Arithmetic/Rational
You are encouraged to solve this task according to the task description, using any language you may know.
- Task
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.
- Related task
Ada
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 Ada.Text_IO; use Ada.Text_IO;
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
<lang algol68> 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</lang>
- 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
BBC BASIC
<lang bbcbasic> *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%
PROCadd(Sum{}, Kf{})
PROCnormalise(Sum{})
Kf.den = n% DIV k%
PROCadd(Sum{}, Kf{})
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
DEF PROCadd(a{}, b{})
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</lang>
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. <lang c>#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); sum = frac_add(sum, kf);
kf = frac_new(1, n / k); sum = frac_add(sum, kf); } if (frac_cmp_int(sum, 1) == 0) printf("%d\n", n); }
return 0; }</lang> See Rational Arithmetic/C
C#
using System;
struct Fraction : IEquatable<Fraction>, IComparable<Fraction>
{
public readonly long Num;
public readonly long Denom;
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++
Boost provides a rational number template. <lang cpp>#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;
}</lang>
Clojure
Ratios are built in to Clojure and support math operations already. They automatically reduce and become Integers if possible. <lang Clojure>user> 22/7 22/7 user> 34/2 17 user> (+ 37/5 42/9) 181/15</lang>
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. <lang lisp>(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)</lang>
D
<lang 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." : ".");
}
}
}</lang>
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.
EchoLisp
EchoLisp supports rational numbers as native type. "Big" rational i.e bigint/bigint are not supported. <lang lisp>
- 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)))))) </lang>
- Output:
<lang lisp>
- 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. </lang>
Elisa
<lang 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;</lang> Tests <lang Elisa>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)?</lang>
- Output:
6 28 496 8128
Elixir
<lang 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
def add(a, b) do
{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
Rational.add(sum, Rational.new(1, factor))
|> Rational.add(Rational.new(1, div(candidate, factor)))
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)</lang>
- 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
<lang 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
PROCEDURE ADD(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 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
ADD(SUM.,KF.->SUM.)
NORMALIZE(SUM.->SUM.)
KF.DEN=INT(N/K)
ADD(SUM.,KF.->SUM.)
NORMALIZE(SUM.->SUM.)
END IF
END FOR
EQ(SUM.,ONE.->RES%)
IF RES% THEN PRINT(N;" is perfect") END IF
END FOR
END PROGRAM</lang>
- Output:
6 is perfect 28 is perfect 496 is perfect 8128 is perfect
F#
The F# Powerpack library defines the BigRational data type. <lang fsharp>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</lang>
Factor
ratio is a built-in numeric type.
<lang factor>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 .</lang>
- Output:
Perfect numbers <= 2^19:
V{ 6 28 496 8128 }
Forth
<lang 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 . ;
\ normalize: factors out gcd and puts sign into numerator
- 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 ;</lang>
Fortran
<lang fortran>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 (+) module procedure rational_add 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
end function rational_add
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</lang> Example: <lang fortran>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</lang>
- 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. <lang frink> 1/2 + 2/3 // 7/6 (approx. 1.1666666666666667)
1/2 + 1/2 // 1
5/sextillion + 3/quadrillion // 600001/200000000000000000000 (exactly 3.000005e-15)
8^(1/3) // 2 (note the exact integer result.) </lang>
GAP
Rational numbers are built-in. <lang gap>2/3 in Rationals;
- true
2/3 + 3/4;
- 17/12</lang>
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.Add
- 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. <lang go>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 {
sum.Add(sum, recip.SetFrac64(1, factor))
if f2 := candidate / factor; f2 != factor {
sum.Add(sum, recip.SetFrac64(1, f2))
}
}
}
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)
}
}
}</lang>
- 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: <lang groovy>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()
case BigDecimal: return toBigDecimal()
case BigInteger: return toBigInteger()
case [Double, Double.TYPE]: return toDouble()
case [Float, Float.TYPE]: return toFloat()
case [Integer, Integer.TYPE]: return toInteger()
case [Long, Long.TYPE]: return toLong()
case String: return toString()
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}"
}
}</lang>
The following RationalCategory class allows for modification of regular Number behavior when interacting with Rational. <lang groovy>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)
}
}</lang>
Test Program (mixes the RationalCategory methods into the Number class): <lang groovy>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 }</lang>
- 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: <lang groovy>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 }</lang>
- 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
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.
<lang haskell>import Data.Ratio
-- simply prints all the perfect numbers main = mapM_ print [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]</lang>
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 /).
Additional procedures are implemented here to complete the task:
- 'makerat' (make), 'absrat' (abs), 'eqrat' (=), 'nerat' (~=), 'ltrat' (<), 'lerat' (<=), 'gerat' (>=), 'gtrat' (>)
<lang Icon>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
rsum := addrat(rsum,addrat(rational(1,f,1),rational(1,integer(c/f),1)))
if rsum.numer = rsum.denom = 1 then
return c
end</lang>
- 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: <lang Icon>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
link rational</lang>
The
provides rational and gcd in numbers. Record definition and usage is shown below:
<lang Icon> record rational(numer, denom, sign) # rational type
addrat(r1,r2) # Add rational numbers r1 and r2. 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</lang>
J
Rational numbers in J may be formed from fixed precision integers by first upgrading them to arbitrary precision integers and then dividing them: <lang J> (x: 3) % (x: -4) _3r4
3 %&x: -4
_3r4</lang>
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:
<lang J>
| _3r4 NB. absolute value
3r4
-2r5 NB. negation
_2r5
3r4+2r5 NB. addition
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</lang>
You can also coerce numbers directly to rational using x: (or to integer or floating point as appropriate using its inverse)
<lang J> x: 3%4 3r4
x:inv 3%4
0.75</lang>
Increment and decrement are also included in the language, but you could just as easily add or subtract 1:
<lang J> >: 3r4 7r4
<: 3r4
_1r4</lang>
J does not encourage the use of specialized mutators, but those could also be defined:
<lang j>mutadd=:adverb define
(m)=: (".m)+y
)
mutsub=:adverb define
(m)=: (".m)-y
)</lang>
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):
<lang J> n=: 3r4
'n' mutadd 1
7r4
'n' mutsub 1
3r4
'n' mutsub 1
_1r4</lang>
(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: <lang j> is_perfect_rational=: 2 = (1 + i.) +/@:%@([ #~ 0 = |) ]</lang> Faster version (but the problem, as stated, is still tremendously inefficient): <lang j>factors=: */&>@{@((^ i.@>:)&.>/)@q:~&__ is_perfect_rational=: 2= +/@:%@,@factors</lang> Exhaustive testing would take forever: <lang j> 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</lang> More limited testing takes reasonable amounts of time: <lang j> (#~ is_perfect_rational"0) (* <:@+:) 2^i.10x 6 28 496 8128</lang>
Java
Uses BigRational class: Arithmetic/Rational/Java <lang 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)
reciprocalSum = reciprocalSum.add(BigRational.valueOf(i).reciprocal());
int maxDivisor = (int) Math.sqrt(i);
if (maxDivisor >= i)
maxDivisor--;
for (int divisor = 2; divisor <= maxDivisor; divisor++) {
if (i % divisor == 0) {
reciprocalSum = reciprocalSum.add(BigRational.valueOf(divisor).reciprocal());
int dividend = i / divisor;
if (divisor != dividend)
reciprocalSum = reciprocalSum.add(BigRational.valueOf(dividend).reciprocal());
}
}
if (reciprocalSum.equals(TWO))
System.out.println(String.valueOf(i) + " is a perfect number");
}
}
}</lang>
- 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
- 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
// variadic, modifies receiver Rational.prototype.add = function() {
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();
};
// variadic, modifies receiver Rational.prototype.subtract = function() {
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);
};
// variadic, modifies receiver Rational.prototype.multiply = function() {
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++)
sum.add(new Rational(1, fctrs[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
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. <lang Julia> function isperfect{T<:Integer}(n::T)
!isprime(n) || return false
tal = 1//n
hi = isqrt(n)
if hi^2 == n
tal += 1//hi
hi -= 1
end
for i in 2:hi
(d, r) = divrem(n, i)
if r == 0
tal += (1//i + 1//d)
end
end
return tal == 1//1
end
lo = 2 hi = 2^19 println("Searching for perfect numbers from ", lo, " to ", hi, ".") for i in 2:2^19
isperfect(i) || continue
println(@sprintf("%8d", i))
end </lang>
- Output:
Searching for perfect numbers from 2 to 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. <lang scala>// 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()
}</lang>
- 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
Lingo
A new 'frac' data type can be implemented like this: <lang lingo>-- 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</lang>
Lingo does not support overwriting built-in operators, so 'frac'-operators must be implemented as functions: <lang lingo>-- 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</lang> Usage: <lang lingo>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"</lang>
Finding perfect numbers: <lang lingo>-- 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</lang> <lang lingo>findPerfects(power(2, 19)) -- 6 -- 28 -- 496 -- 8128</lang>
Lua
<lang 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({
__add = function(a, b)
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))</lang>
Liberty BASIC
Testing all numbers up to 2 ^ 19 takes an excessively long time. <lang lb> n=2^19 for testNumber=1 to n
sum$=castToFraction$(0)
for factorTest=1 to sqr(testNumber)
if GCD(factorTest,testNumber)=factorTest then sum$=add$(sum$,add$(reciprocal$(castToFraction$(factorTest)),reciprocal$(castToFraction$(testNumber/factorTest))))
next factorTest
if equal(sum$,castToFraction$(2))=1 then print testNumber
next testNumber end
function abs$(a$)
aNumerator=val(word$(a$,1,"/")) aDenominator=val(word$(a$,2,"/")) bNumerator=abs(aNumerator) bDenominator=abs(aDenominator) b$=str$(bNumerator)+"/"+str$(bDenominator) abs$=simplify$(b$)
end function
function negate$(a$)
aNumerator=val(word$(a$,1,"/")) aDenominator=val(word$(a$,2,"/")) bNumerator=-1*aNumerator bDenominator=aDenominator b$=str$(bNumerator)+"/"+str$(bDenominator) negate$=simplify$(b$)
end function
function add$(a$,b$)
aNumerator=val(word$(a$,1,"/")) aDenominator=val(word$(a$,2,"/")) bNumerator=val(word$(b$,1,"/")) bDenominator=val(word$(b$,2,"/")) cNumerator=(aNumerator*bDenominator+bNumerator*aDenominator) cDenominator=aDenominator*bDenominator c$=str$(cNumerator)+"/"+str$(cDenominator) add$=simplify$(c$)
end function
function subtract$(a$,b$)
aNumerator=val(word$(a$,1,"/")) aDenominator=val(word$(a$,2,"/")) bNumerator=val(word$(b$,1,"/")) bDenominator=val(word$(b$,2,"/")) cNumerator=(aNumerator*bDenominator-bNumerator*aDenominator) cDenominator=aDenominator*bDenominator c$=str$(cNumerator)+"/"+str$(cDenominator) subtract$=simplify$(c$)
end function
function multiply$(a$,b$)
aNumerator=val(word$(a$,1,"/")) aDenominator=val(word$(a$,2,"/")) bNumerator=val(word$(b$,1,"/")) bDenominator=val(word$(b$,2,"/")) cNumerator=aNumerator*bNumerator cDenominator=aDenominator*bDenominator 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,"/")) aDenominator=val(word$(a$,2,"/")) gcd=GCD(aNumerator,aDenominator) if aNumerator<0 and aDenominator<0 then gcd=-1*gcd bNumerator=aNumerator/gcd bDenominator=aDenominator/gcd b$=str$(bNumerator)+"/"+str$(bDenominator) simplify$=b$
end function
function reciprocal$(a$)
aNumerator=val(word$(a$,1,"/")) aDenominator=val(word$(a$,2,"/")) reciprocal$=str$(aDenominator)+"/"+str$(aNumerator)
end function
function equal(a$,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$,1,"/")) aDenominator=val(word$(a$,2,"/")) castToReal=aNumerator/aDenominator
end function
function castToInt(a$)
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 </lang>
Maple
Maple has full built-in support for arithmetic with fractions (rational numbers). Fractions are treated like any other number in Maple. <lang Maple> > a := 3 / 5;
a := 3/5
> numer( a );
3
> denom( a );
5
</lang> 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: <lang Maple> > b := 4 / 6;
b := 2/3
</lang> All the standard arithmetic operators work with rational numbers. It is not necessary to call any special routines. <lang Maple> > a + b;
19
--
15
> a * b;
2/5
> a / b;
9/10
> a - b;
-1
--
15
> a + 1;
8/5
> a - 1;
-2/5
</lang> 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. <lang Maple> > evalf( 22 / 7 ); # default is 10 digits
3.142857143
> evalf[100]( 22 / 7 ); # 100 digits 3.142857142857142857142857142857142857142857142857142857142857142857\
142857142857142857142857142857143
</lang>
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: <lang Mathematica>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]</lang> 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: <lang Mathematica>c/(2 c) (b^2 - c^2)/(b - c) // Cancel 1/2 + b/c // Together</lang> gives back: <lang Mathematica>1/2 b+c (2 b+c) / (2 c)</lang> 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: <lang Mathematica>1+2*{1,2,3}^3</lang> gives back: <lang Mathematica>{3, 17, 55}</lang> To check for perfect numbers in the range 1 to 2^25 we can use: <lang Mathematica>found={}; CheckPerfect[num_Integer]:=If[Total[1/Divisors[num]]==2,AppendTo[found,num]]; Do[CheckPerfect[i],{i,1,2^25}]; found</lang> gives back: <lang Mathematica>{6, 28, 496, 8128, 33550336}</lang> Final note; approximations of fractions to any precision can be found using the function N.
Maxima
<lang 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</lang>
Modula-2
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 );
END RAdd;
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); b := RAdd( b, RAdd( RCreate( 2,3), RCreate( 4,2 ))); 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>
Nim
<lang 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: ""</lang>
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
Objective-C
File frac.h
#import <Foundation/Foundation.h>
@interface RCRationalNumber : NSObject
{
@private
int numerator;
int denominator;
BOOL autoSimplify;
BOOL withSign;
}
+(instancetype)valueWithNumerator:(int)num andDenominator: (int)den;
+(instancetype)valueWithDouble: (double)fnum;
+(instancetype)valueWithInteger: (int)inum;
+(instancetype)valueWithRational: (RCRationalNumber *)rnum;
-(instancetype)initWithNumerator: (int)num andDenominator: (int)den;
-(instancetype)initWithDouble: (double)fnum precision: (int)prec;
-(instancetype)initWithInteger: (int)inum;
-(instancetype)initWithRational: (RCRationalNumber *)rnum;
-(NSComparisonResult)compare: (RCRationalNumber *)rnum;
-(id)simplify: (BOOL)act;
-(void)setAutoSimplify: (BOOL)v;
-(void)setWithSign: (BOOL)v;
-(BOOL)autoSimplify;
-(BOOL)withSign;
-(NSString *)description;
// ops
-(id)multiply: (RCRationalNumber *)rnum;
-(id)divide: (RCRationalNumber *)rnum;
-(id)add: (RCRationalNumber *)rnum;
-(id)sub: (RCRationalNumber *)rnum;
-(id)abs;
-(id)neg;
-(id)mod: (RCRationalNumber *)rnum;
-(int)sign;
-(BOOL)isNegative;
-(id)reciprocal;
// getter
-(int)numerator;
-(int)denominator;
//setter
-(void)setNumerator: (int)num;
-(void)setDenominator: (int)num;
// defraction
-(double)number;
-(int)integer;
@end
- File frac.m
#import <Foundation/Foundation.h>
#import <math.h>
#import "frac.h"
// gcd: [[Greatest common divisor#Recursive_Euclid_algorithm]]
// if built in as "private" function, add static.
static int lcm(int a, int b)
{
return a / gcd(a,b) * b;
}
@implementation RCRationalNumber
// initializers
-(instancetype)init
{
NSLog(@"initialized to unity");
return [self initWithInteger: 1];
}
-(instancetype)initWithNumerator: (int)num andDenominator: (int)den
{
if ((self = [super init]) != nil) {
if (den == 0) {
NSLog(@"denominator is zero");
return nil;
}
[self setNumerator: num];
[self setDenominator: den];
[self setWithSign: YES];
[self setAutoSimplify: YES];
[self simplify: YES];
}
return self;
}
-(instancetype)initWithInteger:(int)inum
{
return [self initWithNumerator: inum andDenominator: 1];
}
-(instancetype)initWithDouble: (double)fnum precision: (int)prec
{
if ( prec > 9 ) prec = 9;
double p = pow(10.0, (double)prec);
int nd = (int)(fnum * p);
return [self initWithNumerator: nd andDenominator: (int)p ];
}
-(instancetype)initWithRational: (RCRationalNumber *)rnum
{
return [self initWithNumerator: [rnum numerator] andDenominator: [rnum denominator]];
}
// comparing
-(NSComparisonResult)compare: (RCRationalNumber *)rnum
{
if ( [self number] > [rnum number] ) return NSOrderedDescending;
if ( [self number] < [rnum number] ) return NSOrderedAscending;
return NSOrderedSame;
}
// string rapresentation of the Q
-(NSString *)description
{
[self simplify: [self autoSimplify]];
return [NSString stringWithFormat: @"%@%d/%d", [self isNegative] ? @"-" :
( [self withSign] ? @"+" : @"" ),
abs([self numerator]), [self denominator]];
}
// setter options
-(void)setAutoSimplify: (BOOL)v
{
autoSimplify = v;
[self simplify: v];
}
-(void)setWithSign: (BOOL)v
{
withSign = v;
}
// getter for options
-(BOOL)autoSimplify
{
return autoSimplify;
}
-(BOOL)withSign
{
return withSign;
}
// "simplify" the fraction ...
-(id)simplify: (BOOL)act
{
if ( act ) {
int common = gcd([self numerator], [self denominator]);
[self setNumerator: [self numerator]/common];
[self setDenominator: [self denominator]/common];
}
return self;
}
// diadic operators
-(id)multiply: (RCRationalNumber *)rnum
{
int newnum = [self numerator] * [rnum numerator];
int newden = [self denominator] * [rnum denominator];
return [RCRationalNumber valueWithNumerator: newnum
andDenominator: newden];
}
-(id)divide: (RCRationalNumber *)rnum
{
return [self multiply: [rnum reciprocal]];
}
-(id)add: (RCRationalNumber *)rnum
{
int common = lcm([self denominator], [rnum denominator]);
int resnum = common / [self denominator] * [self numerator] +
common / [rnum denominator] * [rnum numerator];
return [RCRationalNumber valueWithNumerator: resnum andDenominator: common];
}
-(id)sub: (RCRationalNumber *)rnum
{
return [self add: [rnum neg]];
}
-(id)mod: (RCRationalNumber *)rnum
{
return [[self divide: rnum]
sub: [RCRationalNumber valueWithInteger: [[self divide: rnum] integer]]];
}
// unary operators
-(id)neg
{
return [RCRationalNumber valueWithNumerator: -1*[self numerator]
andDenominator: [self denominator]];
}
-(id)abs
{
return [RCRationalNumber valueWithNumerator: abs([self numerator])
andDenominator: [self denominator]];
}
-(id)reciprocal
{
return [RCRationalNumber valueWithNumerator: [self denominator]
andDenominator: [self numerator]];
}
// get the sign
-(int)sign
{
return ([self numerator] < 0) ? -1 : 1;
}
// or just test if negative
-(BOOL)isNegative
{
return [self numerator] < 0;
}
// Q as real floating point
-(double)number
{
return (double)[self numerator] / (double)[self denominator];
}
// Q as (truncated) integer
-(int)integer
{
return [self numerator] / [self denominator];
}
// set num and den indipendently, fixing sign accordingly
-(void)setNumerator: (int)num
{
numerator = num;
}
-(void)setDenominator: (int)num
{
if ( num < 0 ) numerator = -numerator;
denominator = abs(num);
}
// getter
-(int)numerator
{
return numerator;
}
-(int)denominator
{
return denominator;
}
// class method
+(instancetype)valueWithNumerator:(int)num andDenominator: (int)den
{
return [[self alloc] initWithNumerator: num andDenominator: den];
}
+(instancetype)valueWithDouble: (double)fnum
{
return [[self alloc] initWithDouble: fnum];
}
+(instancetype)valueWithInteger: (int)inum
{
return [[self alloc] initWithInteger: inum];
}
+(instancetype)valueWithRational: (RCRationalNumber *)rnum
{
return [[self alloc] initWithRational: rnum];
}
@end
- Testing
#import <Foundation/Foundation.h>
#import "frac.h"
#import <math.h>
int main()
{
@autoreleasepool {
int i;
for(i=2; i < 0x80000; i++) {
int candidate = i;
RCRationalNumber *sum = [RCRationalNumber valueWithNumerator: 1
andDenominator: candidate];
int factor;
for(factor=2; factor < sqrt((double)candidate); factor++) {
if ( (candidate % factor) == 0 ) {
sum = [[sum add: [RCRationalNumber valueWithNumerator: 1
andDenominator: factor]]
add: [RCRationalNumber valueWithNumerator: 1
andDenominator: (candidate/factor)]];
}
}
if ( [sum denominator] == 1 ) {
printf("Sum of recipr. factors of %d = %d exactly %s\n",
candidate, [sum integer], ([sum integer]==1) ? "perfect!" : "");
}
}
}
return 0;
}
OCaml
OCaml's Num library implements arbitrary-precision rational numbers: <lang ocaml>#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;;</lang> Delimited overloading can be used to make the arithmetic expressions more readable: <lang ocaml>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</lang>
A type for rational numbers might be implemented like this:
First define the interface, hiding implementation details: <lang ocaml>(* 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</lang>
then implement the module: <lang ocaml>(* 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</lang>
Finally the use type defined by the module to perform the perfect number calculation: <lang ocaml>(* 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</lang>
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
ooRexx
<lang 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 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 -- read only attributes
- 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)
- method add
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
forward message("ADD")
- 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
</lang> 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. <lang parigp>for(n=2,1<<19,
s=0; fordiv(n,d,s+=1/d); if(s==2,print(n))
)</lang>
Perl
Perl's Math::BigRat core module implements arbitrary-precision rational numbers. The bigrat pragma can be used to turn on transparent BigRat support:
<lang perl>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";
}
}</lang> It might be implemented like this:
[insert implementation here]
Perl 6
Perl 6 supports rational arithmetic natively. <lang perl6>for 2..2**19 -> $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!" !! ".");
}
}</lang> 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: <lang perl6>for 1.0, 1.1, 1.2 ... 10 { .say }</lang> 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.
Phix
Phix does not support operator overloading (I am strongly opposed to such nonsense), nor does it have a fraction library, but it might look a bit like this.
See also Bernoulli_numbers for a couple of these routines adapted to use bigatoms.
<lang Phix>without warning -- (several unused routines in this code)
constant NUM = 1, DEN = 2
type frac(object r)
return sequence(r) and integer(r[NUM]) and integer(r[DEN]) and length(r)=2
end type
function normalise(object n, atom d=0) atom g
if sequence(n) then
{n,d} = n
end if
if d<0 then
n = -n
d = -d
end if
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
function frac_add(frac a, frac b) integer {an,ad} = a,
{bn,bd} = b
return normalise(an*bd+bn*ad,ad*bd)
end function
function frac_sub(frac a, frac b) integer {an,ad} = a,
{bn,bd} = b
return normalise(an*bd-bn*ad,ad*bd)
end function
function frac_mul(frac a, frac b) integer {an,ad} = a,
{bn,bd} = b
return normalise(an*bn,ad*bd)
end function
function frac_div(frac a, frac b) integer {an,ad} = a,
{bn,bd} = b
return normalise(an*bd,ad*bn)
end function
function frac_eq(frac a, frac b)
return a==b
end function
function frac_ne(frac a, frac b)
return a!=b
end function
function frac_lt(frac a, frac b)
return frac_sub(a,b)[NUM]<0
end function
function frac_gt(frac a, frac b)
return frac_sub(a,b)[NUM]>0
end function
function frac_le(frac a, frac b)
return frac_sub(a,b)[NUM]<=0
end function
function frac_ge(frac a, frac b)
return frac_sub(a,b)[NUM]>=0
end function
function is_perfect(integer num) frac sum = frac_new(0) sequence f = factors(num,1)
for i=1 to length(f) do
sum = frac_add(sum,frac_new(1,f[i]))
end for
return frac_eq(sum,frac_new(2))
end function
procedure get_perfect_numbers() atom t0 = time()
for i=2 to power(2,19) 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()</lang>
- Output:
perfect: 6 perfect: 28 perfect: 496 perfect: 8128 elapsed: 13.56 seconds perfect: 33550336
PicoLisp
<lang 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") ) ) ) )</lang>
- 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
<lang pli>*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;
sum=fr_add(sum,mk_fr(1,factor));
sum=fr_add(sum,mk_fr(1,candidate/factor));
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);
s=fr_add(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;
fr_add: Proc(a,b) Returns(Dec Fixed(15) Complex); /*-------------------------------------------------------------------- * add 'fractions' a and b *-------------------------------------------------------------------*/ Dcl (a,b,res) Dec Fixed(15) Complex; Dcl (an,ad,bn,bd) Dec Fixed(15); Dcl (rd,rn) Dec Fixed(15); Dcl x Dec Fixed(15); an=real(a); ad=imag(a); bn=real(b); bd=imag(b); rd=ad*bd; rn=an*bd+bn*ad; 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); Return(fr_add(a,b2)); 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 (an,ad,bn,bd) Dec Fixed(15);
Dcl (a2,b2) Dec Fixed(15);
Dcl (rd) Dec Fixed(15);
Dcl res Char(1);
an=real(a);
ad=imag(a);
If ad=0 Then Do;
Put Edit('ad=',ad,'candidate=',candidate)(Skip,a,f(10));
Signal Error;
End;
bn=real(b);
bd=imag(b);
rd=ad*bd;
a2=abs(an*bd)*sign(an)*sign(ad);
b2=abs(bn*ad)*sign(bn)*sign(bd);
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;</lang>
- 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
Python
Python 3's standard library already implements a Fraction class: <lang python>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 ""))</lang>
It might be implemented like this: <lang python>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
def __add__(self, frac):
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):
return self.__add__(-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)</lang>
Racket
Racket always had support for exact rational numbers as a native numeric type.
Example: <lang racket> -> (* 1/7 14) 2 </lang>
REXX
<lang 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)</lang> 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. Before that it was in standard library: <lang ruby>require 'rational' #Only needed in Ruby < 1.9
for candidate in 2 .. 2**19
sum = Rational(1, candidate)
for factor in 2 ... candidate**0.5
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</lang>
- 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
<lang rust>use std::cmp::Ordering; use std::ops::{Add, AddAssign, Sub, SubAssign, Mul, MulAssign, Div, DivAssign, Neg};
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,
}
}
}
impl Add for Rational {
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)
}
}
impl AddAssign for Rational {
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);
}
}
} </lang>
Scheme
Scheme has native rational numbers.
<lang scheme>; 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)))))</lang>
It might be implemented like this:
[insert implementation here]
Scala
<lang 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)
}</lang>
<lang scala>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)
}
}</lang>
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.
<lang seed7>$ 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;</lang>
- Output:
6 is perfect 28 is perfect 496 is perfect 8128 is perfect
Sidef
Sidef has built-in support for rational numbers. <lang ruby>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!' :
}"
}
}</lang>
- 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. <lang slate>54 / 7. 20 reciprocal. (5 / 6) reciprocal. (5 / 6) as: Float.</lang>
Smalltalk
Smalltalk uses naturally and transparently 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
<lang smalltalk>| 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
]
].</lang>
Tcl
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 {
lassign [split $arg] an ad
set n [expr {$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] {
lassign [split $arg] an ad
set n [expr {$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 {
lassign [split $arg] an ad
set n [expr {$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
While TI-89 BASIC has built-in rational and symbolic arithmetic, it does not have user-defined data types.
zkl
Enough of a Rational class for this task (ie implement the testing code "nicely"). <lang zkl>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
fcn __opAdd(n){
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;
}
}</lang> <lang zkl>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 ".")); }</lang>
- 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.
- Programming Tasks
- Arithmetic operations
- Arithmetic
- Ada
- ALGOL 68
- BBC BASIC
- C
- C sharp
- C++
- Boost
- Clojure
- Common Lisp
- D
- EchoLisp
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- Seed7
- Sidef
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- TI-89 BASIC
- TI-89 BASIC examples needing attention
- Examples needing attention
- Zkl