
```   // note, fctrs includes 1, so sum should be 2
return sum.toFloat() == 2.0;
```

}

// find perfect numbers less than 2^19 for (var n = 2; n < Math.pow(2,19); n++)

```   if (isPerfect(n))
print("perfect: " + n);
```

// test 5th perfect number var n = Math.pow(2,12) * (Math.pow(2,13) - 1); if (isPerfect(n))

```   print("perfect: " + n);</lang>
```
Output:
```perfect: 6
perfect: 28
perfect: 496
perfect: 8128
perfect: 33550336```

## jq

Works with: jq

Works with gojq, the Go implementation of jq

In this entry, a jq module for rational arithmetic is first presented. It can be included or imported using jq's "include" or "import" directives. The module includes functions for taking square roots and for converting a rational to a decimal string or decimal string approximation, and is sufficient for the jq solution at Faulhaber's_triangle#jq.

The constructor is named "r" rather than "frac", mainly because "r" is short and handy for what is here more than a simple constructor (it can also be used for normalization and to divide one rational by another), and because name conflicts can easily be resolved using jq's module system.

The other operators for working with rationals also begin with the letter "r":

Comparisons

• `requal`, `rgreaterthan`, `rgreaterthanOrEqual`, `rlessthan`, `rlessthanOrEqual`

Printing

• `rpp` for pretty-printing
• `r_to_decimal` for a decimal string representation

Unary

• `rabs` for unary `abs`
• `rfloor` like `floor`
• `rinv` for unary inverse
• `rminus` for unary minus
• `rround` for rounding

Arithmetic

• `rminus` for subtraction
• `rmult` for multiplication
• `rdiv` for division
• `rsqrt` for square roots

In the following notes, "Rational" refers to a reduced-form rational represented by a JSON object of the form {n:\$n, d:\$d} where n signifies the numerator and d the denominator, and \$d > 0.

The notation `\$p // \$q` is also used, and this is the form used for pretty-printing by the filter rpp/0.

All the "r"-prefixed functions defined here are polymorphic in the sense that an integer or rational can be used where a Rational would normally be expected. This may be especially useful in the case of requal/2.

module {"name": "Rational"};

```# a and b are assumed to be non-zero integers
def gcd(a; b):
# subfunction expects [a,b] as input
# i.e. a ~ .[0] and b ~ .[1]
def rgcd: if .[1] == 0 then .[0]
else [.[1], .[0] % .[1]] | rgcd
end;
[a,b] | rgcd;

# To take advantage of gojq's support for accurate integer division:
def idivide(\$j):
. as \$i
| (\$i % \$j) as \$mod
| (\$i - \$mod) / \$j ;

# To take advantage of gojq's arbitrary-precision integer arithmetic:
def power(\$b): . as \$in | reduce range(0;\$b) as \$i (1; . * \$in);

# \$p should be an integer or a rational
# \$q should be a non-zero integer or a rational
# Output:  a Rational: \$p // \$q
def r(\$p;\$q):
def r: if type == "number" then {n: ., d: 1} else . end;
# The remaining subfunctions assume all args are Rational
def n: if .d < 0 then {n: -.n, d: -.d} else . end;
def rdiv(\$a;\$b):
(\$a.d * \$b.n) as \$denom
| if \$denom==0 then "r: division by 0" | error
else r(\$a.n * \$b.d; \$denom)
end;
if \$q == 1 and (\$p|type) == "number" then {n: \$p, d: 1}
elif \$q == 0 then "r: denominator cannot be 0" | error
else if (\$p|type == "number") and (\$q|type == "number")
then gcd(\$p;\$q) as \$g
| {n: (\$p/\$g), d: (\$q/\$g)} | n
else rdiv(\$p|r; \$q|r)
end
end;

# Polymorphic (integers and rationals in general)
def requal(\$a; \$b):
if \$a | type == "number" and \$b | type == "number" then \$a == \$b
else r(\$a;1) == r(\$b;1)
end;

# Input: a Rational
# Output: a Rational with a denominator that has no more than \$digits digits
# and such that |rBefore - rAfter| < 1/(10|power(\$digits)
# where \$digits should be a positive integer.
def rround(\$digits):
if .d | length > \$digits
then (10|power(\$digits)) as \$p
| .d as \$d
| r(\$p * .n | idivide(\$d); \$p)
else . end;

def r: if type == "number" then {n: ., d: 1} else . end;
(\$a|r) as {n: \$na, d: \$da}
| (\$b|r) as {n: \$nb, d: \$db}
| r( (\$na * \$db) + (\$nb * \$da); \$da * \$db );

def rmult(\$a; \$b):
def r: if type == "number" then {n: ., d: 1} else . end;
(\$a|r) as {n: \$na, d: \$da}
| (\$b|r) as {n: \$nb, d: \$db}
| r( \$na * \$nb; \$da * \$db ) ;

# Input: an array of rationals (integers and/or Rationals)
# Output: a Rational computed using left-associativity
def rmult:
if length == 0 then r(1;1)
elif length == 1 then r(.[0]; 1)  # ensure the result is Rational
else .[0] as \$first
| reduce .[1:][] as \$x (\$first; rmult(.; \$x))
end;

# Input: an array of rationals (integers and/or Rationals)
# Output: a Rational computed using left-associativity
if length == 0 then r(0;1)
elif length == 1 then r(.[0]; 1) # ensure the result is Rational
else .[0] as \$first
| reduce .[1:][] as \$x (\$first; radd(. ; \$x))
end;

def rabs: r(.;1) | r(.n|length; .d|length);

def rminus: r(-1 * .n; .d);

def rminus(\$a; \$b): radd(\$a; rmult(-1; \$b));

# Note that rinv does not check for division by 0
def rinv: r(1; .);

def rdiv(\$a; \$b): r(\$a; \$b);

# Input: an integer or a Rational, \$p
# Output: \$p < \$q
def rlessthan(\$q):
# lt(\$b) assumes . and \$b have the same sign
def lt(\$b):
. as \$a
| (\$a.n * \$b.d) < (\$b.n * \$a.d);

if \$q|type == "number" then rlessthan(r(\$q;1))
else if type == "number" then r(.;1) else . end
| if .n < 0
then if (\$q.n >= 0) then true
else . as \$p | (\$q|rminus | rlessthan(\$p|rminus))
end
else lt(\$q)
end
end;

def rgreaterthan(\$q):
. as \$p | \$q | rlessthan(\$p);

def rlessthanOrEqual(\$q): requal(.;\$q) or rlessthan(\$q);
def rgreaterthanOrEqual(\$q): requal(.;\$q) or rgreaterthan(\$q);

# Input: non-negative integer or Rational
def rsqrt(precision):
r(.;1) as \$n
| (precision + 1) as \$digits
| def update: rmult( r(1;2); radd(.x; rdiv(\$n; .x))) | rround(\$digits);

| def update: rmult( r(1;2); radd(.x; rdiv(\$n; .x)));

r(1; 10|power(precision)) as \$p
| { x: .}
| .root = update
| until( rminus(.root; .x) | rabs | rlessthan(\$p);
.x = .root
| .root = update )
| .root ;

# Use native floats
# q.v. r_to_decimal(precision)
def r_to_decimal: .n / .d;

# Input: a Rational, or {n, d} in general, or an integer.
# Output: a string representation of the input as a decimal number.
# If the input is a number, it is simply converted to a string.
# Otherwise, \$precision determines the number of digits after the decimal point,
# obtained by truncating, but trailing 0s are omitted.
# Examples assuming \$digits is 5:
#  -0//1 => "0"
#   2//1 => "2"
#   1//2 => "0.5"
#   1//3 => "0.33333"
#   7//9 => "0.77777"
#   1//100 => "0.01"
#  -1//10 => "-0.1"
#   1//1000000 => "0."
def r_to_decimal(\$digits):
if .n == 0 # captures the annoying case of -0
then "0"
elif type == "number" then tostring
elif .d < 0 then {n: -.n, d: -.d}|r_to_decimal(\$digits)
elif .n < 0
then "-" + ((.n = -.n) | r_to_decimal(\$digits))
else (10|power(\$digits)) as \$p
| .d as \$d
| if \$d == 1 then .n|tostring
else (\$p * .n | idivide(\$d) | tostring) as \$n
| (\$n|length) as \$nlength
| (if \$nlength > \$digits then \$n[0:\$nlength-\$digits] + "." + \$n[\$nlength-\$digits:]
else "0." + ("0"*(\$digits - \$nlength) + \$n)
end) | sub("0+\$";"")
end
end;

# Assume . is an integer or in canonical form
def rfloor:
if type == "number" then r(.;1)
elif 0 == .n or (0 < .n and .n < .d) then r(0;1)
elif 0 < .n or (.n % .d == 0) then .d as \$d | r(.n | idivide(\$d); 1)
else rminus( r( - .n; .d) | rfloor | rminus; 1)
end;

# pretty print ala Julia
def rpp: "\(.n) // \(.d)";```

Perfect Numbers

```# divisors as an unsorted stream
def divisors:
if . == 1 then 1
else . as \$n
| label \$out
| range(1; \$n) as \$i
| (\$i * \$i) as \$i2
| if \$i2 > \$n then break \$out
else if \$i2 == \$n
then \$i
elif (\$n % \$i) == 0
then \$i, (\$n/\$i)
else empty
end
end
end;

def is_perfect:
requal(2; [divisors | r(1;. )] | radd);

# Example:
range(1;pow(2;19)) | select( is_perfect )```
Output:
```6
28
496
8128
```

## Julia

Julia has native support for rational numbers. Rationals are expressed as m//n, where m and n are integers. In addition to supporting most of the usual mathematical functions in a natural way on rationals, the methods num and den provide the fully reduced numerator and denominator of a rational value.

Works with: Julia version 1.2
```using Primes
divisors(n) = foldl((a, (p, e)) -> vcat((a * [p^i for i in 0:e]')...), factor(n), init=[1])

isperfect(n) = sum(1 // d for d in divisors(n)) == 2

lo, hi = 2, 2^19
println("Perfect numbers between ", lo, " and ", hi, ": ", collect(filter(isperfect, lo:hi)))
```
Output:
```Perfect numbers between 2 and 524288: [6, 28, 496, 8128]
```

## Kotlin

As it's not possible to define arbitrary symbols such as // to be operators in Kotlin, we instead use infix functions idiv (for Ints) and ldiv (for Longs) as a shortcut to generate Frac instances.

```// version 1.1.2

fun gcd(a: Long, b: Long): Long = if (b == 0L) a else gcd(b, a % b)

infix fun Long.ldiv(denom: Long) = Frac(this, denom)

infix fun Int.idiv(denom: Int) = Frac(this.toLong(), denom.toLong())

fun Long.toFrac() = Frac(this, 1)

fun Int.toFrac() = Frac(this.toLong(), 1)

class Frac : Comparable<Frac> {
val num: Long
val denom: Long

companion object {
val ZERO = Frac(0, 1)
val ONE  = Frac(1, 1)
}

constructor(n: Long, d: Long) {
require(d != 0L)
var nn = n
var dd = d
if (nn == 0L) {
dd = 1
}
else if (dd < 0) {
nn = -nn
dd = -dd
}
val g = Math.abs(gcd(nn, dd))
if (g > 1) {
nn /= g
dd /= g
}
num = nn
denom = dd
}

constructor(n: Int, d: Int) : this(n.toLong(), d.toLong())

operator fun plus(other: Frac) =
Frac(num * other.denom + denom * other.num, other.denom * denom)

operator fun unaryPlus() = this

operator fun unaryMinus() = Frac(-num, denom)

operator fun minus(other: Frac) = this + (-other)

operator fun times(other: Frac) = Frac(this.num * other.num, this.denom * other.denom)

operator fun rem(other: Frac) = this - Frac((this / other).toLong(), 1) * other

operator fun inc() = this + ONE
operator fun dec() = this - ONE

fun inverse(): Frac {
require(num != 0L)
return Frac(denom, num)
}

operator fun div(other: Frac) = this * other.inverse()

fun abs() = if (num >= 0) this else -this

override fun compareTo(other: Frac): Int {
val diff = this.toDouble() - other.toDouble()
return when {
diff < 0.0  -> -1
diff > 0.0  -> +1
else        ->  0
}
}

override fun equals(other: Any?): Boolean {
if (other == null || other !is Frac) return false
return this.compareTo(other) == 0
}

override fun hashCode() = num.hashCode() xor denom.hashCode()

override fun toString() = if (denom == 1L) "\$num" else "\$num/\$denom"

fun toDouble() = num.toDouble() / denom

fun toLong() = num / denom
}

fun isPerfect(n: Long): Boolean {
var sum = Frac(1, n)
val limit = Math.sqrt(n.toDouble()).toLong()
for (i in 2L..limit) {
if (n % i == 0L) sum += Frac(1, i) + Frac(1, n / i)
}
return sum == Frac.ONE
}

fun main(args: Array<String>) {
var frac1 = Frac(12, 3)
println ("frac1 = \$frac1")
var frac2 = 15 idiv 2
println("frac2 = \$frac2")
println("frac1 <= frac2 is \${frac1 <= frac2}")
println("frac1 >= frac2 is \${frac1 >= frac2}")
println("frac1 == frac2 is \${frac1 == frac2}")
println("frac1 != frac2 is \${frac1 != frac2}")
println("frac1 + frac2 = \${frac1 + frac2}")
println("frac1 - frac2 = \${frac1 - frac2}")
println("frac1 * frac2 = \${frac1 * frac2}")
println("frac1 / frac2 = \${frac1 / frac2}")
println("frac1 % frac2 = \${frac1 % frac2}")
println("inv(frac1)    = \${frac1.inverse()}")
println("abs(-frac1)   = \${-frac1.abs()}")
println("inc(frac2)    = \${++frac2}")
println("dec(frac2)    = \${--frac2}")
println("dbl(frac2)    = \${frac2.toDouble()}")
println("lng(frac2)    = \${frac2.toLong()}")
println("\nThe Perfect numbers less than 2^19 are:")
// We can skip odd numbers as no known perfect numbers are odd
for (i in 2 until (1 shl 19) step 2) {
if (isPerfect(i.toLong())) print("  \$i")
}
println()
}
```
Output:
```frac1 = 4
frac2 = 15/2
frac1 <= frac2 is true
frac1 >= frac2 is false
frac1 == frac2 is false
frac1 != frac2 is true
frac1 + frac2 = 23/2
frac1 - frac2 = -7/2
frac1 * frac2 = 30
frac1 / frac2 = 8/15
frac1 % frac2 = 4
inv(frac1)    = 1/4
abs(-frac1)   = -4
inc(frac2)    = 17/2
dec(frac2)    = 15/2
dbl(frac2)    = 7.5
lng(frac2)    = 7

The Perfect numbers less than 2^19 are:
6  28  496  8128
```

## Liberty BASIC

Testing all numbers up to 2 ^ 19 takes an excessively long time.

```n=2^19
for testNumber=1 to n
sum\$=castToFraction\$(0)
for factorTest=1 to sqr(testNumber)
next factorTest
if equal(sum\$,castToFraction\$(2))=1 then print testNumber
next testNumber
end

function abs\$(a\$)
aNumerator=val(word\$(a\$,1,"/"))
bNumerator=abs(aNumerator)
b\$=str\$(bNumerator)+"/"+str\$(bDenominator)
abs\$=simplify\$(b\$)
end function

function negate\$(a\$)
aNumerator=val(word\$(a\$,1,"/"))
bNumerator=-1*aNumerator
b\$=str\$(bNumerator)+"/"+str\$(bDenominator)
negate\$=simplify\$(b\$)
end function

aNumerator=val(word\$(a\$,1,"/"))
bNumerator=val(word\$(b\$,1,"/"))
bDenominator=val(word\$(b\$,2,"/"))
c\$=str\$(cNumerator)+"/"+str\$(cDenominator)
end function

function subtract\$(a\$,b\$)
aNumerator=val(word\$(a\$,1,"/"))
bNumerator=val(word\$(b\$,1,"/"))
bDenominator=val(word\$(b\$,2,"/"))
c\$=str\$(cNumerator)+"/"+str\$(cDenominator)
subtract\$=simplify\$(c\$)
end function

function multiply\$(a\$,b\$)
aNumerator=val(word\$(a\$,1,"/"))
bNumerator=val(word\$(b\$,1,"/"))
bDenominator=val(word\$(b\$,2,"/"))
cNumerator=aNumerator*bNumerator
c\$=str\$(cNumerator)+"/"+str\$(cDenominator)
multiply\$=simplify\$(c\$)
end function

function divide\$(a\$,b\$)
divide\$=multiply\$(a\$,reciprocal\$(b\$))
end function

function simplify\$(a\$)
aNumerator=val(word\$(a\$,1,"/"))
if aNumerator<0 and aDenominator<0 then gcd=-1*gcd
bNumerator=aNumerator/gcd
b\$=str\$(bNumerator)+"/"+str\$(bDenominator)
simplify\$=b\$
end function

function reciprocal\$(a\$)
aNumerator=val(word\$(a\$,1,"/"))
end function

function equal(a\$,b\$)
if simplify\$(a\$)=simplify\$(b\$) then equal=1:else equal=0
end function

function castToFraction\$(a)
do
exp=exp+1
a=a*10
loop until a=int(a)
castToFraction\$=simplify\$(str\$(a)+"/"+str\$(10^exp))
end function

function castToReal(a\$)
aNumerator=val(word\$(a\$,1,"/"))
end function

function castToInt(a\$)
castToInt=int(castToReal(a\$))
end function

function GCD(a,b)
if a=0 then
GCD=1
else
if a>=b then
while b
c = a
a = b
b = c mod b
GCD = abs(a)
wend
else
GCD=GCD(b,a)
end if
end if
end function```

## Lingo

A new 'frac' data type can be implemented like this:

```-- parent script "Frac"
property num
property denom

----------------------------------------
-- @constructor
-- @param {integer} numerator
-- @param {integer} [denominator=1]
----------------------------------------
on new (me, numerator, denominator)
if voidP(denominator) then denominator = 1
if denominator=0 then return VOID -- rule out division by zero
g = me._gcd(numerator, denominator)
if g<>0 then
numerator = numerator/g
denominator = denominator/g
else
numerator = 0
denominator = 1
end if
if denominator<0 then
numerator = -numerator
denominator = -denominator
end if
me.num = numerator
me.denom = denominator
return me
end

----------------------------------------
-- Returns string representation "<num>/<denom>"
-- @return {string}
----------------------------------------
on toString (me)
return me.num&"/"&me.denom
end

----------------------------------------
--
----------------------------------------
on _gcd (me, a, b)
if a = 0 then return b
if b = 0 then return a
if a > b then return me._gcd(b, a mod b)
return me._gcd(a, b mod a)
end```

Lingo does not support overwriting built-in operators, so 'frac'-operators must be implemented as functions:

```-- Frac library (movie script)

----------------------------------------
-- Shortcut for creating 'frac' values
-- @param {integer} numerator
-- @param {integer} denominator
-- @return {instance}
----------------------------------------
on frac (numerator, denominator)
return script("Frac").new(numerator, denominator)
end

----------------------------------------
-- All functions below this comment only support 'fracs', i.e. instances
-- of the Frac Class, as arguments. An integer n is casted to frac via frac(n).
----------------------------------------

-- Optionally supports more than 2 arguments
on fAdd (a, b) -- ...
res = a
repeat with i = 2 to the paramCount
p = param(i)
num = res.num * p.denom + res.denom * p.num
denom = res.denom * p.denom
res = frac(num, denom)
end repeat
return res
end

on fSub (a, b)
return frac(a.num * b.den - a.den * b.num, a.den * b.den)
end

-- Optionally supports more than 2 arguments
on fMul (a, b) -- ...
res = a
repeat with i = 2 to the paramCount
p = param(i)
res = frac(res.num * p.num, res.denom * p.denom)
end repeat
return res
end

on fDiv (a, b)
return frac(a.num * b.denom, a.denom * b.num)
end

on fAbs (f)
return frac(abs(f.num), f.denom)
end

on fNeg (f)
return frac(-f.num, f.denom)
end

on fEQ (a, b)
diff = fSub(a, b)
return diff.num=0
end

on fNE (a, b)
return not fEQ (a, b)
end

on fGT (a, b)
diff = fSub(a, b)
return diff.num>0
end

on fLT (a, b)
diff = fSub(a, b)
return diff.num<0
end

on fGE (a, b)
diff = fSub(a, b)
return diff.num>=0
end

on fLE (a, b)
diff = fSub(a, b)
return diff.num<=0
end```

Usage:

```f = frac(2,3)
put f.toString()
-- "2/3"

-- fractions are normalized on the fly
f = frac(4,6)
put f.toString()
-- "2/3"

-- casting integer to frac
f = frac(23)
put f.toString()
-- "23/1"```

Finding perfect numbers:

```-- in some movie script
----------------------------------------
-- Prints all perfect numbers up to n
-- @param {integer|float} n
----------------------------------------
on findPerfects (n)
repeat with i = 2 to n
sum = frac(1, i)
cnt = sqrt(i)
repeat with fac = 2 to cnt
if i mod fac = 0 then sum = fAdd(sum, frac(1, fac), frac(fac, i))
end repeat
if sum.denom = sum.num then put i
end repeat
end```
```findPerfects(power(2, 19))
-- 6
-- 28
-- 496
-- 8128```

## Lua

```function gcd(a,b) return a == 0 and b or gcd(b % a, a) end

do
local function coerce(a, b)
if type(a) == "number" then return rational(a, 1), b end
if type(b) == "number" then return a, rational(b, 1) end
return a, b
end
rational = setmetatable({
local a, b = coerce(a, b)
return rational(a.num * b.den + a.den * b.num, a.den * b.den)
end,
__sub = function(a, b)
local a, b = coerce(a, b)
return rational(a.num * b.den - a.den * b.num, a.den * b.den)
end,
__mul = function(a, b)
local a, b = coerce(a, b)
return rational(a.num * b.num, a.den * b.den)
end,
__div = function(a, b)
local a, b = coerce(a, b)
return rational(a.num * b.den, a.den * b.num)
end,
__pow = function(a, b)
if type(a) == "number" then return a ^ (b.num / b.den) end
return rational(a.num ^ b, a.den ^ b) --runs into a problem if these aren't integers
end,
__concat = function(a, b)
if getmetatable(a) == rational then return a.num .. "/" .. a.den .. b end
return a .. b.num .. "/" .. b.den
end,
__unm = function(a) return rational(-a.num, -a.den) end}, {
__call = function(z, a, b) return setmetatable({num = a / gcd(a, b),den = b / gcd(a, b)}, z) end} )
end

print(rational(2, 3) + rational(3, 5) - rational(1, 10) .. "") --> 7/6
print((rational(4, 5) * rational(5, 9)) ^ rational(1, 2) .. "") --> 2/3
print(rational(45, 60) / rational(5, 2) .. "") --> 3/10
print(5 + rational(1, 3) .. "") --> 16/3

function findperfs(n)
local ret = {}
for i = 1, n do
sum = rational(1, i)
for fac = 2, i^.5 do
if i % fac == 0 then
sum = sum + rational(1, fac) + rational(fac, i)
end
end
if sum.den == sum.num then
ret[#ret + 1] = i
end
end
return table.concat(ret, '\n')
end
print(findperfs(2^19))
```

## M2000 Interpreter

```Class Rational {
\\ this is a compact version for this task
numerator as decimal, denominator as decimal
gcd=lambda->0
lcm=lambda->0
operator "+" {
denom=.lcm(l.denominator, .denominator)
.numerator<=denom/l.denominator*l.numerator+denom/.denominator*.numerator
if .numerator==0 then denom=1
.denominator<=denom
}
Group Real {
value {
link parent numerator, denominator to n, d
=n/d
}
}
Group ToString\$ {
value {
link parent numerator, denominator to n, d
=Str\$(n)+"/"+Str\$(d,"")
}
}
class:
Module Rational (.numerator, .denominator) {
if .denominator=0 then Error "Zero denominator"
sgn=Sgn(.numerator)*Sgn(.denominator)
.denominator<=abs(.denominator)
.numerator<=abs(.numerator)*sgn
gcd1=lambda (a as decimal, b as decimal) -> {
if a<b then swap a,b
g=a mod b
while g {
a=b:b=g: g=a mod b
}
=abs(b)
}
gdcval=gcd1(abs(.numerator), .denominator)
if gdcval<.denominator and gdcval<>0 then
.denominator/=gdcval
.numerator/=gdcval
end if
.gcd<=gcd1
.lcm<=lambda gcd=gcd1 (a as decimal, b as decimal) -> {
=a/gcd(a,b)*b
}
}
}
sum=rational(1, 1)
onediv=rational(1,1)
divcand=rational(1,1)
Profiler
For sum.denominator= 2 to 2**15 {
divcand.denominator=sum.denominator
For onediv.denominator=2 to sqrt(sum.denominator) {
if sum.denominator mod onediv.denominator = 0 then {
divcand.numerator=onediv.denominator
sum=sum+onediv+divcand
}
}
if sum.real=1 then Print sum.denominator;" is perfect"
sum.numerator=1
}
Print timecount```

## Maple

Maple has full built-in support for arithmetic with fractions (rational numbers). Fractions are treated like any other number in Maple.

```> a := 3 / 5;
a := 3/5

> numer( a );
3

> denom( a );
5```

However, while you can enter a fraction such as "4/6", it will automatically be reduced so that the numerator and denominator have no common factor:

```> b := 4 / 6;
b := 2/3```

All the standard arithmetic operators work with rational numbers. It is not necessary to call any special routines.

```> a + b;
19
--
15

> a * b;
2/5

> a / b;
9/10

> a - b;
-1
--
15

> a + 1;
8/5

> a - 1;
-2/5```

Notice that fractions are treated as exact quantities; they are not converted to floats. However, you can get a floating point approximation to any desired accuracy by applying the function evalf to a fraction.

```> evalf( 22 / 7 ); # default is 10 digits
3.142857143

> evalf[100]( 22 / 7 ); # 100 digits
3.142857142857142857142857142857142857142857142857142857142857142857\
142857142857142857142857142857143```

## Mathematica / Wolfram Language

Mathematica has full support for fractions built-in. If one divides two exact numbers it will be left as a fraction if it can't be simplified. Comparison, addition, division, product et cetera are built-in:

```4/16
3/8
8/4
4Pi/2
16!/10!
Sqrt[9/16]
Sqrt[3/4]
(23/12)^5
2 + 1/(1 + 1/(3 + 1/4))

1/2+1/3+1/5
8/Pi+Pi/8 //Together
13/17 + 7/31
Sum[1/n,{n,1,100}]      (*summation of 1/1 + 1/2 + 1/3 + 1/4+ .........+ 1/99 + 1/100*)

1/2-1/3
a=1/3;a+=1/7

1/4==2/8
1/4>3/8
Pi/E >23/20
1/3!=123/370
Sin[3]/Sin[2]>3/20

Numerator[6/9]
Denominator[6/9]
```

gives back:

```1/4
3/8
2
2 Pi
5765760
3/4
Sqrt[3]/2
6436343 / 248832
47/17

31/30
(64+Pi^2) / (8 Pi)
522 / 527
14466636279520351160221518043104131447711 / 2788815009188499086581352357412492142272

1/6
10/21

True
False
True
True
True

2
3```

As you can see, Mathematica automatically handles fraction as exact things, it doesn't evaluate the fractions to a float. It only does this when either the numerator or the denominator is not exact. I only showed integers above, but Mathematica can handle symbolic fraction in the same and complete way:

```c/(2 c)
(b^2 - c^2)/(b - c)  // Cancel
1/2 + b/c // Together
```

gives back:

```1/2
b+c
(2 b+c) / (2 c)
```

Moreover it does simplification like Sin[x]/Cos[x] => Tan[x]. Division, addition, subtraction, powering and multiplication of a list (of any dimension) is automatically threaded over the elements:

```1+2*{1,2,3}^3
```

gives back:

```{3, 17, 55}
```

To check for perfect numbers in the range 1 to 2^25 we can use:

```found={};
CheckPerfect[num_Integer]:=If[Total[1/Divisors[num]]==2,AppendTo[found,num]];
Do[CheckPerfect[i],{i,1,2^25}];
found
```

gives back:

```{6, 28, 496, 8128, 33550336}
```

Final note; approximations of fractions to any precision can be found using the function N.

## Maxima

```/* Rational numbers are builtin */
a: 3 / 11;
3/11

b: 117 / 17;
117/17

a + b;
1338/187

a - b;
-1236/187

a * b;
351/187

a / b;
17/429

a^5;
243/161051

num(a);
3

denom(a);
11

ratnump(a);
true
```

## Modula-2

[This section is included from a subpage and should be edited there, not here.]
Works with: FST Modula-2 v4.0 version no object oriented code used

This is incomplete as the Perfect Numbers task has not been addressed.

Definition Module

<lang modula2>DEFINITION MODULE Rational;

```   TYPE RAT =  RECORD
numerator : INTEGER;
denominator : INTEGER;
END;
```
```   PROCEDURE IGCD( i : INTEGER; j : INTEGER ) : INTEGER;
PROCEDURE ILCM( i : INTEGER; j : INTEGER ) : INTEGER;
PROCEDURE IABS( i : INTEGER ) : INTEGER;
```
```   PROCEDURE RNormalize( i : RAT ) : RAT;
PROCEDURE RCreate( num : INTEGER; dem : INTEGER ) : RAT;
PROCEDURE RAdd( i : RAT; j : RAT ) : RAT;
PROCEDURE RSubtract( i : RAT; j : RAT ) : RAT;
PROCEDURE RMultiply( i : RAT; j : RAT ) : RAT;
PROCEDURE RDivide( i : RAT; j : RAT ) : RAT;
PROCEDURE RAbs( i : RAT ) : RAT;
PROCEDURE RInv( i : RAT ) : RAT;
PROCEDURE RNeg( i : RAT ) : RAT;
```
```   PROCEDURE RInc( i : RAT ) : RAT;
PROCEDURE RDec( i : RAT ) : RAT;

PROCEDURE REQ( i : RAT; j : RAT ) : BOOLEAN;
PROCEDURE RNE( i : RAT; j : RAT ) : BOOLEAN;
PROCEDURE RLT( i : RAT; j : RAT ) : BOOLEAN;
PROCEDURE RLE( i : RAT; j : RAT ) : BOOLEAN;
PROCEDURE RGT( i : RAT; j : RAT ) : BOOLEAN;
PROCEDURE RGE( i : RAT; j : RAT ) : BOOLEAN;
```
```   PROCEDURE RIsZero( i : RAT ) : BOOLEAN;
PROCEDURE RIsNegative( i : RAT ) : BOOLEAN;
PROCEDURE RIsPositive( i : RAT ) : BOOLEAN;
```
```   PROCEDURE RToString( i : RAT; VAR S : ARRAY OF CHAR );
PROCEDURE RToRational( s : ARRAY OF CHAR ) : RAT;
```
```   PROCEDURE WriteRational( i : RAT );
```

END Rational.</lang>

Implementation Module

<lang modula2>IMPLEMENTATION MODULE Rational;

```   FROM Strings IMPORT Assign, Append, Pos, Copy, Length;
FROM NumberConversion IMPORT IntToString, StringToInt;
```
```   FROM InOut IMPORT WriteString (*, WriteCard,WriteLine, WriteInt, WriteLn *);
```
```   PROCEDURE IGCD( i : INTEGER; j : INTEGER ) : INTEGER;
VAR
res : INTEGER;
BEGIN
IF j = 0 THEN
res := i;
ELSE
res := IGCD( j, i MOD j );
END;
```
```       RETURN res;
END IGCD;
```
```   PROCEDURE ILCM( i : INTEGER; j : INTEGER ) : INTEGER;
VAR
res : INTEGER;
BEGIN
res := (i DIV IGCD( i, j ) ) * j;
RETURN res;
END ILCM;
```
```   PROCEDURE IABS( i : INTEGER ) : INTEGER;
VAR
res : INTEGER;
BEGIN
IF i < 0 THEN
res := i * (-1);
ELSE
res := i;
END;
RETURN res;
END IABS;
```
```   PROCEDURE RNormalize( i : RAT ) : RAT;
VAR
gcd : INTEGER;
res : RAT;
BEGIN
gcd := IGCD( ABS( i.numerator ), ABS( i.denominator ) );
IF gcd <> 0 THEN
res.numerator := i.numerator DIV gcd;
res.denominator := i.denominator DIV gcd;
IF ( res.denominator < 0 ) THEN
res.numerator := res.numerator * (-1);
res.denominator := res.denominator * (-1);
END;
ELSE
WITH res DO
numerator := 0;
denominator := 0;
END;
END;
RETURN res;
END RNormalize;
```
```   PROCEDURE RCreate( num : INTEGER; dem : INTEGER ) : RAT;
VAR
rat : RAT;
BEGIN
WITH rat DO
numerator := num;
denominator := dem;
END;
RETURN RNormalize(rat);
END RCreate;
```
```   PROCEDURE RAdd( i : RAT; j : RAT ) : RAT;
BEGIN
RETURN RCreate( i.numerator * j.denominator + j.numerator * i.denominator, i.denominator * j.denominator );
```
```   PROCEDURE RSubtract( i : RAT; j : RAT ) : RAT;
BEGIN
RETURN RCreate( i.numerator * j.denominator - j.numerator * i.denominator, i.denominator * j.denominator );
END RSubtract;
```
```   PROCEDURE RMultiply( i : RAT; j : RAT ) : RAT;
BEGIN
RETURN RCreate( i.numerator * j.numerator, i.denominator * j.denominator );
END RMultiply;
```
```   PROCEDURE RDivide( i : RAT; j : RAT ) : RAT;
BEGIN
RETURN RCreate( i.numerator * j.denominator, i.denominator * j.numerator );
END RDivide;
```
```   PROCEDURE RAbs( i : RAT ) : RAT;
BEGIN
RETURN RCreate( IABS( i.numerator ), i.denominator );
END RAbs;
```
```   PROCEDURE RInv( i : RAT ) : RAT;
BEGIN
RETURN RCreate( i.denominator, i.numerator );
END RInv;
```
```   PROCEDURE RNeg( i : RAT ) : RAT;
BEGIN
RETURN RCreate( i.numerator * (-1), i.denominator );
END RNeg;
```
```   PROCEDURE RInc( i : RAT ) : RAT;
BEGIN
RETURN RCreate( i.numerator + i.denominator, i.denominator );
END RInc;
```
```   PROCEDURE RDec( i : RAT ) : RAT;
BEGIN
RETURN RCreate( i.numerator - i.denominator, i.denominator );
END RDec;
```
```   PROCEDURE REQ( i : RAT; j : RAT ) : BOOLEAN;
VAR
ii : RAT;
jj : RAT;
BEGIN
ii := RNormalize( i );
jj := RNormalize( j );
RETURN ( ( ii.numerator = jj.numerator ) AND ( ii.denominator = jj.denominator ) );
END REQ;
```
```   PROCEDURE RNE( i : RAT; j : RAT ) : BOOLEAN;
BEGIN
RETURN NOT REQ( i, j );
END RNE;
```
```   PROCEDURE RLT( i : RAT; j : RAT ) : BOOLEAN;
BEGIN
RETURN RIsNegative( RSubtract( i, j ) );
END RLT;
```
```   PROCEDURE RLE( i : RAT; j : RAT ) : BOOLEAN;
BEGIN
RETURN NOT RGT( i, j );
END RLE;
```
```   PROCEDURE RGT( i : RAT; j : RAT ) : BOOLEAN;
BEGIN
RETURN RIsPositive( RSubtract( i, j ) );
END RGT;
```
```   PROCEDURE RGE( i : RAT; j : RAT ) : BOOLEAN;
BEGIN
RETURN NOT RLT( i, j );
END RGE;
```
```   PROCEDURE RIsZero( i : RAT ) : BOOLEAN;
BEGIN
RETURN i.numerator = 0;
END RIsZero;
```
```   PROCEDURE RIsNegative( i : RAT ) : BOOLEAN;
BEGIN
RETURN i.numerator < 0;
END RIsNegative;
```
```   PROCEDURE RIsPositive( i : RAT ) : BOOLEAN;
BEGIN
RETURN i.numerator > 0;
END RIsPositive;
```
```   PROCEDURE RToString( i : RAT; VAR S : ARRAY OF CHAR );
VAR
num : ARRAY [1..15] OF CHAR;
den : ARRAY [1..15] OF CHAR;
BEGIN
IF RIsZero( i ) THEN
Assign("0", S );
ELSE
IntToString( i.numerator, num, 1 );
Assign( num, S );
IF ( i.denominator <> 1 ) THEN
IntToString( i.denominator, den, 1 );
Append( S, "/" );
Append( S, den );
END;
END;
END RToString;
```
```   PROCEDURE RToRational( s : ARRAY OF CHAR ) : RAT;
VAR
n : CARDINAL;
numer : INTEGER;
denom : INTEGER;
LHS, RHS : ARRAY [ 1..20 ] OF CHAR;
Flag : BOOLEAN;
BEGIN
numer := 0;
denom := 0;
n := Pos( "/", s );
```
```       IF n > HIGH( s ) THEN
StringToInt( s, numer, Flag );
IF Flag THEN
denom := 1;
END;
ELSE
Copy( s, 0, n, LHS );
Copy( s, n+1, Length( s ), RHS );
StringToInt( LHS, numer, Flag );
IF Flag THEN
StringToInt( RHS, denom, Flag );
END;
END;
RETURN RCreate( numer, denom );
END RToRational;
```
```   PROCEDURE WriteRational( i : RAT );
VAR
res : ARRAY [0 .. 80] OF CHAR;
BEGIN
RToString( i, res );
WriteString( res );
END WriteRational;
```

END Rational.</lang>

Test Program

<lang modula2>MODULE TestRat;

```      FROM InOut IMPORT WriteString, WriteLine;
FROM Terminal IMPORT KeyPressed;
FROM Strings IMPORT CompareStr;
FROM Rational IMPORT RAT, IGCD, RCreate, RToString, RIsZero, RNormalize,
RToRational, REQ, RNE, RLT, RLE, RGT, RGE, WriteRational,
RAdd, RSubtract, RMultiply, RDivide, RAbs, RNeg, RInv;
```

VAR

```   res : INTEGER;
a, b, c, d, e, f : RAT;
ans : ARRAY [1..100] OF CHAR;
```

PROCEDURE Assert( F : BOOLEAN; S : ARRAY OF CHAR ); BEGIN

```   IF ( NOT F) THEN
WriteLine( S );
END;
```

END Assert;

BEGIN

```   a := RCreate( 0, 0 );
Assert( RIsZero( a ), "RIsZero( a )");
```
```   a := RToRational("2");
RToString( a, ans );
res := CompareStr( ans, "2" );
Assert( (res = 0), "CompareStr( RToString( a ), '2' ) = 0");
```
```   a := RToRational("1/2");
RToString( a, ans );
res := CompareStr( ans, "1/2");
Assert( res = 0, "CompareStr( RToString( a, ans ), '1/2') = 0");
```
```   b := RToRational( "2/-12" );
RToString( b, ans );
res := CompareStr( ans, "-1/6");
Assert( res = 0, "CompareStr( RToString( b, ans ), '-1/6') = 0");
```
```   f := RCreate( 0, 9 ); (* rationalizes internally to zero *)
```
```   a := RToRational("1/3");
b := RToRational("1/2");
c := RCreate( 1, 3 );
```
```   Assert( NOT REQ( a, b ), "1/3 == 1/2" );
Assert( REQ( a, c ), "1/3 == 1/3" );
Assert( RNE( a, b ), "1/3 != 1/2" );
Assert( RLT( a, b ), "1/3 < 1/2" );
Assert( NOT RLT(b,a), "1/2 < 1/3" );
Assert( NOT RLT(a,c), "1/3 < 1/3" );
Assert( NOT RGT(a,b), "1/3 > 1/2" );
Assert( RGT(b,a), "1/2 > 1/3" );
Assert( NOT RGT(a,c), "1/3 > 1/3" );
```
```   Assert( RLE( a, b ), "1/3 <= 1/2" );
Assert( NOT RLE( b, a ), "1/2 <= 1/3" );
Assert( RLE( a, c ), "1/3 <= 1/3" );
Assert( NOT RGE(a,b), "1/3 >= 1/2" );
Assert( RGE(b,a), "1/2 >= 1/3" );
Assert( RGE( a,c ), "1/3 >= 1/3" );
```
```   a := RCreate(1,2);
b := RCreate(1,6);
a := RAdd( a, b );
Assert( REQ( a, RToRational("2/3")), "1/2 + 1/6 == 2/3" );
```
```   c := RNeg( a );
Assert( REQ( a, RCreate( 2,3)), "2/3 == 2/3" );
Assert( REQ( c, RCreate( 2,-3)), "Neg 1/2 == -1/2" );
a := RCreate( 2,-3);
```
```   d := RAbs( c );
Assert( REQ( d, RCreate( 2,3 ) ), "abs(neg(1/2))==1/2" );
```
```   a := RToRational( "1/2");
b := RSubtract( b, a );
```
```   Assert( REQ( b, RCreate(-1,3) ), "1/6 - 1/2 == -1/3" );
```
```   c := RInv(b);
RToString( c, ans );
res := CompareStr( ans, "-3" );
Assert( res = 0, "inv(1/6 - 1/2) == -3" );
```
```   f := RInv( f ); (* as f normalized to zero, the reciprocal is still zero *)
```

```   b := RCreate( 1, 6);
RToString( b, ans );
res := CompareStr( ans, "17/6" );
Assert( res = 0, "1/6 + 2/3 + 4/2 = 17/6" );
```
```   a := RCreate(1,3);
b := RCreate(1,6);
c := RCreate(5,6);
d := RToRational("1/5");
e := RToRational("2");
f := RToRational("0/9");
```
```   Assert( REQ( RMultiply( c, d ), b ), "5/6 * 1/5 = 1/6" );
Assert( REQ( RMultiply( c, RMultiply( d, e ) ), a ), "5/6 * 1/5 * 2 = 1/3" );
Assert( REQ( RMultiply( c, RMultiply( d, RMultiply( e, f ) ) ), f ), "5/6 * 1/5 * 2 * 0" );
Assert( REQ( b, RDivide( c, RToRational("5" ) ) ), "5/6 / 5 = 1/6" );
```
```   e := RDivide( c, f ); (* RDivide multiplies so no divide by zero *)
```
```   WriteString("Press any key..."); WHILE NOT KeyPressed() DO END;
```

END TestRat.</lang>

## Modula-3

[This section is included from a subpage and should be edited there, not here.]

This implementation implements the task in a module named `Frac` that exports an opaque type named `T`, a standard Modula-3 practice. Although the task does not require it, the module also defines the exception `ZeroDenominator` for occasions when one might attempt division by zero, which includes at initialization. It provides two functions that return rational numbers for 0 and 1.

Interface module

<lang modula3>INTERFACE Frac;

EXCEPTION ZeroDenominator;

TYPE

``` T <: Public;
Public = OBJECT
METHODS
(* initialization and conversion *)
init(a, b: INTEGER): T RAISES { ZeroDenominator };
fromInt(a: INTEGER): T;
(* unary operators *)
abs(): T;
opposite(): T;
(* binary operators *)
plus(other: T): T;
minus(other: T): T;
times(other: T): T;
dividedBy(other: T): T RAISES { ZeroDenominator };
integerDivision(other: INTEGER): T RAISES { ZeroDenominator };
(* relations *)
equals(other: T): BOOLEAN;
notEqualTo(other: T): BOOLEAN;
lessThan(other: T): BOOLEAN;
lessEqual(other: T): BOOLEAN;
greaterEqual(other: T): BOOLEAN;
greater(other: T): BOOLEAN;
(* other easily-checked properties *)
isInt(): BOOLEAN;
(* inc/decrement *)
inc(step: CARDINAL := 1);
dec(step: CARDINAL := 1);
END;
```
``` PROCEDURE One():  T;
PROCEDURE Zero(): T;
```
``` (* I/O *)
PROCEDURE PutRat(a: T);
```

END Frac.</lang>

Implementation module

The implementation keeps all rational numbers in simplest form.

<lang modula3>MODULE Frac;

IMPORT IO;

TYPE

``` REVEAL T = Public BRANDED OBJECT
num: INTEGER := 0;
den: INTEGER := 1;
OVERRIDES
init := Init;
fromInt := FromInt;
abs := Abs;
opposite := Opposite;
plus := Plus;
minus := Minus;
times := Times;
dividedBy := DividedBy;
integerDivision := IntegerDivision;
equals := Equals;
notEqualTo := NotEqualTo;
lessThan := LessThan;
lessEqual := LessEqual;
greaterEqual := GreaterEqual;
greater := Greater;
isInt := IsInt;
inc := Inc;
dec := Dec;
END;
```

PROCEDURE Gcd(a, b: CARDINAL): CARDINAL = VAR

``` m := MAX(a, b);
n := MIN(a, b);
r: CARDINAL;
```

BEGIN

``` WHILE n # 0 DO
r := m MOD n;
m := n;
n := r;
END;
RETURN m;
```

END Gcd;

PROCEDURE Simplify(VAR a: T) = VAR

``` d := Gcd(ABS(a.num), ABS(a.den));
```

BEGIN

``` a.num := a.num DIV d;
a.den := a.den DIV d;
```

END Simplify;

PROCEDURE Init(self: T; a, b: INTEGER): T RAISES { ZeroDenominator } = BEGIN

``` IF (b > 0) THEN
self.num := a;
self.den := b;
ELSIF (b < 0) THEN
self.num := -a;
self.den := -b;
ELSE
RAISE ZeroDenominator;
END;
Simplify(self);
RETURN self;
```

END Init;

PROCEDURE FromInt(self: T; a: INTEGER): T = BEGIN

``` self.num := a;
self.den := 1;
RETURN self;
```

END FromInt;

PROCEDURE Abs(self: T): T = BEGIN

``` RETURN NEW(T, num := ABS(self.num), den := self.den);
```

END Abs;

PROCEDURE Opposite(self: T): T = BEGIN

``` RETURN NEW(T, num := -self.num, den := self.den);
```

END Opposite;

PROCEDURE Plus(self, other: T): T = VAR

``` result := NEW( T,
num := self.num * other.den + self.den * other.num,
den := self.den * other.den
);
```

BEGIN

``` Simplify(result);
RETURN result;
```

END Plus;

PROCEDURE Minus(self, other: T): T = VAR

``` result := NEW( T,
num := self.num * other.den - self.den * other.num,
den := self.den * other.den
);
```

BEGIN

``` Simplify(result);
RETURN result;
```

END Minus;

PROCEDURE Times(self, other: T): T = VAR

``` result := NEW( T,
num := self.num * other.num,
den := self.den * other.den
);
```

BEGIN

``` Simplify(result);
RETURN result;
```

END Times;

PROCEDURE DividedBy(self, other: T): T RAISES { ZeroDenominator } = VAR

``` result := NEW( T,
num := self.num * other.den,
den := self.den * other.num
);
```

BEGIN

``` IF result.den = 0 THEN RAISE ZeroDenominator; END;
IF result.den < 0 THEN
result.num := -result.num;
result.den := -result.den;
END;
Simplify(result);
RETURN result;
```

END DividedBy;

PROCEDURE IntegerDivision(self: T; other: INTEGER): T RAISES { ZeroDenominator } = VAR

``` result := NEW( T, num := self.num, den := self.den * other );
```

BEGIN

``` IF other = 0 THEN RAISE ZeroDenominator; END;
Simplify(result);
RETURN result;
```

END IntegerDivision;

PROCEDURE Equals(self, other: T): BOOLEAN = BEGIN

``` (* this trick works only because we simplify after each operation *)
RETURN (self.num = other.num) AND (self.den = other.den);
```

END Equals;

PROCEDURE NotEqualTo(self, other: T): BOOLEAN = BEGIN

``` (* this trick works only because we simplify after each operation *)
RETURN (self.num # other.num) OR (self.den # other.den);
```

END NotEqualTo;

PROCEDURE LessThan(self, other: T): BOOLEAN = BEGIN

``` RETURN self.num * other.den < self.den * other.num;
```

END LessThan;

PROCEDURE LessEqual(self, other: T): BOOLEAN = BEGIN

``` RETURN self.num * other.den <= self.den * other.num;
```

END LessEqual;

PROCEDURE GreaterEqual(self, other: T): BOOLEAN = BEGIN

``` RETURN self.num * other.den >= self.den * other.num;
```

END GreaterEqual;

PROCEDURE Greater(self, other: T): BOOLEAN = BEGIN

``` RETURN self.num * other.den > self.den * other.num;
```

END Greater;

PROCEDURE Inc(self: T; step: CARDINAL) = BEGIN

``` INC(self.num, step * self.den);
```

END Inc;

PROCEDURE Dec(self: T; step: CARDINAL) = BEGIN

``` DEC(self.num, step * self.den);
```

END Dec;

PROCEDURE IsInt(self: T): BOOLEAN = BEGIN

``` RETURN self.den = 1;
```

END IsInt;

PROCEDURE One(): T = BEGIN

``` TRY
RETURN NEW(T).init(1, 1);
EXCEPT ZeroDenominator =>
IO.Put("Something went very wrong.\n");
RETURN NIL;
END;
```

END One;

PROCEDURE Zero(): T = BEGIN

``` TRY
RETURN NEW(T).init(0, 1);
EXCEPT ZeroDenominator =>
IO.Put("Something went very wrong.\n");
RETURN NIL;
END;
```

END Zero;

PROCEDURE PutRat(a: T) = BEGIN

``` IO.PutInt(a.num);
IF a.den # 1 THEN
IO.Put(" / "); IO.PutInt(a.den);
END;
```

END PutRat;

BEGIN END Frac.</lang>

Test Program
Translation of: Nim

This module performs a few additional tests. The test for perfect numbers is based on that of Nim above. <lang modula3>MODULE TestRational EXPORTS Main;

IMPORT IO, Frac AS R, Word;

FROM Math IMPORT sqrt;

PROCEDURE PutBool(b: BOOLEAN) = BEGIN

``` IF b THEN IO.Put("true");
ELSE IO.Put("false");
END;
```

END PutBool;

VAR

``` a, b: R.T;
```
``` n: Word.T := 2;
limit: Word.T := 1;
```

BEGIN

``` TRY
a := NEW(R.T).init(2,3);
b := NEW(R.T).init(-3,4);
EXCEPT | R.ZeroDenominator =>
IO.Put("Zero division!\n");
END;
```
``` R.PutRat(a); IO.Put(" op "); R.PutRat(b); IO.Put(" = \n");
```
``` IO.Put("  + : "); R.PutRat(a.plus(b));  IO.PutChar('\n');
IO.Put("  - : "); R.PutRat(a.minus(b)); IO.PutChar('\n');
IO.Put("  * : "); R.PutRat(a.times(b)); IO.PutChar('\n');
```
``` TRY
IO.Put("  /: "); R.PutRat(a.dividedBy(b)); IO.PutChar('\n');
EXCEPT | R.ZeroDenominator =>
IO.Put("Zero division\n");
END;
```
``` IO.Put("  <  : "); PutBool(a.lessThan(b));     IO.PutChar('\n');
IO.Put("  <= : "); PutBool(a.lessEqual(b));    IO.PutChar('\n');
IO.Put("  >= : "); PutBool(a.greaterEqual(b)); IO.PutChar('\n');
IO.Put("  >  : "); PutBool(a.greater(b));      IO.PutChar('\n');
```
``` IO.PutChar('\n');
```
``` IO.Put("Increasing "); R.PutRat(a); IO.Put(" by\n");
IO.Put("  1 gives "); a.inc(1); R.PutRat(a); IO.PutChar('\n');
IO.Put("  3 gives "); a.inc(3); R.PutRat(a); IO.PutChar('\n');
```
``` IO.Put("Decreasing "); R.PutRat(a); IO.Put(" by\n");
IO.Put("  1 gives "); a.dec(1); R.PutRat(a); IO.PutChar('\n');
IO.Put("  3 gives "); a.dec(3); R.PutRat(a); IO.PutChar('\n');
```
``` limit := Word.LeftShift(limit, 19);
TRY
WHILE n < limit DO
VAR
sum := NEW(R.T).init(1, n);
maxFac := FLOOR(sqrt(FLOAT(n, LONGREAL)));
BEGIN
FOR i := 2 TO maxFac DO
IF n MOD i = 0 THEN
sum := sum.plus(NEW(R.T).init(1, i))
.plus(NEW(R.T).init(1, n DIV i));
END;
END;
IF sum.isInt() THEN
IO.Put("sum of reciprocal factors of "); IO.PutInt(n);
IO.Put(" is exactly "); R.PutRat(sum);
IF sum.equals(R.One()) THEN
IO.Put(" perfect!");
END;
IO.PutChar('\n');
END;
END;
INC(n, 2);
END;
EXCEPT R.ZeroDenominator =>
IO.Put("Something went very wrong\n");
END;
```

END TestRational.</lang>

Output:
```2 / 3 op -3 / 4 =
+ : -1 / 12
- : 17 / 12
* : -1 / 2
/: -8 / 9
<  : false
<= : false
>= : true
>  : true

Increasing 2 / 3 by
1 gives 5 / 3
3 gives 14 / 3
Decreasing 14 / 3 by
1 gives 11 / 3
3 gives 2 / 3
sum of reciprocal factors of 6 is exactly 1 perfect!
sum of reciprocal factors of 28 is exactly 1 perfect!
sum of reciprocal factors of 120 is exactly 2
sum of reciprocal factors of 496 is exactly 1 perfect!
sum of reciprocal factors of 672 is exactly 2
sum of reciprocal factors of 8128 is exactly 1 perfect!
sum of reciprocal factors of 30240 is exactly 3
sum of reciprocal factors of 32760 is exactly 3
sum of reciprocal factors of 523776 is exactly 2
```

## Nim

```import math

proc `^`[T](base, exp: T): T =
var (base, exp) = (base, exp)
result = 1

while exp != 0:
if (exp and 1) != 0:
result *= base
exp = exp shr 1
base *= base

proc gcd[T](u, v: T): T =
if v != 0:
gcd(v, u mod v)
else:
u.abs

proc lcm[T](a, b: T): T =
a div gcd(a, b) * b

type Rational* = tuple[num, den: int64]

proc fromInt*(x: SomeInteger): Rational =
result.num = x
result.den = 1

proc frac*(x: var Rational) =
let common = gcd(x.num, x.den)
x.num = x.num div common
x.den = x.den div common

proc `+` *(x, y: Rational): Rational =
let common = lcm(x.den, y.den)
result.num = common div x.den * x.num + common div y.den * y.num
result.den = common
result.frac

proc `+=` *(x: var Rational, y: Rational) =
let common = lcm(x.den, y.den)
x.num = common div x.den * x.num + common div y.den * y.num
x.den = common
x.frac

proc `-` *(x: Rational): Rational =
result.num = -x.num
result.den = x.den

proc `-` *(x, y: Rational): Rational =
x + -y

proc `-=` *(x: var Rational, y: Rational) =
x += -y

proc `*` *(x, y: Rational): Rational =
result.num = x.num * y.num
result.den = x.den * y.den
result.frac

proc `*=` *(x: var Rational, y: Rational) =
x.num *= y.num
x.den *= y.den
x.frac

proc reciprocal*(x: Rational): Rational =
result.num = x.den
result.den = x.num

proc `div`*(x, y: Rational): Rational =
x * y.reciprocal

proc toFloat*(x: Rational): float =
x.num.float / x.den.float

proc toInt*(x: Rational): int64 =
x.num div x.den

proc cmp*(x, y: Rational): int =
cmp x.toFloat, y.toFloat

proc `<` *(x, y: Rational): bool =
x.toFloat < y.toFloat

proc `<=` *(x, y: Rational): bool =
x.toFloat <= y.toFloat

proc abs*(x: Rational): Rational =
result.num = abs x.num
result.den = abs x.den

for candidate in 2'i64 .. <((2'i64)^19):
var sum: Rational = (1'i64, candidate)
for factor in 2'i64 .. pow(candidate.float, 0.5).int64:
if candidate mod factor == 0:
sum += (1'i64, factor) + (1'i64, candidate div factor)
if sum.den == 1:
echo "Sum of recipr. factors of ",candidate," = ",sum.num," exactly ",
if sum.num == 1: "perfect!" else: ""
```

Output:

```Sum of recipr. factors of 6 = 1 exactly perfect!
Sum of recipr. factors of 28 = 1 exactly perfect!
Sum of recipr. factors of 120 = 2 exactly
Sum of recipr. factors of 496 = 1 exactly perfect!
Sum of recipr. factors of 672 = 2 exactly
Sum of recipr. factors of 8128 = 1 exactly perfect!
Sum of recipr. factors of 30240 = 3 exactly
Sum of recipr. factors of 32760 = 3 exactly
Sum of recipr. factors of 523776 = 2 exactly ```

## OCaml

OCaml's Num library implements arbitrary-precision rational numbers:

```#load "nums.cma";;
open Num;;

for candidate = 2 to 1 lsl 19 do
let sum = ref (num_of_int 1 // num_of_int candidate) in
for factor = 2 to truncate (sqrt (float candidate)) do
if candidate mod factor = 0 then
sum := !sum +/ num_of_int 1 // num_of_int factor
+/ num_of_int 1 // num_of_int (candidate / factor)
done;
if is_integer_num !sum then
Printf.printf "Sum of recipr. factors of %d = %d exactly %s\n%!"
candidate (int_of_num !sum) (if int_of_num !sum = 1 then "perfect!" else "")
done;;
```

```let () =
for candidate = 2 to 1 lsl 19 do
let sum = ref Num.(1 / of_int candidate) in
for factor = 2 to truncate (sqrt (float candidate)) do
if candidate mod factor = 0 then
sum := Num.(!sum + 1 / of_int factor + of_int factor / of_int candidate)
done;
if Num.is_integer_num !sum then
Printf.printf "Sum of recipr. factors of %d = %d exactly %s\n%!"
candidate Num.(to_int !sum) (if Num.(!sum = 1) then "perfect!" else "")
done
```

A type for rational numbers might be implemented like this:

First define the interface, hiding implementation details:

```(* interface *)
module type RATIO =
sig
type t
(* construct *)
val frac : int -> int -> t
val from_int : int -> t

(* integer test *)
val is_int : t -> bool

(* output *)
val to_string : t -> string

(* arithmetic *)
val cmp : t -> t -> int
val ( +/ ) : t -> t -> t
val ( -/ ) : t -> t -> t
val ( */ ) : t -> t -> t
val ( // ) : t -> t -> t
end
```

then implement the module:

```(* implementation conforming to signature *)
module Frac : RATIO =
struct
open Big_int

type t = { num : big_int; den : big_int }

(* short aliases for big_int values and functions *)
let zero, one = zero_big_int, unit_big_int
let big, to_int, eq = big_int_of_int, int_of_big_int, eq_big_int
let (+~), (-~), ( *~) = add_big_int, sub_big_int, mult_big_int

(* helper function *)
let rec norm ({num=n;den=d} as k) =
if lt_big_int d zero then
norm {num=minus_big_int n;den=minus_big_int d}
else
let rec hcf a b =
let q,r = quomod_big_int a b in
if eq r zero then b else hcf b r in
let f = hcf n d in
if eq f one then k else
let div = div_big_int in
{ num=div n f; den = div d f } (* inefficient *)

(* public functions *)
let frac a b = norm { num=big a; den=big b }

let from_int a = norm { num=big a; den=one }

let is_int {num=n; den=d} =
eq d one ||
eq (mod_big_int n d) zero

let to_string ({num=n; den=d} as r) =
let r1 = norm r in
let str = string_of_big_int in
if is_int r1 then
str (r1.num)
else
str (r1.num) ^ "/" ^ str (r1.den)

let cmp a b =
let a1 = norm a and b1 = norm b in
compare_big_int (a1.num*~b1.den) (b1.num*~a1.den)

let ( */ ) {num=n1; den=d1} {num=n2; den=d2} =
norm { num = n1*~n2; den = d1*~d2 }

let ( // ) {num=n1; den=d1} {num=n2; den=d2} =
norm { num = n1*~d2; den = d1*~n2 }

let ( +/ ) {num=n1; den=d1} {num=n2; den=d2} =
norm { num = n1*~d2 +~ n2*~d1; den = d1*~d2 }

let ( -/ ) {num=n1; den=d1} {num=n2; den=d2} =
norm { num = n1*~d2 -~ n2*~d1; den = d1*~d2 }
end
```

Finally the use type defined by the module to perform the perfect number calculation:

```(* use the module to calculate perfect numbers *)
let () =
for i = 2 to 1 lsl 19 do
let sum = ref (Frac.frac 1 i) in
for factor = 2 to truncate (sqrt (float i)) do
if i mod factor = 0 then
Frac.(
sum := !sum +/ frac 1 factor +/ frac 1 (i / factor)
)
done;
if Frac.is_int !sum then
Printf.printf "Sum of reciprocal factors of %d = %s exactly %s\n%!"
i (Frac.to_string !sum) (if Frac.to_string !sum = "1" then "perfect!" else "")
done
```

which produces this output:

```Sum of reciprocal factors of 6 = 1 exactly perfect!
Sum of reciprocal factors of 28 = 1 exactly perfect!
Sum of reciprocal factors of 120 = 2 exactly
Sum of reciprocal factors of 496 = 1 exactly perfect!
Sum of reciprocal factors of 672 = 2 exactly
Sum of reciprocal factors of 8128 = 1 exactly perfect!
Sum of reciprocal factors of 30240 = 3 exactly
Sum of reciprocal factors of 32760 = 3 exactly
Sum of reciprocal factors of 523776 = 2 exactly
```

## Ol

Otus Lisp has rational numbers built-in and integrated with all other number types.

```(define x 3/7)
(define y 9/11)
(define z -2/5)

; demonstrate builtin functions:

(print "(abs " z ") = " (abs z))
(print "- " z " = " (- z))
(print x " + " y " = " (+ x y))
(print x " - " y " = " (- x y))
(print x " * " y " = " (* x y))
(print x " / " y " = " (/ x y))
(print x " < " y " = " (< x y))
(print x " > " y " = " (> x y))

; introduce new functions:

(define (+:= x) (+ x 1))
(define (-:= x) (- x 1))

(print "+:= " z " = " (+:= z))
(print "-:= " z " = " (-:= z))

; finally, find all perfect numbers less than 2^15:

(lfor-each (lambda (candidate)
(let ((sum (lfold (lambda (sum factor)
(if (= 0 (modulo candidate factor))
(+ sum (/ 1 factor) (/ factor candidate))
sum))
(/ 1 candidate)
(liota 2 1 (+ (isqrt candidate) 1)))))
(if (= 1 (denominator sum))
(print candidate (if (eq? sum 1) ", perfect" "")))))
(liota 2 1 (expt 2 15)))
```
Output:
```(abs -2/5) = 2/5
- -2/5 = 2/5
3/7 + 9/11 = 96/77
3/7 - 9/11 = -30/77
3/7 * 9/11 = 27/77
3/7 / 9/11 = 11/21
3/7 < 9/11 = #true
3/7 > 9/11 = #false
+:= -2/5 = 3/5
-:= -2/5 = -7/5
6, perfect
28, perfect
120
496, perfect
672
8128, perfect
30240
32760
```

## ooRexx

```loop candidate = 6 to 2**19
sum = .fraction~new(1, candidate)
max2 = rxcalcsqrt(candidate)~trunc

loop factor = 2 to max2
if candidate // factor == 0 then do
sum += .fraction~new(1, factor)
sum += .fraction~new(1, candidate / factor)
end
end
if sum == 1 then say candidate "is a perfect number"
end

::class fraction public inherit orderable
::method init
expose numerator denominator
use strict arg numerator, denominator = 1

if denominator == 0 then raise syntax 98.900 array("Fraction denominator cannot be zero")

-- if the denominator is negative, make the numerator carry the sign
if denominator < 0 then do
numerator = -numerator
denominator = - denominator
end

-- find the greatest common denominator and reduce to
-- the simplest form
gcd = self~gcd(numerator~abs, denominator~abs)

numerator /= gcd
denominator /= gcd

-- fraction instances are immutable, so these are
::attribute numerator GET
::attribute denominator GET

-- calculate the greatest common denominator of a numerator/denominator pair
::method gcd private
use arg x, y

loop while y \= 0
-- check if they divide evenly
temp = x // y
x = y
y = temp
end
return x

-- calculate the least common multiple of a numerator/denominator pair
::method lcm private
use arg x, y
return x / self~gcd(x, y) * y

::method abs
expose numerator denominator
-- the denominator is always forced to be positive
return self~class~new(numerator~abs, denominator)

::method reciprocal
expose numerator denominator
return self~class~new(denominator, numerator)

-- convert a fraction to regular Rexx number
::method toNumber
expose numerator denominator

if numerator == 0 then return 0
return numerator/denominator

::method negative
expose numerator denominator
return self~class~new(-numerator, denominator)

expose numerator denominator
use strict arg other
-- convert to a fraction if a regular number
if \other~isa(.fraction) then other = self~class~new(other, 1)

multiple = self~lcm(denominator, other~denominator)
newa = numerator * multiple / denominator
newb = other~numerator * multiple / other~denominator
return self~class~new(newa + newb, multiple)

::method subtract
use strict arg other
return self + (-other)

::method times
expose numerator denominator
use strict arg other
-- convert to a fraction if a regular number
if \other~isa(.fraction) then other = self~class~new(other, 1)
return self~class~new(numerator * other~numerator, denominator * other~denominator)

::method divide
use strict arg other
-- convert to a fraction if a regular number
if \other~isa(.fraction) then other = self~class~new(other, 1)
-- and multiply by the reciprocal
return self * other~reciprocal

-- compareTo method used by the orderable interface to implement
-- the operator methods
::method compareTo
expose numerator denominator
-- convert to a fraction if a regular number
if \other~isa(.fraction) then other = self~class~new(other, 1)

return (numerator * other~denominator - denominator * other~numerator)~sign

-- we still override "==" and "\==" because we want to bypass the
-- checks for not being an instance of the class
::method "=="
expose numerator denominator
use strict arg other

-- convert to a fraction if a regular number
if \other~isa(.fraction) then other = self~class~new(other, 1)
-- Note:  these are numeric comparisons, so we're using the "="
-- method so those are handled correctly
return numerator = other~numerator & denominator = other~denominator

::method "\=="
use strict arg other
return \self~"\=="(other)

-- some operator overrides -- these only work if the left-hand-side of the
-- subexpression is a quaternion
::method "*"
forward message("TIMES")

::method "/"
forward message("DIVIDE")

::method "-"
-- need to check if this is a prefix minus or a subtract
if arg() == 0 then
forward message("NEGATIVE")
else
forward message("SUBTRACT")

::method "+"
-- need to check if this is a prefix plus or an addition
if arg() == 0 then
return self  -- we can return this copy since it is imutable
else

::method string
expose numerator denominator
if denominator == 1 then return numerator
return numerator"/"denominator

-- override hashcode for collection class hash uses
::method hashCode
expose numerator denominator
return numerator~hashcode~bitxor(numerator~hashcode)

::requires rxmath library
```

Output:

```6 is a perfect number
28 is a perfect number
496 is a perfect number
8128 is a perfect number
```

## PARI/GP

Pari handles rational arithmetic natively.

```for(n=2,1<<19,
s=0;
fordiv(n,d,s+=1/d);
if(s==2,print(n))
)```

## Perl

Perl's `Math::BigRat` core module implements arbitrary-precision rational numbers. The `bigrat` pragma can be used to turn on transparent BigRat support:

```use bigrat;

foreach my \$candidate (2 .. 2**19) {
my \$sum = 1 / \$candidate;
foreach my \$factor (2 .. sqrt(\$candidate)+1) {
if (\$candidate % \$factor == 0) {
\$sum += 1 / \$factor + 1 / (\$candidate / \$factor);
}
}
if (\$sum->denominator() == 1) {
print "Sum of recipr. factors of \$candidate = \$sum exactly ", (\$sum == 1 ? "perfect!" : ""), "\n";
}
}
```

## Phix

Translation of: Tcl

Phix does not support operator overloading (I am strongly opposed to such nonsense), nor does it have a native fraction library, but it might look a bit like this.

```with javascript_semantics
without warning  -- (several unused routines in this code)

constant NUM = 1, DEN = 2

type frac(object r)
return sequence(r) and length(r)=2 and integer(r[NUM]) and integer(r[DEN])
end type

function normalise(object n, atom d=0)
if sequence(n) then
{n,d} = n
end if
if d<0 then
n = -n
d = -d
end if
atom g = gcd(n,d)
return {n/g,d/g}
end function

function frac_new(integer n, d=1)
return normalise(n,d)
end function

function frac_abs(frac r)
return {abs(r[NUM]),r[DEN]}
end function

function frac_inv(frac r)
return reverse(r)
end function

{bn,bd} = b
end function

function frac_sub(frac a, b)
{bn,bd} = b
end function

function frac_mul(frac a, b)
{bn,bd} = b
end function

function frac_div(frac a, b)
{bn,bd} = b
end function

function frac_eq(frac a, b)
return a==b
end function

function frac_ne(frac a, b)
return a!=b
end function

function frac_lt(frac a, b)
return frac_sub(a,b)[NUM]<0
end function

function frac_gt(frac a, b)
return frac_sub(a,b)[NUM]>0
end function

function frac_le(frac a, b)
return frac_sub(a,b)[NUM]<=0
end function

function frac_ge(frac a, b)
return frac_sub(a,b)[NUM]>=0
end function

function is_perfect(integer num)
frac total = frac_new(0)
sequence f = factors(num,1)
for i=1 to length(f) do
end for
return frac_eq(total,frac_new(2))
end function

procedure get_perfect_numbers()
atom t0 = time()
integer lim = power(2,iff(platform()=JS?13:19))
for i=2 to lim do
if is_perfect(i) then
printf(1,"perfect: %d\n",i)
end if
end for
printf(1,"elapsed: %3.2f seconds\n",time()-t0)

integer pn5 = power(2,12)*(power(2,13)-1) -- 5th perfect number
if is_perfect(pn5) then
printf(1,"perfect: %d\n",pn5)
end if
end procedure

get_perfect_numbers()
```
Output:
```perfect: 6
perfect: 28
perfect: 496
perfect: 8128
elapsed: 13.56 seconds
perfect: 33550336
```

### mpq

Library: Phix/mpfr

Turned out to be slightly slower than native, but worth it for large number support.

```include builtins/mpfr.e
function is_perfect(integer num)
mpq tot = mpq_init(),
fth = mpq_init()
sequence f = factors(num,1)
for i=1 to length(f) do
mpq_set_si(fth,1,f[i])
end for
return mpq_cmp_si(tot,2,1)=0
end function

procedure get_perfect_numbers()
atom t0 = time()
integer lim = power(2,iff(platform()=JS?13:19))
for i=2 to lim do
if is_perfect(i) then
printf(1,"perfect: %d\n",i)
end if
end for
printf(1,"elapsed: %3.2f seconds\n",time()-t0)

integer pn5 = power(2,12)*(power(2,13)-1) -- 5th perfect number
if is_perfect(pn5) then
printf(1,"perfect: %d\n",pn5)
end if
end procedure

get_perfect_numbers()
```
Output:
```perfect: 6
perfect: 28
perfect: 496
perfect: 8128
elapsed: 17.31 seconds
perfect: 33550336
```

Note that power(2,19) took over 270s under mpfr.js, so reduced to power(2,13) on that platform, making it finish in 0.99s

## Picat

A naive addition algorithm is used, so the program is slow.

```main =>
foreach (I in 2..2**19, is_perfect(I))
println(I)
end.

is_perfect(N) => sum_rationals([\$frac(1,D) : D in divisors(N)]) == \$frac(2,1).

divisors(N) = [I : I in 1..N, N mod I == 0].

new_fract(A,B) = \$frac(Num, Den) =>
G = gcd(A,B),
Num = A // G,
Den = B // G.

sum_rationals([X]) = X.
Output:
```6
28
496
8128
```

## PicoLisp

```(load "@lib/frac.l")

(for (N 2  (> (** 2 19) N)  (inc N))
(let (Sum (frac 1 N)  Lim (sqrt N))
(for (F 2  (>= Lim F) (inc F))
(when (=0 (% N F))
(setq Sum
(f+ Sum
(f+ (frac 1 F) (frac 1 (/ N F))) ) ) ) )
(when (= 1 (cdr Sum))
(prinl
"Perfect " N
", sum is " (car Sum)
(and (= 1 (car Sum)) ": perfect") ) ) ) )```
Output:
```Perfect 6, sum is 1: perfect
Perfect 28, sum is 1: perfect
Perfect 120, sum is 2
Perfect 496, sum is 1: perfect
Perfect 672, sum is 2
Perfect 8128, sum is 1: perfect
Perfect 30240, sum is 3
Perfect 32760, sum is 3
Perfect 523776, sum is 2```

## PL/I

```*process source attributes xref or(!);
arat: Proc Options(main);
/*--------------------------------------------------------------------
* Rational Arithmetic
* (Mis)use the Complex data type to represent fractions
* real(x) is used as numerator
* imag(x) is used as denominator
* Output:
* a=-3/7 b=9/2
* a*b=-27/14
* a+b=57/14
* a-b=-69/14
* a/b=-2/21
* -3/7<9/2
* 9/2>-3/7
* -3/7=-3/7
* 26.01.2015 handle 0/0
*-------------------------------------------------------------------*/
Dcl (abs,imag,mod,real,sign,trim) Builtin;
Dcl sysprint Print;
Dcl (candidate,max2,factor) Dec Fixed(15);
Dcl sum complex Dec Fixed(15);
Dcl one complex Dec Fixed(15);

one=mk_fr(1,1);
Put Edit('First solve the task at hand')(Skip,a);
Do candidate = 2 to 10000;
sum = mk_fr(1, candidate);
max2 = sqrt(candidate);
Do factor = 2 to max2;
If mod(candidate,factor)=0 Then Do;
End;
End;
If fr_cmp(sum,one)='=' Then Do;
Put Edit(candidate,' is a perfect number')(Skip,f(7),a);
Do factor = 2 to candidate-1;
If mod(candidate,factor)=0 Then
Put Edit(factor)(f(5));
End;
End;
End;

Put Edit('','Then try a few things')(Skip,a);
Dcl a Complex Dec Fixed(15);
Dcl b Complex Dec Fixed(15);
Dcl p Complex Dec Fixed(15);
Dcl s Complex Dec Fixed(15);
Dcl d Complex Dec Fixed(15);
Dcl q Complex Dec Fixed(15);
Dcl zero Complex Dec Fixed(15);
zero=mk_fr(0,1); Put Edit('zero=',fr_rep(zero))(Skip,2(a));
a=mk_fr(0,0);    Put Edit('a=',fr_rep(a))(Skip,2(a));
/*--------------------------------------------------------------------
a=mk_fr(-3333,0); Put Edit('a=',fr_rep(a))(Skip,2(a));
=>  Request mk_fr(-3333,0)
Denominator must not be 0
IBM0280I  ONCODE=0009  The ERROR condition was raised
by a SIGNAL statement.
At offset +00000276 in procedure with entry FT
*-------------------------------------------------------------------*/
a=mk_fr(0,3333); Put Edit('a=',fr_rep(a))(Skip,2(a));
Put Edit('-3,7')(Skip,a);
a=mk_fr(-3,7);
b=mk_fr(9,2);
p=fr_mult(a,b);
d=fr_sub(a,b);
q=fr_div(a,b);
r=fr_div(b,a);
Put Edit('a=',fr_rep(a))(Skip,2(a));
Put Edit('b=',fr_rep(b))(Skip,2(a));
Put Edit('a*b=',fr_rep(p))(Skip,2(a));
Put Edit('a+b=',fr_rep(s))(Skip,2(a));
Put Edit('a-b=',fr_rep(d))(Skip,2(a));
Put Edit('a/b=',fr_rep(q))(Skip,2(a));
Put Edit('b/a=',fr_rep(r))(Skip,2(a));
Put Edit(fr_rep(a),fr_cmp(a,b),fr_rep(b))(Skip,3(a));
Put Edit(fr_rep(b),fr_cmp(b,a),fr_rep(a))(Skip,3(a));
Put Edit(fr_rep(a),fr_cmp(a,a),fr_rep(a))(Skip,3(a));

mk_fr: Proc(n,d) Recursive Returns(Dec Fixed(15) Complex);
/*--------------------------------------------------------------------
* make a Complex number
* normalize and cancel
*-------------------------------------------------------------------*/
Dcl (n,d) Dec Fixed(15);
Dcl (na,da) Dec Fixed(15);
Dcl res Dec Fixed(15) Complex;
Dcl x   Dec Fixed(15);
na=abs(n);
da=abs(d);
Select;
When(n=0) Do;
real(res)=0;
imag(res)=1;
End;
When(d=0) Do;
Put Edit('Request mk_fr('!!n_rep(n)!!','!!n_rep(d)!!')')
(Skip,a);
Put Edit('Denominator must not be 0')(Skip,a);
Signal error;
End;
Otherwise Do;
x=gcd(na,da);
real(res)=sign(n)*sign(d)*na/x;
imag(res)=da/x;
End;
End;
Return(res);
End;

/*--------------------------------------------------------------------
* add 'fractions' a and b
*-------------------------------------------------------------------*/
Dcl (a,b,res)     Dec Fixed(15) Complex;
Dcl (rd,rn)       Dec Fixed(15);
Dcl x             Dec Fixed(15);
an=real(a);
bn=real(b);
bd=imag(b);
x=gcd(rd,rn);
real(res)=rn/x;
imag(res)=rd/x;
Return(res);
End;

fr_sub: Proc(a,b) Returns(Dec Fixed(15) Complex);
/*--------------------------------------------------------------------
* subtract 'fraction' b from a
*-------------------------------------------------------------------*/
Dcl (a,b) Dec Fixed(15) Complex;
Dcl b2    Dec Fixed(15) Complex;
real(b2)=-real(b);
imag(b2)=imag(b);
End;

fr_mult: Proc(a,b) Returns(Dec Fixed(15) Complex);
/*--------------------------------------------------------------------
* multiply 'fractions' a and b
*-------------------------------------------------------------------*/
Dcl (a,b,res) Dec Fixed(15) Complex;
real(res)=real(a)*real(b);
imag(res)=imag(a)*imag(b);
Return(res);
End;

fr_div: Proc(a,b) Returns(Dec Fixed(15) Complex);
/*--------------------------------------------------------------------
* divide 'fraction' a by b
*-------------------------------------------------------------------*/
Dcl (a,b) Dec Fixed(15) Complex;
Dcl b2    Dec Fixed(15) Complex;
real(b2)=imag(b);
imag(b2)=real(b);
If real(a)=0 & real(b)=0 Then
Return(mk_fr(1,1));
Return(fr_mult(a,b2));
End;

fr_cmp: Proc(a,b) Returns(char(1));
/*--------------------------------------------------------------------
* compare 'fractions' a and b
*-------------------------------------------------------------------*/
Dcl (a,b)         Dec Fixed(15) Complex;
Dcl (a2,b2)       Dec Fixed(15);
Dcl (rd)          Dec Fixed(15);
Dcl res           Char(1);
an=real(a);
Signal Error;
End;
bn=real(b);
bd=imag(b);
Select;
When(a2<b2) res='<';
When(a2>b2) res='>';
Otherwise Do;
res='=';
End;
End;
Return(res);
End;

fr_rep: Proc(f) Returns(char(15) Var);
/*--------------------------------------------------------------------
* Return the representation of 'fraction' f
*-------------------------------------------------------------------*/
Dcl f     Dec Fixed(15) Complex;
Dcl res   Char(15) Var;
Dcl (n,d) Pic'(14)Z9';
Dcl x     Dec Fixed(15);
Dcl s     Dec Fixed(15);
n=abs(real(f));
d=abs(imag(f));
x=gcd(n,d);
s=sign(real(f))*sign(imag(f));
res=trim(n/x)!!'/'!!trim(d/x);
If s<0 Then
res='-'!!res;
Return(res);
End;

n_rep: Proc(x) Returns(char(15) Var);
/*--------------------------------------------------------------------
* Return the representation of x
*-------------------------------------------------------------------*/
Dcl x     Dec Fixed(15);
Dcl res   Char(15) Var;
Put String(res) List(x);
res=trim(res);
Return(res);
End;

gcd: Proc(a,b) Returns(Dec Fixed(15)) Recursive;
/*--------------------------------------------------------------------
* Compute the greatest common divisor
*-------------------------------------------------------------------*/
Dcl (a,b) Dec Fixed(15) Nonassignable;
If b=0 then Return (abs(a));
Return(gcd(abs(b),mod(abs(a),abs(b))));
End gcd;

lcm: Proc(a,b) Returns(Dec Fixed(15));
/*--------------------------------------------------------------------
* Compute the least common multiple
*-------------------------------------------------------------------*/
Dcl (a,b) Dec Fixed(15) Nonassignable;
if a=0 ! b=0 then Return (0);
Return(abs(a*b)/gcd(a,b));
End lcm;

End;```
Output:
```First solve the task at hand
6 is a perfect number    2    3
28 is a perfect number    2    4    7   14
496 is a perfect number    2    4    8   16   31   62  124  248
8128 is a perfect number    2    4    8   16   32   64  127  254  508 1016 2032 4064

Then try a few things
zero=0/1
a=0/1
a=0/1
-3,7
a=-3/7
b=9/2
a*b=-27/14
a+b=57/14
a-b=-69/14
a/b=-2/21
b/a=1/0
-3/7<9/2
9/2>-3/7
-3/7=-3/7```

## Prolog

Prolog supports rational numbers, where P/Q is written as P rdiv Q.

```divisor(N, Div) :-
Max is floor(sqrt(N)),
between(1, Max, D),
divmod(N, D, _, 0),
(Div = D; Div is N div D, Div =\= D).

divisors(N, Divs) :-
setof(M, divisor(N, M), Divs).

recip(A, B) :- B is 1 rdiv A.

sumrecip(N, A) :-
divisors(N, [1 | Ds]),
maplist(recip, Ds, As),
sum_list(As, A).

perfect(X) :- sumrecip(X, 1).

main :-
Limit is 1 << 19,
forall(
(between(1, Limit, N), perfect(N)),
(format("~w~n", [N]))),
halt.

?- main.
```
Output:
```6
28
496
8128
```

## Python

Works with: Python version 3.0

Python 3's standard library already implements a Fraction class:

```from fractions import Fraction

for candidate in range(2, 2**19):
sum = Fraction(1, candidate)
for factor in range(2, int(candidate**0.5)+1):
if candidate % factor == 0:
sum += Fraction(1, factor) + Fraction(1, candidate // factor)
if sum.denominator == 1:
print("Sum of recipr. factors of %d = %d exactly %s" %
(candidate, int(sum), "perfect!" if sum == 1 else ""))
```

It might be implemented like this:

```def lcm(a, b):
return a // gcd(a,b) * b

def gcd(u, v):
return gcd(v, u%v) if v else abs(u)

class Fraction:
def __init__(self, numerator, denominator):
common = gcd(numerator, denominator)
self.numerator = numerator//common
self.denominator = denominator//common
common = lcm(self.denominator, frac.denominator)
n = common // self.denominator * self.numerator + common // frac.denominator * frac.numerator
return Fraction(n, common)
def __sub__(self, frac):
def __neg__(self):
return Fraction(-self.numerator, self.denominator)
def __abs__(self):
return Fraction(abs(self.numerator), abs(self.denominator))
def __mul__(self, frac):
return Fraction(self.numerator * frac.numerator, self.denominator * frac.denominator)
def __div__(self, frac):
return self.__mul__(frac.reciprocal())
def reciprocal(self):
return Fraction(self.denominator, self.numerator)
def __cmp__(self, n):
return int(float(self) - float(n))
def __float__(self):
return float(self.numerator / self.denominator)
def __int__(self):
return (self.numerator // self.denominator)
```

## Quackery

Quackery comes with a rational arithmetic library coded in Quackery, `bigrat.qky`, and documented in The Book of Quackery.pdf. Both are available at the Quackery Github repository.

`factors` is defined at Factors of an integer#Quackery.

```  [ \$ "bigrat.qky" loadfile ] now!

[ -2 n->v rot
factors witheach
[ n->v 1/v v+ ]
v0= ]             is perfect ( n -> b )

19 bit times [ i^ perfect if [ i^ echo cr ] ]```
Output:
```6
28
496
8128
```

## Racket

Racket always had support for exact rational numbers as a native numeric type.

Example:

```-> (* 1/7 14)
2
```

## Raku

(formerly Perl 6)

Works with: rakudo version 2016.08

Raku supports rational arithmetic natively.

```(2..2**19).hyper.map: -> \$candidate {
my \$sum = 1 / \$candidate;
for 2 .. ceiling(sqrt(\$candidate)) -> \$factor {
if \$candidate %% \$factor {
\$sum += 1 / \$factor + 1 / (\$candidate / \$factor);
}
}
if \$sum.nude[1] == 1 {
say "Sum of reciprocal factors of \$candidate = \$sum exactly", (\$sum == 1 ?? ", perfect!" !! ".");
}
}
```

Note also that ordinary decimal literals are stored as Rats, so the following loop always stops exactly on 10 despite 0.1 not being exactly representable in floating point:

```for 1.0, 1.1, 1.2 ... 10 { .say }
```

The arithmetic is all done in rationals, which are converted to floating-point just before display so that people don't have to puzzle out what 53/10 means.

## REXX

```/*REXX program implements a reasonably complete  rational arithmetic  (using fractions).*/
L=length(2**19 - 1)                              /*saves time by checking even numbers. */
do j=2  by 2  to 2**19 - 1;       s=0       /*ignore unity (which can't be perfect)*/
mostDivs=eDivs(j);                @=        /*obtain divisors>1; zero sum; null @. */
do k=1  for  words(mostDivs)              /*unity isn't return from  eDivs  here.*/
r='1/'word(mostDivs, k);        @=@ r;         s=\$fun(r, , s)
end   /*k*/
if s\==1  then iterate                      /*Is sum not equal to unity?   Skip it.*/
say 'perfect number:'       right(j, L)       "   fractions:"            @
end   /*j*/
exit                                             /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
\$div: procedure;  parse arg x;   x=space(x,0);   f= 'fractional division'
parse var x n '/' d;       d=p(d 1)
if d=0               then call err  'division by zero:'            x
if \datatype(n,'N')  then call err  'a non─numeric numerator:'     x
if \datatype(d,'N')  then call err  'a non─numeric denominator:'   x
return n/d
/*──────────────────────────────────────────────────────────────────────────────────────*/
\$fun: procedure;  parse arg z.1,,z.2 1 zz.2;    arg ,op;  op=p(op '+')
F= 'fractionalFunction';        do j=1  for 2;  z.j=translate(z.j, '/', "_");   end  /*j*/
if abbrev('ADD'      , op)                               then op= "+"
if abbrev('DIVIDE'   , op)                               then op= "/"
if abbrev('INTDIVIDE', op, 4)                            then op= "÷"
if abbrev('MODULUS'  , op, 3) | abbrev('MODULO', op, 3)  then op= "//"
if abbrev('MULTIPLY' , op)                               then op= "*"
if abbrev('POWER'    , op)                               then op= "^"
if abbrev('SUBTRACT' , op)                               then op= "-"
if z.1==''                                               then z.1= (op\=="+" & op\=='-')
if z.2==''                                               then z.2= (op\=="+" & op\=='-')
z_=z.2
/* [↑]  verification of both fractions.*/
do j=1  for 2
if pos('/', z.j)==0    then z.j=z.j"/1";         parse var  z.j  n.j  '/'  d.j
if \datatype(n.j,'N')  then call err  'a non─numeric numerator:'     n.j
if \datatype(d.j,'N')  then call err  'a non─numeric denominator:'   d.j
if d.j=0               then call err  'a denominator of zero:'       d.j
n.j=n.j/1;          d.j=d.j/1
do  while \datatype(n.j,'W');     n.j=(n.j*10)/1;     d.j=(d.j*10)/1
end  /*while*/                      /* [↑]   {xxx/1}  normalizes a number. */
g=gcd(n.j, d.j);    if g=0  then iterate;  n.j=n.j/g;          d.j=d.j/g
end    /*j*/

select
when op=='+' | op=='-' then do;  l=lcm(d.1,d.2);    do j=1  for 2;  n.j=l*n.j/d.j;  d.j=l
end   /*j*/
if op=='-'  then n.2= -n.2;        t=n.1 + n.2;    u=l
end
when op=='**' | op=='↑'  |,
op=='^'  then do;  if \datatype(z_,'W')  then call err 'a non─integer power:'  z_
t=1;  u=1;     do j=1  for abs(z_);  t=t*n.1;  u=u*d.1
end   /*j*/
if z_<0  then parse value   t  u   with   u  t      /*swap  U and T */
end
when op=='/'  then do;      if n.2=0   then call err  'a zero divisor:'   zz.2
t=n.1*d.2;    u=n.2*d.1
end
when op=='÷'  then do;      if n.2=0   then call err  'a zero divisor:'   zz.2
t=trunc(\$div(n.1 '/' d.1));    u=1
end                           /* [↑]  this is integer division.     */
when op=='//' then do;      if n.2=0   then call err  'a zero divisor:'   zz.2
_=trunc(\$div(n.1 '/' d.1));     t=_ - trunc(_) * d.1;            u=1
end                          /* [↑]  modulus division.              */
when op=='ABS'  then do;   t=abs(n.1);       u=abs(d.1);        end
when op=='*'    then do;   t=n.1 * n.2;      u=d.1 * d.2;       end
when op=='EQ' | op=='='                then return \$div(n.1 '/' d.1)  = fDiv(n.2 '/' d.2)
when op=='NE' | op=='\=' | op=='╪' | ,
op=='¬='    then return \$div(n.1 '/' d.1) \= fDiv(n.2 '/' d.2)
when op=='GT' | op=='>'                then return \$div(n.1 '/' d.1) >  fDiv(n.2 '/' d.2)
when op=='LT' | op=='<'                then return \$div(n.1 '/' d.1) <  fDiv(n.2 '/' d.2)
when op=='GE' | op=='≥'  | op=='>='    then return \$div(n.1 '/' d.1) >= fDiv(n.2 '/' d.2)
when op=='LE' | op=='≤'  | op=='<='    then return \$div(n.1 '/' d.1) <= fDiv(n.2 '/' d.2)
otherwise       call err  'an illegal function:'   op
end   /*select*/

if t==0  then return 0;            g=gcd(t, u);             t=t/g;                   u=u/g
if u==1  then return t
return t'/'u
/*──────────────────────────────────────────────────────────────────────────────────────*/
eDivs: procedure; parse arg x 1 b,a
do j=2  while j*j<x;       if x//j\==0  then iterate;   a=a j;   b=x%j b;     end
if j*j==x  then return a j b;                                            return a b
/*───────────────────────────────────────────────────────────────────────────────────────────────────*/
err:   say;   say '***error*** '    f     " detected"   arg(1);    say;         exit 13
gcd:   procedure; parse arg x,y; if x=0  then return y;  do until _==0; _=x//y; x=y; y=_; end; return x
lcm:   procedure; parse arg x,y; if y=0  then return 0; x=x*y/gcd(x, y);        return x
p:     return word( arg(1), 1)
```

Programming note:   the   eDivs, gcd, lcm   functions are optimized functions for this program only.

output

```perfect number:      6    fractions:  1/2 1/3 1/6
perfect number:     28    fractions:  1/2 1/4 1/7 1/14 1/28
perfect number:    496    fractions:  1/2 1/4 1/8 1/16 1/31 1/62 1/124 1/248 1/496
perfect number:   8128    fractions:  1/2 1/4 1/8 1/16 1/32 1/64 1/127 1/254 1/508 1/1016 1/2032 1/4064 1/8128
```

## Ruby

Ruby has a Rational class in it's core since 1.9.

```for candidate in 2 .. 2**19
sum = Rational(1, candidate)
for factor in 2 .. Integer.sqrt(candidate)
if candidate % factor == 0
sum += Rational(1, factor) + Rational(1, candidate / factor)
end
end
if sum.denominator == 1
puts "Sum of recipr. factors of %d = %d exactly %s" %
[candidate, sum.to_i, sum == 1 ? "perfect!" : ""]
end
end
```
Output:
```Sum of recipr. factors of 6 = 1 exactly perfect!
Sum of recipr. factors of 28 = 1 exactly perfect!
Sum of recipr. factors of 120 = 2 exactly
Sum of recipr. factors of 496 = 1 exactly perfect!
Sum of recipr. factors of 672 = 2 exactly
Sum of recipr. factors of 8128 = 1 exactly perfect!
Sum of recipr. factors of 30240 = 3 exactly
Sum of recipr. factors of 32760 = 3 exactly
Sum of recipr. factors of 523776 = 2 exactly
```

## Rust

```use std::cmp::Ordering;

fn gcd(a: i64, b: i64) -> i64 {
match b {
0 => a,
_ => gcd(b, a % b),
}
}

fn lcm(a: i64, b: i64) -> i64 {
a / gcd(a, b) * b
}

#[derive(Clone, Copy, Debug, Eq, PartialEq, Hash, Ord)]
pub struct Rational {
numerator: i64,
denominator: i64,
}

impl Rational {
fn new(numerator: i64, denominator: i64) -> Self {
let divisor = gcd(numerator, denominator);
Rational {
numerator: numerator / divisor,
denominator: denominator / divisor,
}
}
}

type Output = Self;

fn add(self, other: Self) -> Self {
let multiplier = lcm(self.denominator, other.denominator);
Rational::new(self.numerator * multiplier / self.denominator +
other.numerator * multiplier / other.denominator,
multiplier)
}
}

fn add_assign(&mut self, other: Self) {
*self = *self + other;
}
}

impl Sub for Rational {
type Output = Self;

fn sub(self, other: Self) -> Self {
self + -other
}
}

impl SubAssign for Rational {
fn sub_assign(&mut self, other: Self) {
*self = *self - other;
}
}

impl Mul for Rational {
type Output = Self;

fn mul(self, other: Self) -> Self {
Rational::new(self.numerator * other.numerator,
self.denominator * other.denominator)
}
}

impl MulAssign for Rational {
fn mul_assign(&mut self, other: Self) {
*self = *self * other;
}
}

impl Div for Rational {
type Output = Self;

fn div(self, other: Self) -> Self {
self *
Rational {
numerator: other.denominator,
denominator: other.numerator,
}
}
}

impl DivAssign for Rational {
fn div_assign(&mut self, other: Self) {
*self = *self / other;
}
}

impl Neg for Rational {
type Output = Self;

fn neg(self) -> Self {
Rational {
numerator: -self.numerator,
denominator: self.denominator,
}
}
}

impl PartialOrd for Rational {
fn partial_cmp(&self, other: &Self) -> Option<Ordering> {
(self.numerator * other.denominator).partial_cmp(&(self.denominator * other.numerator))
}
}

impl<T: Into<i64>> From<T> for Rational {
fn from(value: T) -> Self {
Rational::new(value.into(), 1)
}
}

fn main() {
let max = 1 << 19;
for candidate in 2..max {
let mut sum = Rational::new(1, candidate);
for factor in 2..(candidate as f64).sqrt().ceil() as i64 {
if candidate % factor == 0 {
sum += Rational::new(1, factor);
sum += Rational::new(1, candidate / factor);
}
}

if sum == 1.into() {
println!("{} is perfect", candidate);
}
}
}
```

## Scala

```class Rational(n: Long, d:Long) extends Ordered[Rational]
{
require(d!=0)
private val g:Long = gcd(n, d)
val numerator:Long = n/g
val denominator:Long = d/g

def this(n:Long)=this(n,1)

def +(that:Rational):Rational=new Rational(
numerator*that.denominator + that.numerator*denominator,
denominator*that.denominator)

def -(that:Rational):Rational=new Rational(
numerator*that.denominator - that.numerator*denominator,
denominator*that.denominator)

def *(that:Rational):Rational=
new Rational(numerator*that.numerator, denominator*that.denominator)

def /(that:Rational):Rational=
new Rational(numerator*that.denominator, that.numerator*denominator)

def unary_~ :Rational=new Rational(denominator, numerator)

def unary_- :Rational=new Rational(-numerator, denominator)

def abs :Rational=new Rational(Math.abs(numerator), Math.abs(denominator))

override def compare(that:Rational):Int=
(this.numerator*that.denominator-that.numerator*this.denominator).toInt

override def toString()=numerator+"/"+denominator

private def gcd(x:Long, y:Long):Long=
if(y==0) x else gcd(y, x%y)
}

object Rational
{
def apply(n: Long, d:Long)=new Rational(n,d)
def apply(n:Long)=new Rational(n)
implicit def longToRational(i:Long)=new Rational(i)
}
```
```def find_perfects():Unit=
{
for (candidate <- 2 until 1<<19)
{
var sum= ~Rational(candidate)
for (factor <- 2 until (Math.sqrt(candidate)+1).toInt)
{
if (candidate%factor==0)
sum+= ~Rational(factor)+ ~Rational(candidate/factor)
}

if (sum.denominator==1 && sum.numerator==1)
printf("Perfect number %d sum is %s\n", candidate, sum)
}
}
```

## Scheme

Scheme has native rational numbers.

Works with: Scheme version R5RS
```; simply prints all the perfect numbers
(do ((candidate 2 (+ candidate 1))) ((>= candidate (expt 2 19)))
(let ((sum (/ 1 candidate)))
(do ((factor 2 (+ factor 1))) ((>= factor (sqrt candidate)))
(if (= 0 (modulo candidate factor))
(set! sum (+ sum (/ 1 factor) (/ factor candidate)))))
(if (= 1 (denominator sum))
(begin (display candidate) (newline)))))
```

It might be implemented like this:

[insert implementation here]

## Seed7

The library rational.s7i defines the type rational, which supports the required functionality. Rational numbers are based on the type integer. For rational numbers, which are based on integers with unlimited precision, use bigRational, which is defined in the library bigrat.s7i.

```\$ include "seed7_05.s7i";
include "rational.s7i";

const func boolean: isPerfect (in integer: candidate) is func
result
var boolean: isPerfect is FALSE;
local
var integer: divisor is 0;
var rational: sum is rational.value;
begin
sum := 1 / candidate;
for divisor range 2 to sqrt(candidate) do
if candidate mod divisor = 0 then
sum +:= 1 / divisor + 1 / (candidate div divisor);
end if;
end for;
isPerfect := sum = rat(1);
end func;

const proc: main is func
local
var integer: candidate is 0;
begin
for candidate range 2 to 2 ** 19 - 1 do
if isPerfect(candidate) then
writeln(candidate <& " is perfect");
end if;
end for;
end func;```
Output:
```6 is perfect
28 is perfect
496 is perfect
8128 is perfect
```

## Sidef

Sidef has built-in support for rational numbers.

```for n in (1 .. 2**19) {
var frac = 0

n.divisors.each {|d|
frac += 1/d
}

if (frac.is_int) {
say "Sum of reciprocal divisors of #{n} = #{frac} exactly #{
frac == 2 ? '- perfect!' : ''
}"
}
}
```
Output:
```Sum of reciprocal divisors of 1 = 1 exactly
Sum of reciprocal divisors of 6 = 2 exactly - perfect!
Sum of reciprocal divisors of 28 = 2 exactly - perfect!
Sum of reciprocal divisors of 120 = 3 exactly
Sum of reciprocal divisors of 496 = 2 exactly - perfect!
Sum of reciprocal divisors of 672 = 3 exactly
Sum of reciprocal divisors of 8128 = 2 exactly - perfect!
Sum of reciprocal divisors of 30240 = 4 exactly
Sum of reciprocal divisors of 32760 = 4 exactly
Sum of reciprocal divisors of 523776 = 3 exactly
```

## Slate

Slate uses infinite-precision fractions transparently.

```54 / 7.
20 reciprocal.
(5 / 6) reciprocal.
(5 / 6) as: Float.```

## Smalltalk

Smalltalk uses naturally and transparently infinite precision fractions (through the class Fraction):

```st> 54/7
54/7
st> 54/7 + 1
61/7
st> 54/7 < 50
true
st> 20 reciprocal
1/20
st> (5/6) reciprocal
6/5
st> (5/6) asFloat
0.8333333333333334
```
Works with: GNU Smalltalk

(and all others)

```| sum |
2 to: (2 raisedTo: 19) do: [ :candidate |
sum := candidate reciprocal.
2 to: (candidate sqrt) do: [ :factor |
( (candidate \\ factor) = 0 )
ifTrue: [
sum := sum + (factor reciprocal) + ((candidate / factor) reciprocal)
]
].
( (sum denominator) = 1 )
ifTrue: [
('Sum of recipr. factors of %1 = %2 exactly %3' %
{ candidate printString .
(sum asInteger) printString .
( sum = 1 ) ifTrue: [ 'perfect!' ]
ifFalse: [ ' ' ] }) displayNl
]
].
```

## Swift

```import Foundation

extension BinaryInteger {
@inlinable
public func gcd(with other: Self) -> Self {
var gcd = self
var b = other

while b != 0 {
(gcd, b) = (b, gcd % b)
}

return gcd
}

@inlinable
public func lcm(with other: Self) -> Self {
let g = gcd(with: other)

return self / g * other
}
}

public struct Frac<NumType: BinaryInteger & SignedNumeric>: Equatable {
@usableFromInline
var _num: NumType

@usableFromInline
var _dom: NumType

@usableFromInline
init(_num: NumType, _dom: NumType) {
self._num = _num
self._dom = _dom
}

@inlinable
public init(numerator: NumType, denominator: NumType) {
let divisor = numerator.gcd(with: denominator)

self._num = numerator / divisor
self._dom = denominator / divisor
}

@inlinable
public static func + (lhs: Frac, rhs: Frac) -> Frac {
let multiplier = lhs._dom.lcm(with: rhs.denominator)

return Frac(
numerator: lhs._num * multiplier / lhs._dom + rhs._num * multiplier / rhs._dom,
denominator: multiplier
)
}

@inlinable
public static func += (lhs: inout Frac, rhs: Frac) {
lhs = lhs + rhs
}

@inlinable
public static func - (lhs: Frac, rhs: Frac) -> Frac {
return lhs + -rhs
}

@inlinable
public static func -= (lhs: inout Frac, rhs: Frac) {
lhs = lhs + -rhs
}

@inlinable
public static func * (lhs: Frac, rhs: Frac) -> Frac {
return Frac(numerator: lhs._num * rhs._num, denominator: lhs._dom * rhs._dom)
}

@inlinable
public static func *= (lhs: inout Frac, rhs: Frac) {
lhs = lhs * rhs
}

@inlinable
public static func / (lhs: Frac, rhs: Frac) -> Frac {
return lhs * Frac(_num: rhs._dom, _dom: rhs._num)
}

@inlinable
public static func /= (lhs: inout Frac, rhs: Frac) {
lhs = lhs / rhs
}

@inlinable
prefix static func - (rhs: Frac) -> Frac {
return Frac(_num: -rhs._num, _dom: rhs._dom)
}
}

extension Frac {
@inlinable
public var numerator: NumType {
get { _num }
set {
let divisor = newValue.gcd(with: denominator)

_num = newValue / divisor
_dom = denominator / divisor
}
}

@inlinable
public var denominator: NumType {
get { _dom }
set {
let divisor = newValue.gcd(with: numerator)

_num = numerator / divisor
_dom = newValue / divisor
}
}
}

extension Frac: CustomStringConvertible {
public var description: String {
let neg = numerator < 0 || denominator < 0

return "Frac(\(neg ? "-" : "")\(abs(numerator)) / \(abs(denominator)))"
}
}

extension Frac: Comparable {
@inlinable
public static func <(lhs: Frac, rhs: Frac) -> Bool {
return lhs._num * rhs._dom < lhs._dom * rhs._num
}
}

extension Frac: ExpressibleByIntegerLiteral {
public init(integerLiteral value: Int) {
self._num = NumType(value)
self._dom = 1
}
}

for candidate in 2..<1<<19 {
var sum = Frac(numerator: 1, denominator: candidate)

let m = Int(ceil(Double(candidate).squareRoot()))

for factor in 2..<m where candidate % factor == 0 {
sum += Frac(numerator: 1, denominator: factor)
sum += Frac(numerator: 1, denominator: candidate / factor)
}

if sum == 1 {
print("\(candidate) is perfect")
}
}
```
Output:
```6 is perfect
28 is perfect
496 is perfect
8128 is perfect```

## Tcl

[This section is included from a subpage and should be edited there, not here.]

Code to find factors of a number not shown:

```namespace eval rat {}

proc rat::new {args} {
if {[llength \$args] == 0} {
set args {0}
}
lassign [split {*}\$args] n d
if {\$d == 0} {
error "divide by zero"
}
if {\$d < 0} {
set n [expr {-1 * \$n}]
set d [expr {abs(\$d)}]
}
return [normalize \$n \$d]
}

proc rat::split {args} {
if {[llength \$args] == 1} {
lassign [::split \$args /] n d
if {\$d eq ""} {
set d 1
}
} else {
lassign \$args n d
}
return [list \$n \$d]
}

proc rat::join {rat} {
lassign \$rat n d
if {\$n == 0} {
return 0
} elseif {\$d == 1} {
return \$n
} else {
return \$n/\$d
}
}

proc rat::normalize {n d} {
set gcd [gcd \$n \$d]
return [join [list [expr {\$n/\$gcd}] [expr {\$d/\$gcd}]]]
}

proc rat::gcd {a b} {
while {\$b != 0} {
lassign [list \$b [expr {\$a % \$b}]] a b
}
return \$a
}

proc rat::abs {rat} {
lassign [split \$rat] n d
return [join [list [expr {abs(\$n)}] \$d]]
}

proc rat::inv {rat} {
lassign [split \$rat] n d
return [normalize \$d \$n]
}

proc rat::+ {args} {
set n 0
set d 1
foreach arg \$args {
set n [expr {\$n*\$ad + \$an*\$d}]
set d [expr {\$d * \$ad}]
}
return [normalize \$n \$d]
}

proc rat::- {args} {
lassign [split [lindex \$args 0]] n d
if {[llength \$args] == 1} {
return [join [list [expr {-1 * \$n}] \$d]]
}
foreach arg [lrange \$args 1 end] {
set n [expr {\$n*\$ad - \$an*\$d}]
set d [expr {\$d * \$ad}]
}
return [normalize \$n \$d]
}

proc rat::* {args} {
set n 1
set d 1
foreach arg \$args {
set n [expr {\$n * \$an}]
set d [expr {\$d * \$ad}]
}
return [normalize \$n \$d]
}

proc rat::/ {a b} {
set r [* \$a [inv \$b]]
if {[string match */0 \$r]} {
error "divide by zero"
}
return \$r
}

proc rat::== {a b} {
return [expr {[- \$a \$b] == 0}]
}

proc rat::!= {a b} {
return [expr { ! [== \$a \$b]}]
}

proc rat::< {a b} {
lassign [split [- \$a \$b]] n d
return [expr {\$n < 0}]
}

proc rat::> {a b} {
lassign [split [- \$a \$b]] n d
return [expr {\$n > 0}]
}

proc rat::<= {a b} {
return [expr { ! [> \$a \$b]}]
}

proc rat::>= {a b} {
return [expr { ! [< \$a \$b]}]
}

################################################
proc is_perfect {num} {
set sum [rat::new 0]
foreach factor [all_factors \$num] {
set sum [rat::+ \$sum [rat::new 1/\$factor]]
}
# note, all_factors includes 1, so sum should be 2
return [rat::== \$sum 2]
}

proc get_perfect_numbers {} {
set t [clock seconds]
set limit [expr 2**19]
for {set num 2} {\$num < \$limit} {incr num} {
if {[is_perfect \$num]} {
puts "perfect: \$num"
}
}
puts "elapsed: [expr {[clock seconds] - \$t}] seconds"

set num [expr {2**12 * (2**13 - 1)}] ;# 5th perfect number
if {[is_perfect \$num]} {
puts "perfect: \$num"
}
}

source primes.tcl
get_perfect_numbers
```
Output:
```perfect: 6
perfect: 28
perfect: 496
perfect: 8128
elapsed: 477 seconds
perfect: 33550336
```

## TI-89 BASIC

 This example is incomplete. Please ensure that it meets all task requirements and remove this message.

While TI-89 BASIC has built-in rational and symbolic arithmetic, it does not have user-defined data types.

## Wren

Library: Wren-math
Library: Wren-rat

The latter module already contains support for rational number arithmetic.

```import "./math" for Int
import "./rat" for Rat

System.print("The following numbers (less than 2^19) are perfect:")
for (i in 2...(1<<19)) {
var sum = Rat.new(1, i)
for (j in Int.properDivisors(i)[1..-1]) sum = sum + Rat.new(1, j)
if (sum == Rat.one) System.print("  %(i)")
}
```
Output:
```The following numbers (less than 2^19) are perfect:
6
28
496
8128
```

## zkl

Enough of a Rational class for this task (ie implement the testing code "nicely").

```class Rational{  // Weenie Rational class, can handle BigInts
fcn init(_a,_b){ var a=_a, b=_b; normalize(); }
fcn toString{
if(b==1) a.toString()
else     "%d//%d".fmt(a,b)
}
var [proxy] isZero=fcn{ a==0 };
fcn normalize{  // divide a and b by gcd
g:= a.gcd(b);
a/=g; b/=g;
if(b<0){ a=-a; b=-b; } // denominator > 0
self
}
fcn abs       { a=a.abs(); self }
fcn __opNegate{ a=-a;      self }			    // -Rat
if(Rational.isChildOf(n)) self(a*n.b + b*n.a, b*n.b); // Rat + Rat
else self(b*n + a, b);				    // Rat + Int
}
fcn __opSub(n){ self(a*n.b - b*n.a, b*n.b) }		    // Rat - Rat
fcn __opMul(n){
if(Rational.isChildOf(n)) self(a*n.a, b*n.b);	    // Rat * Rat
else self(a*n, b);				    // Rat * Int
}
fcn __opDiv(n){ self(a*n.b,b*n.a) }			    // Rat / Rat
fcn __opEQ(r){				       // Rat==Rat, Rat==n
if(Rational.isChildOf(r)) a==r.a and b=r.b;
else			b==1   and a==r;
}
}```
```foreach p in ([2 .. (2).pow(19)]){
sum,limit := Rational(1,p), p.toFloat().sqrt();
foreach factor in ([2 .. limit]){
if(p%factor == 0) sum+=Rational(1,factor) + Rational(factor,p);
}
if(sum.b==1) println("Sum of recipr. factors of %6s = %s exactly%s"
.fmt(p, sum, (sum==1) and ", perfect." or "."));
}```
Output:
```Sum of recipr. factors of      6 = 1 exactly, perfect.
Sum of recipr. factors of     28 = 1 exactly, perfect.
Sum of recipr. factors of    120 = 2 exactly.
Sum of recipr. factors of    496 = 1 exactly, perfect.
Sum of recipr. factors of    672 = 2 exactly.
Sum of recipr. factors of   8128 = 1 exactly, perfect.
Sum of recipr. factors of  30240 = 3 exactly.
Sum of recipr. factors of  32760 = 3 exactly.
Sum of recipr. factors of 523776 = 2 exactly.
```