Arithmetic/Rational: Difference between revisions
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[[Category:Arithmetic]]
{{task|Arithmetic operations}}
;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.
Line 18 ⟶ 16:
Finally test the operators:
Use the new type '''frac''' to find all [[Perfect Numbers|perfect numbers]] less
;Related tasks:
* [[Perfect Numbers]]
* [[Check Machin-like formulas]]
<br><br>
=={{header|Action!}}==
Calculations on a real Atari 8-bit computer take quite long time. It is recommended to use an emulator capable with increasing speed of Atari CPU.
{{libheader|Action! Tool Kit}}
<syntaxhighlight lang="action!">INCLUDE "D2:REAL.ACT" ;from the Action! Tool Kit
TYPE Frac=[INT num,den]
REAL half
PROC PrintFrac(Frac POINTER x)
PrintI(x.num) Put('/) PrintI(x.den)
RETURN
INT FUNC Gcd(INT a,b)
INT tmp
IF a<b THEN
tmp=a a=b b=tmp
FI
WHILE b#0
DO
tmp=a MOD b
a=b
b=tmp
OD
RETURN (a)
PROC Init(INT n,d Frac POINTER res)
IF d>0 THEN
res.num=n res.den=d
ELSEIF d<0 THEN
res.num=-n res.den=-d
ELSE
Print("Denominator cannot be zero!")
Break()
FI
RETURN
PROC Assign(Frac POINTER x,res)
Init(x.num,x.den,res)
RETURN
PROC Neg(Frac POINTER x,res)
Init(-x.num,x.den,res)
RETURN
PROC Inverse(Frac POINTER x,res)
Init(x.den,x.num)
RETURN
PROC Abs(Frac POINTER x,res)
IF x.num<0 THEN
Neg(x,res)
ELSE
Assign(x,res)
FI
RETURN
PROC Add(Frac POINTER x,y,res)
INT common,xDen,yDen
common=Gcd(x.den,y.den)
xDen=x.den/common
yDen=y.den/common
Init(x.num*yDen+y.num*xDen,xDen*y.den,res)
RETURN
PROC Sub(Frac POINTER x,y,res)
Frac n
Neg(y,n) Add(x,n,res)
RETURN
PROC Mult(Frac POINTER x,y,res)
Init(x.num*y.num,x.den*y.den,res)
RETURN
PROC Div(Frac POINTER x,y,res)
Frac i
Inverse(y,i) Mult(x,i,res)
RETURN
BYTE FUNC Greater(Frac POINTER x,y)
Frac diff
Sub(x,y,diff)
IF diff.num>0 THEN
RETURN (1)
FI
RETURN (0)
BYTE FUNC Less(Frac POINTER x,y)
RETURN (Greater(y,x))
BYTE FUNC GreaterEqual(Frac POINTER x,y)
Frac diff
Sub(x,y,diff)
IF diff.num>=0 THEN
RETURN (1)
FI
RETURN (0)
BYTE FUNC LessEqual(Frac POINTER x,y)
RETURN (GreaterEqual(y,x))
BYTE FUNC Equal(Frac POINTER x,y)
Frac diff
Sub(x,y,diff)
IF diff.num=0 THEN
RETURN (1)
FI
RETURN (0)
BYTE FUNC NotEqual(Frac POINTER x,y)
IF Equal(x,y) THEN
RETURN (0)
FI
RETURN (1)
INT FUNC Sqrt(INT x)
REAL r1,r2
IF x=0 THEN
RETURN (0)
FI
IntToReal(x,r1)
Power(r1,half,r2)
RETURN (RealToInt(r2))
PROC Main()
DEFINE MAXINT="32767"
INT i,f,max2
Frac sum,tmp1,tmp2,tmp3,one
Put(125) PutE() ;clear screen
ValR("0.5",half)
Init(1,1,one)
FOR i=2 TO MAXINT
DO
Init(1,i,sum) ;sum=1/i
max2=Sqrt(i)
FOR f=2 TO max2
DO
IF i MOD f=0 THEN
Init(1,f,tmp1) ;tmp1=1/f
Add(sum,tmp1,tmp2) ;tmp2=sum+1/f
Init(f,i,tmp3) ;tmp3=f/i
Add(tmp2,tmp3,sum) ;sum=sum+1/f+f/i
FI
OD
IF Equal(sum,one) THEN
PrintF("%I is perfect%E",i)
FI
OD
RETURN</syntaxhighlight>
{{out}}
[https://gitlab.com/amarok8bit/action-rosetta-code/-/raw/master/images/Rational.png Screenshot from Atari 8-bit computer]
<pre>
6 is perfect
28 is perfect
496 is perfect
8128 is perfect
</pre>
=={{header|Ada}}==
<div style="text-align:right;font-size:7pt">''<nowiki>[</nowiki>This section is included from [[Arithmetic/Rational/Ada|a subpage]] and should be edited there, not here.<nowiki>]</nowiki>''</div>
{{:Arithmetic/Rational/Ada}}
=={{header|ALGOL 68}}==
{{works with|ALGOL 68|Standard - no extensions to language used}}
{{works with|ALGOL 68G|Any - tested with release mk15-0.8b.fc9.i386}}
<syntaxhighlight lang="algol68"> MODE FRAC = STRUCT( INT num #erator#, den #ominator#);
FORMAT frac repr = $g(-0)"//"g(-0)$;
Line 101 ⟶ 268:
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 ),
Line 144 ⟶ 311:
MODAB = (REF FRAC a, FRAC b)REF FRAC: ( a %*:= b );
Example: searching for Perfect Numbers.
FRAC sum:= FRACINIT 0;
Line 163 ⟶ 330:
candidate, ENTIER sum, real sum, ENTIER sum = 1))
FI
OD</syntaxhighlight>
{{out}}
<pre>
Sum of reciprocal factors of 6 = 1 exactly, about 1.0000000000000000000000000001 perfect!
Line 175 ⟶ 342:
Sum of reciprocal factors of 32760 = 3 exactly, about 3.0000000000000000000000000003
Sum of reciprocal factors of 523776 = 2 exactly, about 2.0000000000000000000000000005
</pre>
=={{header|Arturo}}==
Arturo comes with built-in support for rational numbers.
<syntaxhighlight lang="rebol">a: to :rational [1 2]
b: to :rational [3 4]
print ["a:" a]
print ["b:" b]
print ["a + b :" a + b]
print ["a - b :" a - b]
print ["a * b :" a * b]
print ["a / b :" a / b]
print ["a // b :" a // b]
print ["a % b :" a % b]
print ["reciprocal b:" reciprocal b]
print ["neg a:" neg a]
print ["pi ~=" to :rational 3.14]</syntaxhighlight>
{{out}}
<pre>a: 1/2
b: 3/4
a + b : 5/4
a - b : -1/4
a * b : 3/8
a / b : 2/3
a // b : 2/3
a % b : 1/2
reciprocal b: 4/3
neg a: -1/2
pi ~= 157/50</pre>
=={{header|BBC BASIC}}==
{{works with|BBC BASIC for Windows}}
<syntaxhighlight 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</syntaxhighlight>
Output:
<pre>
6 is perfect
28 is perfect
496 is perfect
8128 is perfect
</pre>
=={{header|C}}==
C does not have overloadable operators. The following implementation <u>''does not define all operations''</u> 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.
<syntaxhighlight 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;
}</syntaxhighlight>
See [[Rational Arithmetic/C]]
=={{header|
<div style="text-align:right;font-size:7pt">''<nowiki>[</nowiki>This section is included from [[Arithmetic/Rational/C sharp|a subpage]] and should be edited there, not here.<nowiki>]</nowiki>''</div>
{{:Arithmetic/Rational/C sharp}}
=={{header|C++}}==
{{libheader|Boost}}
Boost provides a rational number template.
<syntaxhighlight 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;
}</syntaxhighlight>
===Without using external libraries===
<syntaxhighlight lang="c++">
#include <cmath>
#include <cstdint>
#include <iostream>
#include <numeric>
#include <stdexcept>
class Rational {
public:
/// Constructors ///
Rational() : numer(0), denom(1) {}
Rational(const int64_t number) : numer(number), denom(1) {}
Rational(const int64_t& numerator, const int64_t& denominator) : numer(numerator), denom(denominator) {
if ( numer == 0 ) {
denom = 1;
} else if ( denom == 0 ) {
throw std::invalid_argument("Denominator cannot be zero: " + denom);
} else if ( denom < 0 ) {
numer = -numer;
denom = -denom;
}
int64_t divisor = std::gcd(numerator, denom);
numer = numer / divisor;
denom = denom / divisor;
}
Rational(const Rational& other) : numer(other.numer), denom(other.denom) {}
/// Operators ///
Rational& operator=(const Rational& other) {
if ( *this != other ) { numer = other.numer; denom = other.denom; }
return *this;
}
bool operator!=(const Rational& other) const { return ! ( *this == other ); }
bool operator==(const Rational& other) const {
if ( numer == other.numer && denom == other.denom ) { return true; }
return false;
}
Rational& operator+=(const Rational& other) {
*this = Rational(numer* other.denom + other.numer * denom, denom * other.denom);
return *this;
}
Rational operator+(const Rational& other) const { return Rational(*this) += other; }
Rational& operator-=(const Rational& other) {
Rational temp(other);
temp.numer = -temp.numer;
return *this += temp;
}
Rational operator-(const Rational& other) const { return Rational(*this) -= other; }
Rational& operator*=(const Rational& other) {
*this = Rational(numer * other.numer, denom * other.denom);
return *this;
}
Rational operator*(const Rational& other) const { return Rational(*this) *= other; };
Rational& operator/=(const Rational other) {
Rational temp(other.denom, other.numer);
*this *= temp;
return *this;
}
Rational operator/(const Rational& other) const { return Rational(*this) /= other; };
bool operator<(const Rational& other) const { return numer * other.denom < denom * other.numer; }
bool operator<=(const Rational& other) const { return ! ( other < *this ); }
bool operator>(const Rational& other) const { return other < *this; }
bool operator>=(const Rational& other) const { return ! ( *this < other ); }
Rational operator-() const { return Rational(-numer, denom); }
Rational& operator++() { numer += denom; return *this; }
Rational operator++(int) { Rational temp = *this; ++*this; return temp; }
Rational& operator--() { numer -= denom; return *this; }
Rational operator--(int) { Rational temp = *this; --*this; return temp; }
friend std::ostream& operator<<(std::ostream& outStream, const Rational& other) {
outStream << other.numer << "/" << other.denom;
return outStream;
}
/// Methods ///
Rational reciprocal() const { return Rational(denom, numer); }
Rational positive() const { return Rational(abs(numer), denom); }
int64_t to_integer() const { return numer / denom; }
double to_double() const { return (double) numer / denom; }
int64_t hash() const { return std::hash<int64_t>{}(numer) ^ std::hash<int64_t>{}(denom); }
private:
int64_t numer;
int64_t denom;
};
int main() {
std::cout << "Perfect numbers less than 2^19:" << std::endl;
const int32_t limit = 1 << 19;
for ( int32_t candidate = 2; candidate < limit; ++candidate ) {
Rational sum = Rational(1, candidate);
int32_t square_root = (int32_t) sqrt(candidate);
for ( int32_t factor = 2; factor <= square_root; ++factor ) {
if ( candidate % factor == 0 ) {
sum += Rational(1, factor);
sum += Rational(1, candidate / factor);
}
}
if ( sum == Rational(1) ) {
std::cout << candidate << std::endl;
}
}
}
</syntaxhighlight>
{{ out }}
<pre>
Perfect numbers less than 2^19:
6
28
496
8128
</pre>
=={{header|Clojure}}==
Ratios are built in to Clojure and support math operations already. They automatically reduce and become Integers if possible.
<syntaxhighlight lang="clojure">user> 22/7
22/7
user> 34/2
17
user> (+ 37/5 42/9)
181/15</syntaxhighlight>
=={{header|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.
<syntaxhighlight lang="lisp">(loop for candidate from 2 below (expt 2 19)
for sum = (+ (/ candidate)
(loop for factor from 2 to (isqrt candidate)
Line 190 ⟶ 718:
sum (+ (/ factor) (/ (floor candidate factor)))))
when (= sum 1)
collect candidate)</
=={{header|D}}==
<syntaxhighlight 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." : ".");
}
}
}</syntaxhighlight>
Use the <code>-version=rational_arithmetic_main</code> compiler switch to run the test code.
{{out}}
<pre>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.</pre>
Currently RationalT!BigInt is not fast.
=={{header|Delphi}}==
{{libheader| System.SysUtils}}
{{libheader| Boost.Rational}}[[https://github.com/MaiconSoft/DelphiBoostLib]]
{{Trans|C#}}
<syntaxhighlight lang="delphi">
program Arithmetic_Rational;
{$APPTYPE CONSOLE}
uses
System.SysUtils,
Boost.Rational;
var
sum: TFraction;
max: Integer = 1 shl 19;
candidate, max2, factor: Integer;
begin
for candidate := 2 to max - 1 do
begin
sum := Fraction(1, candidate);
max2 := Trunc(Sqrt(candidate));
for factor := 2 to max2 do
begin
if (candidate mod factor) = 0 then
begin
sum := sum + Fraction(1, factor);
sum := sum + Fraction(1, candidate div factor);
end;
end;
if sum = Fraction(1) then
Writeln(candidate, ' is perfect');
end;
Readln;
end.</syntaxhighlight>
{{out}}
<pre>6 is perfect
28 is perfect
496 is perfect
8128 is perfect</pre>
=={{header|EchoLisp}}==
EchoLisp supports rational numbers as native type. "Big" rational i.e bigint/bigint are not supported.
<syntaxhighlight 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))))))
</syntaxhighlight>
{{out}}
<syntaxhighlight 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.
</syntaxhighlight>
=={{header|Elisa}}==
<syntaxhighlight 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;</syntaxhighlight>
Tests
<syntaxhighlight 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)?</syntaxhighlight>
{{out}}
<pre>
6
28
496
8128
</pre>
=={{header|Elixir}}==
<syntaxhighlight 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)</syntaxhighlight>
{{out}}
<pre>
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
</pre>
=={{header|ERRE}}==
<syntaxhighlight 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</syntaxhighlight>
{{out}}
<pre> 6 is perfect
28 is perfect
496 is perfect
8128 is perfect
</pre>
=={{header|F_Sharp|F#}}==
The F# Powerpack library defines the BigRational data type.
<syntaxhighlight 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</syntaxhighlight>
=={{header|Factor}}==
<code>ratio</code> is a built-in numeric type.
<syntaxhighlight 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 .</syntaxhighlight>
{{out}}
<pre>
Perfect numbers <= 2^19:
V{ 6 28 496 8128 }
</pre>
=={{header|Fermat}}==
Fermat supports rational aritmetic natively.
<syntaxhighlight lang="fermat">
for n=2 to 2^19 by 2 do
s:=3/n;
m:=1;
while m<=n/3 do
if Divides(m,n) then s:=s+1/m; fi;
m:=m+1;
od;
if s=2 then !!n fi;
od;</syntaxhighlight>
{{out}}
<pre>
6
28
496
8128
</pre>
=={{header|Forth}}==
<
\ Uses the stack convention of the built-in "*/" for int * frac -> int
Line 230 ⟶ 1,354:
: rat-inc tuck + swap ;
: rat-dec tuck - swap ;</
=={{header|Fortran}}==
{{works with|Fortran|90 and later}}
<
implicit none
Line 469 ⟶ 1,593:
end function rational_modulo
end module module_rational</
Example:
<
use module_rational
Line 498 ⟶ 1,622:
end do
end program perfect_numbers</
{{out}}
<
28
496
8128</
=={{header|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.
<syntaxhighlight 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.)
</syntaxhighlight>
=={{header|GAP}}==
Rational numbers are built-in.
<syntaxhighlight lang="gap">2/3 in Rationals;
# true
2/3 + 3/4;
# 17/12</syntaxhighlight>
=={{header|Go}}==
Go does not have user defined operators. Go does however have a rational number type in the <code>math/big</code> 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.
<syntaxhighlight 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)
}
}
}</syntaxhighlight>
{{out}}
<pre>
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
</pre>
=={{header|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:
<syntaxhighlight 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}"
}
}</syntaxhighlight>
The following ''RationalCategory'' class allows for modification of regular ''Number'' behavior when interacting with ''Rational''.
<syntaxhighlight 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)
}
}</syntaxhighlight>
Test Program (mixes the ''RationalCategory'' methods into the ''Number'' class):
<syntaxhighlight 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 }</syntaxhighlight>
{{out}}
<pre style="height:30ex;overflow:scroll;">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
</pre>
The following uses the ''Rational'' class, with ''RationalCategory'' mixed into ''Number'', to find all perfect numbers less than 2<sup>19</sup>:
<syntaxhighlight 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 }</syntaxhighlight>
{{out}}
<pre>...................................................................................................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]]</pre>
=={{header|Haskell}}==
Haskell provides a <code>Rational</code> type, which is really an alias for <code>Ratio Integer</code> (<code>Ratio</code> being a polymorphic type implementing rational numbers for any <code>Integral</code> type of numerators and denominators). The fraction is constructed using the <code>%</code> operator.
<syntaxhighlight lang="haskell">import Data.Ratio ((%))
-- Prints the first N perfect numbers.
main = do
let n = 4
take
n
| 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 ]
</syntaxhighlight>
For a sample implementation of <code>Ratio</code>, see [http://www.haskell.org/onlinereport/ratio.html the Haskell 98 Report].
=={{header|Icon}} and {{header|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' (>)
<syntaxhighlight 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</syntaxhighlight>
{{out}}
<pre>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)</pre>
The following task functions are missing from the IPL:
<syntaxhighlight 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</syntaxhighlight>
The {{libheader|Icon Programming Library}} provides [http://www.cs.arizona.edu/icon/library/src/procs/rational.icn rational] and [http://www.cs.arizona.edu/icon/library/src/procs/numbers.icn gcd in numbers]. Record definition and usage is shown below:
<syntaxhighlight 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</syntaxhighlight>
=={{header|J}}==
Rational numbers in J may be formed from fixed precision integers by first upgrading them to arbitrary precision integers and then dividing them:
<syntaxhighlight lang="j"> (x: 3) % (x: -4)
_3r4
3 %&x: -4
_3r4</syntaxhighlight>
Note that the syntax is analogous to the syntax for floating point numbers, but uses <code>r</code> to separate the numerator and denominator instead of <code>e</code> to separate the mantissa and exponent.
Thus:
<syntaxhighlight 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</syntaxhighlight>
You can also coerce numbers directly to rational using x: (or to integer or floating point as appropriate using its inverse)
<syntaxhighlight lang="j"> x: 3%4
3r4
x:inv 3%4
0.75</syntaxhighlight>
Increment and decrement are also included in the language, but you could just as easily add or subtract 1:
<syntaxhighlight lang="j"> >: 3r4
7r4
<: 3r4
_1r4</syntaxhighlight>
J does not encourage the use of specialized mutators, but those could also be defined:
<syntaxhighlight lang="j">mutadd=:adverb define
(m)=: (".m)+y
)
mutsub=:adverb define
(m)=: (".m)-y
)</syntaxhighlight>
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):
<syntaxhighlight lang="j"> n=: 3r4
'n' mutadd 1
7r4
'n' mutsub 1
3r4
'n' mutsub 1
_1r4</syntaxhighlight>
(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:
<syntaxhighlight lang="j"> is_perfect_rational=: 2 = (1 + i.) +/@:%@([ #~ 0 = |) ]</syntaxhighlight>
Faster version (but the problem, as stated, is still tremendously inefficient):
<syntaxhighlight lang="j">factors=: */&>@{@((^ i.@>:)&.>/)@q:~&__
is_perfect_rational=: 2= +/@:%@,@factors</syntaxhighlight>
Exhaustive testing would take forever:
<
6 28 496 8128
I.is_perfect_rational@x:@"0 i.2^19x
6 28 496 8128</
More limited testing takes reasonable amounts of time:
<
6 28 496 8128</
=={{header|Java}}==
Uses BigRational class: [[Arithmetic/Rational/Java]]
<syntaxhighlight 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");
}
}
}</syntaxhighlight>
{{out}}
<pre>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</pre>
=={{header|JavaScript}}==
<div style="text-align:right;font-size:7pt">''<nowiki>[</nowiki>This section is included from [[Arithmetic/Rational/JavaScript|a subpage]] and should be edited there, not here.<nowiki>]</nowiki>''</div>
{{:Arithmetic/Rational/JavaScript}}
=={{header|jq}}==
{{works with|jq}}
'''Works with gojq, the Go implementation of jq'''
In this entry, a jq module for rational arithmetic is first
presented. It can be included or imported using jq's "include" or
"import" directives. The module includes functions for taking
square roots and for converting a rational to a decimal string
or decimal string approximation,
and is sufficient for the jq solution at [[Faulhaber%27s_triangle#jq]].
The constructor is named "r" rather than "frac", mainly because "r" is
short and handy for what is here more than a simple constructor (it
can also be used for normalization and to divide one rational by
another), and because name conflicts can easily be resolved using jq's
module system.
The other operators for working with rationals also begin with the letter "r":
'''Comparisons'''
*`requal`, `rgreaterthan`, `rgreaterthanOrEqual`, `rlessthan`, `rlessthanOrEqual`
'''Printing'''
* `rpp` for pretty-printing
* `r_to_decimal` for a decimal string representation
'''Unary'''
* `rabs` for unary `abs`
* `rfloor` like `floor`
* `rinv` for unary inverse
* `rminus` for unary minus
* `rround` for rounding
'''Arithmetic'''
* `radd` for addition
* `rminus` for subtraction
* `rmult` for multiplication
* `rdiv` for division
* `rsqrt` for square roots
<br>
In the following notes, "Rational" refers to a reduced-form rational
represented by a JSON object of the form {n:$n, d:$d} where
n signifies the numerator and d the denominator, and $d > 0.
The notation `$p // $q` is also used, and this is the form used for
pretty-printing by the filter rpp/0.
All the "r"-prefixed functions defined here are polymorphic in the
sense that an integer or rational can be used where a Rational
would normally be expected. This may be especially useful
in the case of requal/2.
'''module {"name": "Rational"};'''
<syntaxhighlight lang="jq"># a and b are assumed to be non-zero integers
def gcd(a; b):
# subfunction expects [a,b] as input
# i.e. a ~ .[0] and b ~ .[1]
def rgcd: if .[1] == 0 then .[0]
else [.[1], .[0] % .[1]] | rgcd
end;
[a,b] | rgcd;
# To take advantage of gojq's support for accurate integer division:
def idivide($j):
. as $i
| ($i % $j) as $mod
| ($i - $mod) / $j ;
# To take advantage of gojq's arbitrary-precision integer arithmetic:
def power($b): . as $in | reduce range(0;$b) as $i (1; . * $in);
# $p should be an integer or a rational
# $q should be a non-zero integer or a rational
# Output: a Rational: $p // $q
def r($p;$q):
def r: if type == "number" then {n: ., d: 1} else . end;
# The remaining subfunctions assume all args are Rational
def n: if .d < 0 then {n: -.n, d: -.d} else . end;
def rdiv($a;$b):
($a.d * $b.n) as $denom
| if $denom==0 then "r: division by 0" | error
else r($a.n * $b.d; $denom)
end;
if $q == 1 and ($p|type) == "number" then {n: $p, d: 1}
elif $q == 0 then "r: denominator cannot be 0" | error
else if ($p|type == "number") and ($q|type == "number")
then gcd($p;$q) as $g
| {n: ($p/$g), d: ($q/$g)} | n
else rdiv($p|r; $q|r)
end
end;
# Polymorphic (integers and rationals in general)
def requal($a; $b):
if $a | type == "number" and $b | type == "number" then $a == $b
else r($a;1) == r($b;1)
end;
# Input: a Rational
# Output: a Rational with a denominator that has no more than $digits digits
# and such that |rBefore - rAfter| < 1/(10|power($digits)
# where $digits should be a positive integer.
def rround($digits):
if .d | length > $digits
then (10|power($digits)) as $p
| .d as $d
| r($p * .n | idivide($d); $p)
else . end;
# Polymorphic; see also radd/0
def radd($a; $b):
def r: if type == "number" then {n: ., d: 1} else . end;
($a|r) as {n: $na, d: $da}
| ($b|r) as {n: $nb, d: $db}
| r( ($na * $db) + ($nb * $da); $da * $db );
# Polymorphic; see also rmult/0
def rmult($a; $b):
def r: if type == "number" then {n: ., d: 1} else . end;
($a|r) as {n: $na, d: $da}
| ($b|r) as {n: $nb, d: $db}
| r( $na * $nb; $da * $db ) ;
# Input: an array of rationals (integers and/or Rationals)
# Output: a Rational computed using left-associativity
def rmult:
if length == 0 then r(1;1)
elif length == 1 then r(.[0]; 1) # ensure the result is Rational
else .[0] as $first
| reduce .[1:][] as $x ($first; rmult(.; $x))
end;
# Input: an array of rationals (integers and/or Rationals)
# Output: a Rational computed using left-associativity
def radd:
if length == 0 then r(0;1)
elif length == 1 then r(.[0]; 1) # ensure the result is Rational
else .[0] as $first
| reduce .[1:][] as $x ($first; radd(. ; $x))
end;
def rabs: r(.;1) | r(.n|length; .d|length);
def rminus: r(-1 * .n; .d);
def rminus($a; $b): radd($a; rmult(-1; $b));
# Note that rinv does not check for division by 0
def rinv: r(1; .);
def rdiv($a; $b): r($a; $b);
# Input: an integer or a Rational, $p
# Output: $p < $q
def rlessthan($q):
# lt($b) assumes . and $b have the same sign
def lt($b):
. as $a
| ($a.n * $b.d) < ($b.n * $a.d);
if $q|type == "number" then rlessthan(r($q;1))
else if type == "number" then r(.;1) else . end
| if .n < 0
then if ($q.n >= 0) then true
else . as $p | ($q|rminus | rlessthan($p|rminus))
end
else lt($q)
end
end;
def rgreaterthan($q):
. as $p | $q | rlessthan($p);
def rlessthanOrEqual($q): requal(.;$q) or rlessthan($q);
def rgreaterthanOrEqual($q): requal(.;$q) or rgreaterthan($q);
# Input: non-negative integer or Rational
def rsqrt(precision):
r(.;1) as $n
| (precision + 1) as $digits
| def update: rmult( r(1;2); radd(.x; rdiv($n; .x))) | rround($digits);
| def update: rmult( r(1;2); radd(.x; rdiv($n; .x)));
r(1; 10|power(precision)) as $p
| { x: .}
| .root = update
| until( rminus(.root; .x) | rabs | rlessthan($p);
.x = .root
| .root = update )
| .root ;
# Use native floats
# q.v. r_to_decimal(precision)
def r_to_decimal: .n / .d;
# Input: a Rational, or {n, d} in general, or an integer.
# Output: a string representation of the input as a decimal number.
# If the input is a number, it is simply converted to a string.
# Otherwise, $precision determines the number of digits after the decimal point,
# obtained by truncating, but trailing 0s are omitted.
# Examples assuming $digits is 5:
# -0//1 => "0"
# 2//1 => "2"
# 1//2 => "0.5"
# 1//3 => "0.33333"
# 7//9 => "0.77777"
# 1//100 => "0.01"
# -1//10 => "-0.1"
# 1//1000000 => "0."
def r_to_decimal($digits):
if .n == 0 # captures the annoying case of -0
then "0"
elif type == "number" then tostring
elif .d < 0 then {n: -.n, d: -.d}|r_to_decimal($digits)
elif .n < 0
then "-" + ((.n = -.n) | r_to_decimal($digits))
else (10|power($digits)) as $p
| .d as $d
| if $d == 1 then .n|tostring
else ($p * .n | idivide($d) | tostring) as $n
| ($n|length) as $nlength
| (if $nlength > $digits then $n[0:$nlength-$digits] + "." + $n[$nlength-$digits:]
else "0." + ("0"*($digits - $nlength) + $n)
end) | sub("0+$";"")
end
end;
# Assume . is an integer or in canonical form
def rfloor:
if type == "number" then r(.;1)
elif 0 == .n or (0 < .n and .n < .d) then r(0;1)
elif 0 < .n or (.n % .d == 0) then .d as $d | r(.n | idivide($d); 1)
else rminus( r( - .n; .d) | rfloor | rminus; 1)
end;
# pretty print ala Julia
def rpp: "\(.n) // \(.d)";</syntaxhighlight>
'''Perfect Numbers'''
<syntaxhighlight lang="jq">
# divisors as an unsorted stream
def divisors:
if . == 1 then 1
else . as $n
| label $out
| range(1; $n) as $i
| ($i * $i) as $i2
| if $i2 > $n then break $out
else if $i2 == $n
then $i
elif ($n % $i) == 0
then $i, ($n/$i)
else empty
end
end
end;
def is_perfect:
requal(2; [divisors | r(1;. )] | radd);
# Example:
range(1;pow(2;19)) | select( is_perfect )
</syntaxhighlight>
{{out}}
<pre>
6
28
496
8128
</pre>
=={{header|Julia}}==
Julia has native support for rational numbers. Rationals are expressed as <tt>m//n</tt>, where <tt>m</tt> and <tt>n</tt> are integers. In addition to supporting most of the usual mathematical functions in a natural way on rationals, the methods <tt>num</tt> and <tt>den</tt> provide the fully reduced numerator and denominator of a rational value.
{{works with|Julia|1.2}}
<syntaxhighlight lang="julia">using Primes
divisors(n) = foldl((a, (p, e)) -> vcat((a * [p^i for i in 0:e]')...), factor(n), init=[1])
isperfect(n) = sum(1 // d for d in divisors(n)) == 2
lo, hi = 2, 2^19
println("Perfect numbers between ", lo, " and ", hi, ": ", collect(filter(isperfect, lo:hi)))
</syntaxhighlight>
{{out}}
<pre>
Perfect numbers between 2 and 524288: [6, 28, 496, 8128]
</pre>
=={{header|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.
<syntaxhighlight 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(" $i")
}
println()
}</syntaxhighlight>
{{out}}
<pre>
frac1 = 4
frac2 = 15/2
frac1 <= frac2 is true
frac1 >= frac2 is false
frac1 == frac2 is false
frac1 != frac2 is true
frac1 + frac2 = 23/2
frac1 - frac2 = -7/2
frac1 * frac2 = 30
frac1 / frac2 = 8/15
frac1 % frac2 = 4
inv(frac1) = 1/4
abs(-frac1) = -4
inc(frac2) = 17/2
dec(frac2) = 15/2
dbl(frac2) = 7.5
lng(frac2) = 7
The Perfect numbers less than 2^19 are:
6 28 496 8128
</pre>
=={{header|Liberty BASIC}}==
Testing all numbers up to 2 ^ 19 takes an excessively long time.
<syntaxhighlight lang="lb">
n=2^19
for testNumber=1 to n
sum$=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
</syntaxhighlight>
=={{header|Lingo}}==
A new 'frac' data type can be implemented like this:
<syntaxhighlight 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</syntaxhighlight>
Lingo does not support overwriting built-in operators, so 'frac'-operators must be implemented as functions:
<syntaxhighlight 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</syntaxhighlight>
Usage:
<syntaxhighlight 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"</syntaxhighlight>
Finding perfect numbers:
<syntaxhighlight 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</syntaxhighlight>
<syntaxhighlight lang="lingo">findPerfects(power(2, 19))
-- 6
-- 28
-- 496
-- 8128</syntaxhighlight>
=={{header|Lua}}==
<
do
Line 606 ⟶ 3,111:
return table.concat(ret, '\n')
end
print(findperfs(2^19))</
=={{header|M2000 Interpreter}}==
http://www.rosettacode.org/wiki/M2000_Interpreter_rational_numbers
<syntaxhighlight lang="m2000 interpreter">
Class Rational {
\\ this is a compact version for this task
numerator as decimal, denominator as decimal
gcd=lambda->0
lcm=lambda->0
operator "+" {
Read l
denom=.lcm(l.denominator, .denominator)
.numerator<=denom/l.denominator*l.numerator+denom/.denominator*.numerator
if .numerator==0 then denom=1
.denominator<=denom
}
Group Real {
value {
link parent numerator, denominator to n, d
=n/d
}
}
Group ToString$ {
value {
link parent numerator, denominator to n, d
=Str$(n)+"/"+Str$(d,"")
}
}
class:
Module Rational (.numerator, .denominator) {
if .denominator=0 then Error "Zero denominator"
sgn=Sgn(.numerator)*Sgn(.denominator)
.denominator<=abs(.denominator)
.numerator<=abs(.numerator)*sgn
gcd1=lambda (a as decimal, b as decimal) -> {
if a<b then swap a,b
g=a mod b
while g {
a=b:b=g: g=a mod b
}
=abs(b)
}
gdcval=gcd1(abs(.numerator), .denominator)
if gdcval<.denominator and gdcval<>0 then
.denominator/=gdcval
.numerator/=gdcval
end if
.gcd<=gcd1
.lcm<=lambda gcd=gcd1 (a as decimal, b as decimal) -> {
=a/gcd(a,b)*b
}
}
}
sum=rational(1, 1)
onediv=rational(1,1)
divcand=rational(1,1)
Profiler
For sum.denominator= 2 to 2**15 {
divcand.denominator=sum.denominator
For onediv.denominator=2 to sqrt(sum.denominator) {
if sum.denominator mod onediv.denominator = 0 then {
divcand.numerator=onediv.denominator
sum=sum+onediv+divcand
}
}
if sum.real=1 then Print sum.denominator;" is perfect"
sum.numerator=1
}
Print timecount
</syntaxhighlight>
=={{header|Maple}}==
Maple has full built-in support for arithmetic with fractions (rational numbers). Fractions are treated like any other number in Maple.
<syntaxhighlight lang="maple">
> a := 3 / 5;
a := 3/5
> numer( a );
3
> denom( a );
5
</syntaxhighlight>
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:
<syntaxhighlight lang="maple">
> b := 4 / 6;
b := 2/3
</syntaxhighlight>
All the standard arithmetic operators work with rational numbers. It is not necessary to call any special routines.
<syntaxhighlight 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
</syntaxhighlight>
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.
<syntaxhighlight lang="maple">
> evalf( 22 / 7 ); # default is 10 digits
3.142857143
> evalf[100]( 22 / 7 ); # 100 digits
3.142857142857142857142857142857142857142857142857142857142857142857\
142857142857142857142857142857143
</syntaxhighlight>
=={{header|Mathematica}} / {{header|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:
<syntaxhighlight lang="mathematica">4/16
3/8
8/4
Line 635 ⟶ 3,262:
Numerator[6/9]
Denominator[6/9]</
gives back:
<
3/8
2
Line 662 ⟶ 3,289:
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:
<
(b^2 - c^2)/(b - c) // Cancel
1/2 + b/c // Together</
gives back:
<syntaxhighlight lang="mathematica">1/2
b+c
(2 b+c) / (2 c)</
Moreover it does simplification like Sin[x]/Cos[x] => Tan[x]. Division, addition, subtraction, powering and multiplication of a list (of any dimension) is automatically threaded over the elements:
<
gives back:
<syntaxhighlight lang
To check for perfect numbers in the range 1 to 2^25 we can use:
<
CheckPerfect[num_Integer]:=If[Total[1/Divisors[num]]==2,AppendTo[found,num]];
Do[CheckPerfect[i],{i,1,2^25}];
found</
gives back:
<
Final note; approximations of fractions to any precision can be found using the function N.
=={{header|Maxima}}==
<syntaxhighlight 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</syntaxhighlight>
=={{header|Modula-2}}==
<div style="text-align:right;font-size:7pt">''<nowiki>[</nowiki>This section is included from [[Arithmetic/Rational/Modula-2|a subpage]] and should be edited there, not here.<nowiki>]</nowiki>''</div>
{{:Arithmetic/Rational/Modula-2}}
=={{header|Modula-3}}==
<div style="text-align:right;font-size:7pt">''<nowiki>[</nowiki>This section is included from [[Arithmetic/Rational/Modula-3|a subpage]] and should be edited there, not here.<nowiki>]</nowiki>''</div>
{{:Arithmetic/Rational/Modula-3}}
=={{header|Nim}}==
<syntaxhighlight 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: ""</syntaxhighlight>
Output:
<pre>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 </pre>
=={{header|Objective-C}}==
See [[
=={{header|OCaml}}==
OCaml's Num library implements arbitrary-precision rational numbers:
<syntaxhighlight lang="ocaml">#load "nums.cma";;
open Num;;
Line 703 ⟶ 3,479:
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;;</
[http://forge.ocamlcore.org/projects/pa-do/ Delimited overloading] can be used to make the arithmetic expressions more readable:
<syntaxhighlight 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</syntaxhighlight>
<syntaxhighlight 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</syntaxhighlight>
then implement the module:
<syntaxhighlight 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</syntaxhighlight>
Finally the use type defined by the module to perform the perfect number calculation:
<syntaxhighlight 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</syntaxhighlight>
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
=={{header|Ol}}==
Otus Lisp has rational numbers built-in and integrated with all other number types.
<syntaxhighlight lang="scheme">
(define x 3/7)
(define y 9/11)
(define z -2/5)
; demonstrate builtin functions:
(print "(abs " z ") = " (abs z))
(print "- " z " = " (- z))
(print x " + " y " = " (+ x y))
(print x " - " y " = " (- x y))
(print x " * " y " = " (* x y))
(print x " / " y " = " (/ x y))
(print x " < " y " = " (< x y))
(print x " > " y " = " (> x y))
; introduce new functions:
(define (+:= x) (+ x 1))
(define (-:= x) (- x 1))
(print "+:= " z " = " (+:= z))
(print "-:= " z " = " (-:= z))
; finally, find all perfect numbers less than 2^15:
(lfor-each (lambda (candidate)
(let ((sum (lfold (lambda (sum factor)
(if (= 0 (modulo candidate factor))
(+ sum (/ 1 factor) (/ factor candidate))
sum))
(/ 1 candidate)
(liota 2 1 (+ (isqrt candidate) 1)))))
(if (= 1 (denominator sum))
(print candidate (if (eq? sum 1) ", perfect" "")))))
(liota 2 1 (expt 2 15)))
</syntaxhighlight>
{{Out}}
<pre>
(abs -2/5) = 2/5
- -2/5 = 2/5
3/7 + 9/11 = 96/77
3/7 - 9/11 = -30/77
3/7 * 9/11 = 27/77
3/7 / 9/11 = 11/21
3/7 < 9/11 = #true
3/7 > 9/11 = #false
+:= -2/5 = 3/5
-:= -2/5 = -7/5
6, perfect
28, perfect
120
496, perfect
672
8128, perfect
30240
32760
</pre>
=={{header|ooRexx}}==
<syntaxhighlight 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 public inherit orderable
::method init
expose numerator denominator
use strict arg numerator, denominator = 1
if denominator == 0 then raise syntax 98.900 array("Fraction denominator cannot be zero")
-- if the denominator is negative, make the numerator carry the sign
if denominator < 0 then do
numerator = -numerator
denominator = - denominator
end
-- find the greatest common denominator and reduce to
-- the simplest form
gcd = self~gcd(numerator~abs, denominator~abs)
numerator /= gcd
denominator /= gcd
-- fraction instances are immutable, so these are
-- 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
</syntaxhighlight>
Output:
<pre>
6 is a perfect number
28 is a perfect number
496 is a perfect number
8128 is a perfect number
</pre>
=={{header|PARI/GP}}==
Pari handles rational arithmetic natively.
<syntaxhighlight lang="parigp">for(n=2,1<<19,
s=0;
fordiv(n,d,s+=1/d);
if(s==2,print(n))
)</syntaxhighlight>
=={{header|Perl}}==
Perl's <code>Math::BigRat</code> core module implements arbitrary-precision rational numbers. The <code>bigrat</code> pragma can be used to turn on transparent BigRat support:
<syntaxhighlight lang="perl">use bigrat;
foreach my $candidate (2 .. 2**19) {
Line 724 ⟶ 3,861:
print "Sum of recipr. factors of $candidate = $sum exactly ", ($sum == 1 ? "perfect!" : ""), "\n";
}
}</
=={{header|Phix}}==
{{Trans|Tcl}}
Phix does not support operator overloading (I am strongly opposed to such nonsense), nor does it have a native fraction library, but it might look a bit like this.
<!--<syntaxhighlight lang="phix">(phixonline)-->
<span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span>
<span style="color: #008080;">without</span> <span style="color: #000000;">warning</span> <span style="color: #000080;font-style:italic;">-- (several unused routines in this code)</span>
<span style="color: #008080;">constant</span> <span style="color: #000000;">NUM</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">DEN</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">2</span>
<span style="color: #008080;">type</span> <span style="color: #000000;">frac</span><span style="color: #0000FF;">(</span><span style="color: #004080;">object</span> <span style="color: #000000;">r</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">return</span> <span style="color: #004080;">sequence</span><span style="color: #0000FF;">(</span><span style="color: #000000;">r</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">and</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">r</span><span style="color: #0000FF;">)=</span><span style="color: #000000;">2</span> <span style="color: #008080;">and</span> <span style="color: #004080;">integer</span><span style="color: #0000FF;">(</span><span style="color: #000000;">r</span><span style="color: #0000FF;">[</span><span style="color: #000000;">NUM</span><span style="color: #0000FF;">])</span> <span style="color: #008080;">and</span> <span style="color: #004080;">integer</span><span style="color: #0000FF;">(</span><span style="color: #000000;">r</span><span style="color: #0000FF;">[</span><span style="color: #000000;">DEN</span><span style="color: #0000FF;">])</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">type</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">normalise</span><span style="color: #0000FF;">(</span><span style="color: #004080;">object</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">atom</span> <span style="color: #000000;">d</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #004080;">sequence</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span>
<span style="color: #0000FF;">{</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #000000;">d</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">n</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">d</span><span style="color: #0000FF;"><</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">n</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">n</span>
<span style="color: #000000;">d</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">d</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #004080;">atom</span> <span style="color: #000000;">g</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">gcd</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #000000;">d</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">return</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">n</span><span style="color: #0000FF;">/</span><span style="color: #000000;">g</span><span style="color: #0000FF;">,</span><span style="color: #000000;">d</span><span style="color: #0000FF;">/</span><span style="color: #000000;">g</span><span style="color: #0000FF;">}</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">frac_new</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">d</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">normalise</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #000000;">d</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">frac_abs</span><span style="color: #0000FF;">(</span><span style="color: #000000;">frac</span> <span style="color: #000000;">r</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">return</span> <span style="color: #0000FF;">{</span><span style="color: #7060A8;">abs</span><span style="color: #0000FF;">(</span><span style="color: #000000;">r</span><span style="color: #0000FF;">[</span><span style="color: #000000;">NUM</span><span style="color: #0000FF;">]),</span><span style="color: #000000;">r</span><span style="color: #0000FF;">[</span><span style="color: #000000;">DEN</span><span style="color: #0000FF;">]}</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">frac_inv</span><span style="color: #0000FF;">(</span><span style="color: #000000;">frac</span> <span style="color: #000000;">r</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">return</span> <span style="color: #7060A8;">reverse</span><span style="color: #0000FF;">(</span><span style="color: #000000;">r</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">frac_add</span><span style="color: #0000FF;">(</span><span style="color: #000000;">frac</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">integer</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">an</span><span style="color: #0000FF;">,</span><span style="color: #000000;">ad</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">,</span>
<span style="color: #0000FF;">{</span><span style="color: #000000;">bn</span><span style="color: #0000FF;">,</span><span style="color: #000000;">bd</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">b</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">normalise</span><span style="color: #0000FF;">(</span><span style="color: #000000;">an</span><span style="color: #0000FF;">*</span><span style="color: #000000;">bd</span><span style="color: #0000FF;">+</span><span style="color: #000000;">bn</span><span style="color: #0000FF;">*</span><span style="color: #000000;">ad</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">ad</span><span style="color: #0000FF;">*</span><span style="color: #000000;">bd</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">frac_sub</span><span style="color: #0000FF;">(</span><span style="color: #000000;">frac</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">integer</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">an</span><span style="color: #0000FF;">,</span><span style="color: #000000;">ad</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">,</span>
<span style="color: #0000FF;">{</span><span style="color: #000000;">bn</span><span style="color: #0000FF;">,</span><span style="color: #000000;">bd</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">b</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">normalise</span><span style="color: #0000FF;">(</span><span style="color: #000000;">an</span><span style="color: #0000FF;">*</span><span style="color: #000000;">bd</span><span style="color: #0000FF;">-</span><span style="color: #000000;">bn</span><span style="color: #0000FF;">*</span><span style="color: #000000;">ad</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">ad</span><span style="color: #0000FF;">*</span><span style="color: #000000;">bd</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">frac_mul</span><span style="color: #0000FF;">(</span><span style="color: #000000;">frac</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">integer</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">an</span><span style="color: #0000FF;">,</span><span style="color: #000000;">ad</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">,</span>
<span style="color: #0000FF;">{</span><span style="color: #000000;">bn</span><span style="color: #0000FF;">,</span><span style="color: #000000;">bd</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">b</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">normalise</span><span style="color: #0000FF;">(</span><span style="color: #000000;">an</span><span style="color: #0000FF;">*</span><span style="color: #000000;">bn</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">ad</span><span style="color: #0000FF;">*</span><span style="color: #000000;">bd</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">frac_div</span><span style="color: #0000FF;">(</span><span style="color: #000000;">frac</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">integer</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">an</span><span style="color: #0000FF;">,</span><span style="color: #000000;">ad</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">,</span>
<span style="color: #0000FF;">{</span><span style="color: #000000;">bn</span><span style="color: #0000FF;">,</span><span style="color: #000000;">bd</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">b</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">normalise</span><span style="color: #0000FF;">(</span><span style="color: #000000;">an</span><span style="color: #0000FF;">*</span><span style="color: #000000;">bd</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">ad</span><span style="color: #0000FF;">*</span><span style="color: #000000;">bn</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">frac_eq</span><span style="color: #0000FF;">(</span><span style="color: #000000;">frac</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">==</span><span style="color: #000000;">b</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">frac_ne</span><span style="color: #0000FF;">(</span><span style="color: #000000;">frac</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">!=</span><span style="color: #000000;">b</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">frac_lt</span><span style="color: #0000FF;">(</span><span style="color: #000000;">frac</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">frac_sub</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)[</span><span style="color: #000000;">NUM</span><span style="color: #0000FF;">]<</span><span style="color: #000000;">0</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">frac_gt</span><span style="color: #0000FF;">(</span><span style="color: #000000;">frac</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">frac_sub</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)[</span><span style="color: #000000;">NUM</span><span style="color: #0000FF;">]></span><span style="color: #000000;">0</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">frac_le</span><span style="color: #0000FF;">(</span><span style="color: #000000;">frac</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">frac_sub</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)[</span><span style="color: #000000;">NUM</span><span style="color: #0000FF;">]<=</span><span style="color: #000000;">0</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">frac_ge</span><span style="color: #0000FF;">(</span><span style="color: #000000;">frac</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">frac_sub</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)[</span><span style="color: #000000;">NUM</span><span style="color: #0000FF;">]>=</span><span style="color: #000000;">0</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">is_perfect</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">num</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">frac</span> <span style="color: #000000;">total</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">frac_new</span><span style="color: #0000FF;">(</span><span style="color: #000000;">0</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">f</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">factors</span><span style="color: #0000FF;">(</span><span style="color: #000000;">num</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">f</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">total</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">frac_add</span><span style="color: #0000FF;">(</span><span style="color: #000000;">total</span><span style="color: #0000FF;">,</span><span style="color: #000000;">frac_new</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">f</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]))</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">frac_eq</span><span style="color: #0000FF;">(</span><span style="color: #000000;">total</span><span style="color: #0000FF;">,</span><span style="color: #000000;">frac_new</span><span style="color: #0000FF;">(</span><span style="color: #000000;">2</span><span style="color: #0000FF;">))</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">procedure</span> <span style="color: #000000;">get_perfect_numbers</span><span style="color: #0000FF;">()</span>
<span style="color: #004080;">atom</span> <span style="color: #000000;">t0</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">time</span><span style="color: #0000FF;">()</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">lim</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">power</span><span style="color: #0000FF;">(</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,</span><span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">platform</span><span style="color: #0000FF;">()=</span><span style="color: #004600;">JS</span><span style="color: #0000FF;">?</span><span style="color: #000000;">13</span><span style="color: #0000FF;">:</span><span style="color: #000000;">19</span><span style="color: #0000FF;">))</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">2</span> <span style="color: #008080;">to</span> <span style="color: #000000;">lim</span> <span style="color: #008080;">do</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">is_perfect</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"perfect: %d\n"</span><span style="color: #0000FF;">,</span><span style="color: #000000;">i</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"elapsed: %3.2f seconds\n"</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">time</span><span style="color: #0000FF;">()-</span><span style="color: #000000;">t0</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">pn5</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">power</span><span style="color: #0000FF;">(</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">12</span><span style="color: #0000FF;">)*(</span><span style="color: #7060A8;">power</span><span style="color: #0000FF;">(</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">13</span><span style="color: #0000FF;">)-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">-- 5th perfect number</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">is_perfect</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pn5</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"perfect: %d\n"</span><span style="color: #0000FF;">,</span><span style="color: #000000;">pn5</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">procedure</span>
<span style="color: #000000;">get_perfect_numbers</span><span style="color: #0000FF;">()</span>
<!--</syntaxhighlight>-->
{{out}}
<pre>
perfect: 6
perfect: 28
perfect: 496
perfect: 8128
elapsed: 13.56 seconds
perfect: 33550336
</pre>
=== mpq ===
{{libheader|Phix/mpfr}}
Turned out to be slightly slower than native, but worth it for large number support.<br>
See also [[Bernoulli_numbers#Phix|Bernoulli_numbers]] for another example of mpqs in action.
<!--<syntaxhighlight lang="phix">(phixonline)-->
<span style="color: #008080;">include</span> <span style="color: #000000;">builtins</span><span style="color: #0000FF;">/</span><span style="color: #004080;">mpfr</span><span style="color: #0000FF;">.</span><span style="color: #000000;">e</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">is_perfect</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">num</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">mpq</span> <span style="color: #000000;">tot</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpq_init</span><span style="color: #0000FF;">(),</span>
<span style="color: #000000;">fth</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpq_init</span><span style="color: #0000FF;">()</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">f</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">factors</span><span style="color: #0000FF;">(</span><span style="color: #000000;">num</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">f</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
<span style="color: #7060A8;">mpq_set_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">fth</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">f</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">])</span>
<span style="color: #7060A8;">mpq_add</span><span style="color: #0000FF;">(</span><span style="color: #000000;">tot</span><span style="color: #0000FF;">,</span><span style="color: #000000;">tot</span><span style="color: #0000FF;">,</span><span style="color: #000000;">fth</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">return</span> <span style="color: #7060A8;">mpq_cmp_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">tot</span><span style="color: #0000FF;">,</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)=</span><span style="color: #000000;">0</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">procedure</span> <span style="color: #000000;">get_perfect_numbers</span><span style="color: #0000FF;">()</span>
<span style="color: #004080;">atom</span> <span style="color: #000000;">t0</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">time</span><span style="color: #0000FF;">()</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">lim</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">power</span><span style="color: #0000FF;">(</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,</span><span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">platform</span><span style="color: #0000FF;">()=</span><span style="color: #004600;">JS</span><span style="color: #0000FF;">?</span><span style="color: #000000;">13</span><span style="color: #0000FF;">:</span><span style="color: #000000;">19</span><span style="color: #0000FF;">))</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">2</span> <span style="color: #008080;">to</span> <span style="color: #000000;">lim</span> <span style="color: #008080;">do</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">is_perfect</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"perfect: %d\n"</span><span style="color: #0000FF;">,</span><span style="color: #000000;">i</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"elapsed: %3.2f seconds\n"</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">time</span><span style="color: #0000FF;">()-</span><span style="color: #000000;">t0</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">pn5</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">power</span><span style="color: #0000FF;">(</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">12</span><span style="color: #0000FF;">)*(</span><span style="color: #7060A8;">power</span><span style="color: #0000FF;">(</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">13</span><span style="color: #0000FF;">)-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">-- 5th perfect number</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">is_perfect</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pn5</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"perfect: %d\n"</span><span style="color: #0000FF;">,</span><span style="color: #000000;">pn5</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">procedure</span>
<span style="color: #000000;">get_perfect_numbers</span><span style="color: #0000FF;">()</span>
<!--</syntaxhighlight>-->
{{out}}
<pre>
perfect: 6
perfect: 28
perfect: 496
perfect: 8128
elapsed: 17.31 seconds
perfect: 33550336
</pre>
<small>Note that power(2,19) took over 270s under mpfr.js, so reduced to power(2,13) on that platform, making it finish in 0.99s</small>
=={{header|Picat}}==
A naive addition algorithm is used, so the program is slow.
<syntaxhighlight lang="picat">
main =>
foreach (I in 2..2**19, is_perfect(I))
println(I)
end.
is_perfect(N) => sum_rationals([$frac(1,D) : D in divisors(N)]) == $frac(2,1).
divisors(N) = [I : I in 1..N, N mod I == 0].
add(frac(A,B), frac(C,D)) = new_fract(A*D+B*C, B*D).
new_fract(A,B) = $frac(Num, Den) =>
G = gcd(A,B),
Num = A // G,
Den = B // G.
sum_rationals([X]) = X.
sum_rationals([X,Y|T]) = sum_rationals([add(X,Y)|T]).
</syntaxhighlight>
{{out}}
<pre>
6
28
496
8128
</pre>
=={{header|PicoLisp}}==
<syntaxhighlight 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") ) ) ) )</syntaxhighlight>
{{out}}
<pre>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</pre>
=={{header|PL/I}}==
<syntaxhighlight 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;</syntaxhighlight>
{{out}}
<pre>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</pre>
=={{header|Prolog}}==
Prolog supports rational numbers, where P/Q is written as P rdiv Q.
<syntaxhighlight lang="prolog">
divisor(N, Div) :-
Max is floor(sqrt(N)),
between(1, Max, D),
divmod(N, D, _, 0),
(Div = D; Div is N div D, Div =\= D).
divisors(N, Divs) :-
setof(M, divisor(N, M), Divs).
recip(A, B) :- B is 1 rdiv A.
sumrecip(N, A) :-
divisors(N, [1 | Ds]),
maplist(recip, Ds, As),
sum_list(As, A).
perfect(X) :- sumrecip(X, 1).
main :-
Limit is 1 << 19,
forall(
(between(1, Limit, N), perfect(N)),
(format("~w~n", [N]))),
halt.
?- main.
</syntaxhighlight>
{{Out}}
<pre>
6
28
496
8128
</pre>
=={{header|Python}}==
{{works with|Python|3.0}}
Python 3's standard library already implements a Fraction class:
<syntaxhighlight lang="python">from fractions import Fraction
for candidate in range(2, 2**19):
Line 744 ⟶ 4,406:
if sum.denominator == 1:
print("Sum of recipr. factors of %d = %d exactly %s" %
(candidate, int(sum), "perfect!" if sum == 1 else ""))</
It might be implemented like this:
<syntaxhighlight lang="python">def lcm(a, b):
return a // gcd(a,b) * b
Line 780 ⟶ 4,440:
return float(self.numerator / self.denominator)
def __int__(self):
return (self.numerator // self.denominator)</
=={{header|
Quackery comes with a rational arithmetic library coded in Quackery, <code>bigrat.qky</code>, and documented in '''The Book of Quackery.pdf'''. Both are available at [https://github.com/GordonCharlton/Quackery the Quackery Github repository].
<code>factors</code> is defined at [[Factors of an integer#Quackery]].
<syntaxhighlight lang="quackery"> [ $ "bigrat.qky" loadfile ] now!
[ -2 n->v rot
factors witheach
[ n->v 1/v v+ ]
v0= ] is perfect ( n -> b )
19 bit times [ i^ perfect if [ i^ echo cr ] ]</syntaxhighlight>
{{out}}
<pre>6
28
496
8128
</pre>
=={{header|Racket}}==
Racket always had support for exact rational numbers as a native numeric type.
Example:
<syntaxhighlight lang="racket">
-> (* 1/7 14)
2
</syntaxhighlight>
=={{header|Raku}}==
(formerly Perl 6)
{{Works with|rakudo|2016.08}}
Raku supports rational arithmetic natively.
<syntaxhighlight lang="raku" line>(2..2**19).hyper.map: -> $candidate {
my $sum = 1 / $candidate;
for 2 .. ceiling(sqrt($candidate)) -> $factor {
if $candidate %% $factor {
$sum += 1 / $factor + 1 / ($candidate / $factor);
}
}
if $sum.nude[1] == 1 {
say "Sum of reciprocal factors of $candidate = $sum exactly", ($sum == 1 ?? ", perfect!" !! ".");
}
}</syntaxhighlight>
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:
<syntaxhighlight lang="raku" line>for 1.0, 1.1, 1.2 ... 10 { .say }</syntaxhighlight>
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.
=={{header|REXX}}==
<syntaxhighlight 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)</syntaxhighlight>
Programming note: the '''eDivs, gcd, lcm''' functions are optimized functions for this program only.
'''output'''
<pre>
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
</pre>
=={{header|Ruby}}==
Ruby has a Rational class in it's core since 1.9.
<syntaxhighlight lang="ruby">
for candidate in 2 .. 2**19
sum = Rational(1, candidate)
for factor in 2 .. Integer.
if candidate % factor == 0
sum += Rational(1, factor) + Rational(1, candidate / factor)
Line 798 ⟶ 4,606:
[candidate, sum.to_i, sum == 1 ? "perfect!" : ""]
end
end</
{{out}}
<pre>
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
</pre>
=={{header|Rust}}==
<syntaxhighlight 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);
}
}
}
</syntaxhighlight>
=={{header|Scala}}==
<syntaxhighlight 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)
}</syntaxhighlight>
<syntaxhighlight 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)
}
}</syntaxhighlight>
=={{header|Scheme}}==
Scheme has native rational numbers.
{{works with|Scheme|R5RS}}
<syntaxhighlight 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)))))</syntaxhighlight>
It might be implemented like this:
[insert implementation here]
=={{header|
The library [http://seed7.sourceforge.net/libraries/rational.htm rational.s7i] defines the type
[http://seed7.sourceforge.net/manual/types.htm#rational rational], which supports the required functionality.
Rational numbers are based on the type [http://seed7.sourceforge.net/manual/types.htm#integer integer].
For rational numbers, which are based on integers with unlimited precision, use
[http://seed7.sourceforge.net/manual/types.htm#bigRational bigRational], which is defined
in the library [http://seed7.sourceforge.net/libraries/bigrat.htm bigrat.s7i].
<syntaxhighlight 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;</syntaxhighlight>
{{out}}
<pre>
6 is perfect
28 is perfect
496 is perfect
8128 is perfect
</pre>
=={{header|Sidef}}==
Sidef has built-in support for rational numbers.
<syntaxhighlight 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!' : ''
}"
}
}</syntaxhighlight>
{{out}}
<pre>
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
</pre>
=={{header|Slate}}==
Slate uses infinite-precision fractions transparently.
<syntaxhighlight lang="slate">54 / 7.
20 reciprocal.
(5 / 6) reciprocal.
(5 / 6) as: Float.</
=={{header|Smalltalk}}==
Smalltalk uses ''naturally'' and transparently infinite precision fractions (through the class Fraction):
<pre>st> 54/7
54/7
Line 830 ⟶ 4,930:
0.8333333333333334
</pre>
{{works with|GNU Smalltalk}} (and all others)
<syntaxhighlight lang="smalltalk">| sum |
2 to: (2 raisedTo: 19) do: [ :candidate |
sum := candidate reciprocal.
Line 849 ⟶ 4,948:
ifFalse: [ ' ' ] }) displayNl
]
].</
=={{header|Swift}}==
<syntaxhighlight lang="swift">import Foundation
extension BinaryInteger {
@inlinable
public func gcd(with other: Self) -> Self {
var gcd = self
var b = other
while b != 0 {
(gcd, b) = (b, gcd % b)
}
return gcd
}
@inlinable
public func lcm(with other: Self) -> Self {
let g = gcd(with: other)
return self / g * other
}
}
public struct Frac<NumType: BinaryInteger & SignedNumeric>: Equatable {
@usableFromInline
var _num: NumType
@usableFromInline
var _dom: NumType
@usableFromInline
init(_num: NumType, _dom: NumType) {
self._num = _num
self._dom = _dom
}
@inlinable
public init(numerator: NumType, denominator: NumType) {
let divisor = numerator.gcd(with: denominator)
self._num = numerator / divisor
self._dom = denominator / divisor
}
@inlinable
public static func + (lhs: Frac, rhs: Frac) -> Frac {
let multiplier = lhs._dom.lcm(with: rhs.denominator)
return Frac(
numerator: lhs._num * multiplier / lhs._dom + rhs._num * multiplier / rhs._dom,
denominator: multiplier
)
}
@inlinable
public static func += (lhs: inout Frac, rhs: Frac) {
lhs = lhs + rhs
}
@inlinable
public static func - (lhs: Frac, rhs: Frac) -> Frac {
return lhs + -rhs
}
@inlinable
public static func -= (lhs: inout Frac, rhs: Frac) {
lhs = lhs + -rhs
}
@inlinable
public static func * (lhs: Frac, rhs: Frac) -> Frac {
return Frac(numerator: lhs._num * rhs._num, denominator: lhs._dom * rhs._dom)
}
@inlinable
public static func *= (lhs: inout Frac, rhs: Frac) {
lhs = lhs * rhs
}
@inlinable
public static func / (lhs: Frac, rhs: Frac) -> Frac {
return lhs * Frac(_num: rhs._dom, _dom: rhs._num)
}
@inlinable
public static func /= (lhs: inout Frac, rhs: Frac) {
lhs = lhs / rhs
}
@inlinable
prefix static func - (rhs: Frac) -> Frac {
return Frac(_num: -rhs._num, _dom: rhs._dom)
}
}
extension Frac {
@inlinable
public var numerator: NumType {
get { _num }
set {
let divisor = newValue.gcd(with: denominator)
_num = newValue / divisor
_dom = denominator / divisor
}
}
@inlinable
public var denominator: NumType {
get { _dom }
set {
let divisor = newValue.gcd(with: numerator)
_num = numerator / divisor
_dom = newValue / divisor
}
}
}
extension Frac: CustomStringConvertible {
public var description: String {
let neg = numerator < 0 || denominator < 0
return "Frac(\(neg ? "-" : "")\(abs(numerator)) / \(abs(denominator)))"
}
}
extension Frac: Comparable {
@inlinable
public static func <(lhs: Frac, rhs: Frac) -> Bool {
return lhs._num * rhs._dom < lhs._dom * rhs._num
}
}
extension Frac: ExpressibleByIntegerLiteral {
public init(integerLiteral value: Int) {
self._num = NumType(value)
self._dom = 1
}
}
for candidate in 2..<1<<19 {
var sum = Frac(numerator: 1, denominator: candidate)
let m = Int(ceil(Double(candidate).squareRoot()))
for factor in 2..<m where candidate % factor == 0 {
sum += Frac(numerator: 1, denominator: factor)
sum += Frac(numerator: 1, denominator: candidate / factor)
}
if sum == 1 {
print("\(candidate) is perfect")
}
}</syntaxhighlight>
{{out}}
<pre>6 is perfect
28 is perfect
496 is perfect
8128 is perfect</pre>
=={{header|Tcl}}==
<div style="text-align:right;font-size:7pt">''<nowiki>[</nowiki>This section is included from [[Arithmetic/Rational/Tcl|a subpage]] and should be edited there, not here.<nowiki>]</nowiki>''</div>
{{:Arithmetic/Rational/Tcl}}
=={{header|TI-89 BASIC}}==
{{incomplete|TI-89 BASIC}}
While TI-89 BASIC has built-in rational and symbolic arithmetic, it does not have user-defined data types.
=={{header|Wren}}==
{{libheader|Wren-math}}
{{libheader|Wren-rat}}
The latter module already contains support for rational number arithmetic.
<syntaxhighlight lang="wren">import "./math" for Int
import "./rat" for Rat
System.print("The following numbers (less than 2^19) are perfect:")
for (i in 2...(1<<19)) {
var sum = Rat.new(1, i)
for (j in Int.properDivisors(i)[1..-1]) sum = sum + Rat.new(1, j)
if (sum == Rat.one) System.print(" %(i)")
}</syntaxhighlight>
{{out}}
<pre>
The following numbers (less than 2^19) are perfect:
6
28
496
8128
</pre>
=={{header|zkl}}==
Enough of a Rational class for this task (ie implement the testing code "nicely").
<syntaxhighlight 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;
}
}</syntaxhighlight>
<syntaxhighlight 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 "."));
}</syntaxhighlight>
{{out}}
<pre>
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.
</pre>
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