Arithmetic/Complex

From Rosetta Code
Jump to: navigation, search
Task
Arithmetic/Complex
You are encouraged to solve this task according to the task description, using any language you may know.

A complex number is a number which can be written as "a + b \times i" (sometimes shown as "b + a \times i") where a and b are real numbers and i is the square root of -1. Typically, complex numbers are represented as a pair of real numbers called the "imaginary part" and "real part", where the imaginary part is the number to be multiplied by i.

  • Show addition, multiplication, negation, and inversion of complex numbers in separate functions. (Subtraction and division operations can be made with pairs of these operations.) Print the results for each operation tested.
  • Optional: Show complex conjugation. By definition, the complex conjugate of a + bi is abi.

Some languages have complex number libraries available. If your language does, show the operations. If your language does not, also show the definition of this type.

Contents

[edit] Ada

with Ada.Numerics.Generic_Complex_Types;
with Ada.Text_IO.Complex_IO;
 
procedure Complex_Operations is
-- Ada provides a pre-defined generic package for complex types
-- That package contains definitions for composition,
-- negation, addition, subtraction, multiplication, division,
-- conjugation, exponentiation, and absolute value, as well as
-- basic comparison operations.
-- Ada provides a second pre-defined package for sin, cos, tan, cot,
-- arcsin, arccos, arctan, arccot, and the hyperbolic versions of
-- those trigonometric functions.
 
-- The package Ada.Numerics.Generic_Complex_Types requires definition
-- with the real type to be used in the complex type definition.
 
package Complex_Types is new Ada.Numerics.Generic_Complex_Types (Long_Float);
use Complex_Types;
package Complex_IO is new Ada.Text_IO.Complex_IO (Complex_Types);
use Complex_IO;
use Ada.Text_IO;
 
A : Complex := Compose_From_Cartesian (Re => 1.0, Im => 1.0);
B : Complex := Compose_From_Polar (Modulus => 1.0, Argument => 3.14159);
C : Complex;
 
begin
-- Addition
C := A + B;
Put("A + B = "); Put(C);
New_Line;
-- Multiplication
C := A * B;
Put("A * B = "); Put(C);
New_Line;
-- Inversion
C := 1.0 / A;
Put("1.0 / A = "); Put(C);
New_Line;
-- Negation
C := -A;
Put("-A = "); Put(C);
New_Line;
-- Conjugation
C := Conjugate (C);
end Complex_Operations;

[edit] ALGOL 68

Works with: ALGOL 68 version Revision 1 - no extensions to language used
Works with: ALGOL 68G version Any - tested with release 1.18.0-9h.tiny
main:(
FORMAT compl fmt = $g(-7,5)"⊥"g(-7,5)$;
 
PROC compl operations = VOID: (
LONG COMPL a = 1.0 1.0;
LONG COMPL b = 3.14159 1.2;
 
LONG COMPL c;
 
printf(($x"a="f(compl fmt)l$,a));
printf(($x"b="f(compl fmt)l$,b));
 
# addition #
c := a + b;
printf(($x"a+b="f(compl fmt)l$,c));
# multiplication #
c := a * b;
printf(($x"a*b="f(compl fmt)l$,c));
# inversion #
c := 1.0 / a;
printf(($x"1/c="f(compl fmt)l$,c));
# negation #
c := -a;
printf(($x"-a="f(compl fmt)l$,c))
);
compl operations
)
Output:
a=1.00000⊥1.00000
b=3.14159⊥1.20000
a+b=4.14159⊥2.20000
a*b=1.94159⊥4.34159
1/c=0.50000⊥-.50000
-a=-1.0000⊥-1.0000

[edit] APL

 
x←1j1 ⍝assignment
y←5.25j1.5
x+y ⍝addition
6.25J2.5
x×y ⍝multiplication
3.75J6.75
⌹x ⍝inversion
0.5j_0.5
-x ⍝negation
¯1J¯1
 

[edit] App Inventor

App Inventor has native support for complex numbers.
The linked image gives a few examples of complex arithmetic and a custom complex conjugate function.
View the blocks and app screen...

[edit] AutoHotkey

contributed by Laszlo on the ahk forum

Cset(C,1,1)
MsgBox % Cstr(C) ; 1 + i*1
Cneg(C,C)
MsgBox % Cstr(C) ; -1 - i*1
Cadd(C,C,C)
MsgBox % Cstr(C) ; -2 - i*2
Cinv(D,C)
MsgBox % Cstr(D) ; -0.25 + 0.25*i
Cmul(C,C,D)
MsgBox % Cstr(C) ; 1 + i*0
 
Cset(ByRef C, re, im) {
VarSetCapacity(C,16)
NumPut(re,C,0,"double")
NumPut(im,C,8,"double")
}
Cre(ByRef C) {
Return NumGet(C,0,"double")
}
Cim(ByRef C) {
Return NumGet(C,8,"double")
}
Cstr(ByRef C) {
Return Cre(C) ((i:=Cim(C))<0 ? " - i*" . -i : " + i*" . i)
}
Cadd(ByRef C, ByRef A, ByRef B) {
VarSetCapacity(C,16)
NumPut(Cre(A)+Cre(B),C,0,"double")
NumPut(Cim(A)+Cim(B),C,8,"double")
}
Cmul(ByRef C, ByRef A, ByRef B) {
VarSetCapacity(C,16)
t := Cre(A)*Cim(B)+Cim(A)*Cre(B)
NumPut(Cre(A)*Cre(B)-Cim(A)*Cim(B),C,0,"double")
NumPut(t,C,8,"double") ; A or B can be C!
}
Cneg(ByRef C, ByRef A) {
VarSetCapacity(C,16)
NumPut(-Cre(A),C,0,"double")
NumPut(-Cim(A),C,8,"double")
}
Cinv(ByRef C, ByRef A) {
VarSetCapacity(C,16)
d := Cre(A)**2 + Cim(A)**2
NumPut( Cre(A)/d,C,0,"double")
NumPut(-Cim(A)/d,C,8,"double")
}

[edit] BASIC

Works with: QuickBasic version 4.5
TYPE complex
real AS DOUBLE
imag AS DOUBLE
END TYPE
DECLARE SUB add (a AS complex, b AS complex, c AS complex)
DECLARE SUB mult (a AS complex, b AS complex, c AS complex)
DECLARE SUB inv (a AS complex, b AS complex)
DECLARE SUB neg (a AS complex, b AS complex)
CLS
DIM x AS complex
DIM y AS complex
DIM z AS complex
x.real = 1
x.imag = 1
y.real = 2
y.imag = 2
CALL add(x, y, z)
PRINT z.real; "+"; z.imag; "i"
CALL mult(x, y, z)
PRINT z.real; "+"; z.imag; "i"
CALL inv(x, z)
PRINT z.real; "+"; z.imag; "i"
CALL neg(x, z)
PRINT z.real; "+"; z.imag; "i"
 
 
SUB add (a AS complex, b AS complex, c AS complex)
c.real = a.real + b.real
c.imag = a.imag + b.imag
END SUB
 
SUB inv (a AS complex, b AS complex)
denom = a.real ^ 2 + a.imag ^ 2
b.real = a.real / denom
b.imag = -a.imag / denom
END SUB
 
SUB mult (a AS complex, b AS complex, c AS complex)
c.real = a.real * b.real - a.imag * b.imag
c.imag = a.real * b.imag + a.imag * b.real
END SUB
 
SUB neg (a AS complex, b AS complex)
b.real = -a.real
b.imag = -a.imag
END SUB

Output:

 3 + 3 i
 0 + 4 i
 .5 +-.5 i
-1 +-1 i

[edit] BBC BASIC

      DIM Complex{r, i}
 
DIM a{} = Complex{} : a.r = 1.0 : a.i = 1.0
DIM b{} = Complex{} : b.r = PI# : b.i = 1.2
DIM o{} = Complex{}
 
PROCcomplexadd(o{}, a{}, b{})
PRINT "Result of addition is " FNcomplexshow(o{})
PROCcomplexmul(o{}, a{}, b{})
PRINT "Result of multiplication is " ; FNcomplexshow(o{})
PROCcomplexneg(o{}, a{})
PRINT "Result of negation is " ; FNcomplexshow(o{})
PROCcomplexinv(o{}, a{})
PRINT "Result of inversion is " ; FNcomplexshow(o{})
END
 
DEF PROCcomplexadd(dst{}, one{}, two{})
dst.r = one.r + two.r
dst.i = one.i + two.i
ENDPROC
 
DEF PROCcomplexmul(dst{}, one{}, two{})
dst.r = one.r*two.r - one.i*two.i
dst.i = one.i*two.r + one.r*two.i
ENDPROC
 
DEF PROCcomplexneg(dst{}, src{})
dst.r = -src.r
dst.i = -src.i
ENDPROC
 
DEF PROCcomplexinv(dst{}, src{})
LOCAL denom : denom = src.r^2 + src.i^ 2
dst.r = src.r / denom
dst.i = -src.i / denom
ENDPROC
 
DEF FNcomplexshow(src{})
IF src.i >= 0 THEN = STR$(src.r) + " + " +STR$(src.i) + "i"
= STR$(src.r) + " - " + STR$(-src.i) + "i"

Output:

Result of addition is 4.14159265 + 2.2i
Result of multiplication is 1.94159265 + 4.34159265i
Result of negation is -1 - 1i
Result of inversion is 0.5 - 0.5i

[edit] Bracmat

Bracmat recognizes the symbol i as the square root of -1. The results of the functions below are not necessarily of the form a+b*i, but as the last example shows, Bracmat nevertheless can work out that two different representations of the same mathematical object, when subtracted from each other, give zero. You may wonder why in the functions multiply and negate there are terms 1 and -1. These terms are a trick to force Bracmat to expand the products. As it is more costly to factorize a sum than to expand a product into a sum, Bracmat retains isolated products. However, when in combination with a non-zero term, the product is expanded.

  (add=a b.!arg:(?a,?b)&!a+!b)
& ( multiply
= a b.!arg:(?a,?b)&1+!a*!b+-1
)
& (negate=.1+-1*!arg+-1)
& ( conjugate
= a b
.  !arg:i&-i
| !arg:-i&i
| !arg:?a_?b&(conjugate$!a)_(conjugate$!b)
| !arg
)
& ( invert
= conjugated
. conjugate$!arg:?conjugated
& multiply$(!arg,!conjugated)^-1*!conjugated
)
& out$("(a+i*b)+(a+i*b) =" add$(a+i*b,a+i*b))
& out$("(a+i*b)+(a+-i*b) =" add$(a+i*b,a+-i*b))
& out$("(a+i*b)*(a+i*b) =" multiply$(a+i*b,a+i*b))
& out$("(a+i*b)*(a+-i*b) =" multiply$(a+i*b,a+-i*b))
& out$("-1*(a+i*b) =" negate$(a+i*b))
& out$("-1*(a+-i*b) =" negate$(a+-i*b))
& out$("sin$x = " sin$x)
& out$("conjugate sin$x =" conjugate$(sin$x))
& out
$ ("sin$x minus conjugate sin$x =" sin$x+negate$(conjugate$(sin$x)))
& done;

Output:

(a+i*b)+(a+i*b) = 2*a+2*i*b
(a+i*b)+(a+-i*b) = 2*a
(a+i*b)*(a+i*b) = a^2+-1*b^2+2*i*a*b
(a+i*b)*(a+-i*b) = a^2+b^2
-1*(a+i*b) = -1*a+-i*b
-1*(a+-i*b) = -1*a+i*b
sin$x =  i*(-1/2*e^(i*x)+1/2*e^(-i*x))
conjugate sin$x  = -i*(1/2*e^(i*x)+-1/2*e^(-i*x))
sin$x minus conjugate sin$x = 0

[edit] C

Works with: C99

The more recent C99 standard has built-in complex number primitive types, which can be declared with float, double, or long double precision. To use these types and their associated library functions, you must include the <complex.h> header. (Note: this is a different header than the <complex> templates that are defined by C++.) [1] [2]

#include <complex.h>
#include <stdio.h>
 
void cprint(double complex c)
{
printf("%f%+fI", creal(c), cimag(c));
}
void complex_operations() {
double complex a = 1.0 + 1.0I;
double complex b = 3.14159 + 1.2I;
 
double complex c;
 
printf("\na="); cprint(a);
printf("\nb="); cprint(b);
 
// addition
c = a + b;
printf("\na+b="); cprint(c);
// multiplication
c = a * b;
printf("\na*b="); cprint(c);
// inversion
c = 1.0 / a;
printf("\n1/c="); cprint(c);
// negation
c = -a;
printf("\n-a="); cprint(c);
// conjugate
c = conj(a);
printf("\nconj a="); cprint(c); printf("\n");
}
Works with: C89

User-defined type:

typedef struct{
double real;
double imag;
} Complex;
 
Complex add(Complex a, Complex b){
Complex ans;
ans.real = a.real + b.real;
ans.imag = a.imag + b.imag;
return ans;
}
 
Complex mult(Complex a, Complex b){
Complex ans;
ans.real = a.real * b.real - a.imag * b.imag;
ans.imag = a.real * b.imag + a.imag * b.real;
return ans;
}
 
/* it's arguable that things could be better handled if either
a.real or a.imag is +/-inf, but that's much work */

Complex inv(Complex a){
Complex ans;
double denom = a.real * a.real + a.imag * a.imag;
ans.real = a.real / denom;
ans.imag = -a.imag / denom;
return ans;
}
 
Complex neg(Complex a){
Complex ans;
ans.real = -a.real;
ans.imag = -a.imag;
return ans;
}
 
Complex conj(Complex a){
Complex ans;
ans.real = a.real;
ans.imag = -a.imag;
return ans;
}
 
void put(Complex c)
{
printf("%lf%+lfI", c.real, c.imag);
}
 
void complex_ops(void)
{
Complex a = { 1.0, 1.0 };
Complex b = { 3.14159, 1.2 };
 
printf("\na="); put(a);
printf("\nb="); put(b);
printf("\na+b="); put(add(a,b));
printf("\na*b="); put(mult(a,b));
printf("\n1/a="); put(inv(a));
printf("\n-a="); put(neg(a));
printf("\nconj a="); put(conj(a)); printf("\n");
}

[edit] C#

Works with: C sharp version 4.0
namespace RosettaCode.Arithmetic.Complex
{
using System;
using System.Numerics;
 
internal static class Program
{
private static void Main()
{
var number = Complex.ImaginaryOne;
foreach (var result in new[] { number + number, number * number, -number, 1 / number, Complex.Conjugate(number) })
{
Console.WriteLine(result);
}
}
}
}
Works with: C sharp version 1.2
using System;
 
public struct ComplexNumber
{
public static readonly ComplexNumber i = new ComplexNumber(0.0, 1.0);
public static readonly ComplexNumber Zero = new ComplexNumber(0.0, 0.0);
 
public double Re;
public double Im;
 
public ComplexNumber(double re)
{
this.Re = re;
this.Im = 0;
}
 
public ComplexNumber(double re, double im)
{
this.Re = re;
this.Im = im;
}
 
public static ComplexNumber operator *(ComplexNumber n1, ComplexNumber n2)
{
return new ComplexNumber(n1.Re * n2.Re - n1.Im * n2.Im,
n1.Im * n2.Re + n1.Re * n2.Im);
}
 
public static ComplexNumber operator *(double n1, ComplexNumber n2)
{
return new ComplexNumber(n1 * n2.Re, n1 * n2.Im);
}
 
public static ComplexNumber operator /(ComplexNumber n1, ComplexNumber n2)
{
double n2Norm = n2.Re * n2.Re + n2.Im * n2.Im;
return new ComplexNumber((n1.Re * n2.Re + n1.Im * n2.Im) / n2Norm,
(n1.Im * n2.Re - n1.Re * n2.Im) / n2Norm);
}
 
public static ComplexNumber operator /(ComplexNumber n1, double n2)
{
return new ComplexNumber(n1.Re / n2, n1.Im / n2);
}
 
public static ComplexNumber operator +(ComplexNumber n1, ComplexNumber n2)
{
return new ComplexNumber(n1.Re + n2.Re, n1.Im + n2.Im);
}
 
public static ComplexNumber operator -(ComplexNumber n1, ComplexNumber n2)
{
return new ComplexNumber(n1.Re - n2.Re, n1.Im - n2.Im);
}
 
public static ComplexNumber operator -(ComplexNumber n)
{
return new ComplexNumber(-n.Re, -n.Im);
}
 
public static implicit operator ComplexNumber(double n)
{
return new ComplexNumber(n, 0.0);
}
 
public static explicit operator double(ComplexNumber n)
{
return n.Re;
}
 
public static bool operator ==(ComplexNumber n1, ComplexNumber n2)
{
return n1.Re == n2.Re && n1.Im == n2.Im;
}
 
public static bool operator !=(ComplexNumber n1, ComplexNumber n2)
{
return n1.Re != n2.Re || n1.Im != n2.Im;
}
 
public override bool Equals(object obj)
{
return this == (ComplexNumber)obj;
}
 
public override int GetHashCode()
{
return Re.GetHashCode() ^ Im.GetHashCode();
}
 
public override string ToString()
{
return String.Format("{0}+{1}*i", Re, Im);
}
}
 
public static class ComplexMath
{
public static double Abs(ComplexNumber a)
{
return Math.Sqrt(Norm(a));
}
 
public static double Norm(ComplexNumber a)
{
return a.Re * a.Re + a.Im * a.Im;
}
 
public static double Arg(ComplexNumber a)
{
return Math.Atan2(a.Im, a.Re);
}
 
public static ComplexNumber Inverse(ComplexNumber a)
{
double norm = Norm(a);
return new ComplexNumber(a.Re / norm, -a.Im / norm);
}
 
public static ComplexNumber Conjugate(ComplexNumber a)
{
return new ComplexNumber(a.Re, -a.Im);
 
}
 
public static ComplexNumber Exp(ComplexNumber a)
{
double e = Math.Exp(a.Re);
return new ComplexNumber(e * Math.Cos(a.Im), e * Math.Sin(a.Im));
}
 
public static ComplexNumber Log(ComplexNumber a)
{
 
return new ComplexNumber(0.5 * Math.Log(Norm(a)), Arg(a));
}
 
public static ComplexNumber Power(ComplexNumber a, ComplexNumber power)
{
return Exp(power * Log(a));
}
 
public static ComplexNumber Power(ComplexNumber a, int power)
{
bool inverse = false;
if (power < 0)
{
inverse = true; power = -power;
}
 
ComplexNumber result = 1.0;
ComplexNumber multiplier = a;
while (power > 0)
{
if ((power & 1) != 0) result *= multiplier;
multiplier *= multiplier;
power >>= 1;
}
 
if (inverse)
return Inverse(result);
else
return result;
}
 
public static ComplexNumber Sqrt(ComplexNumber a)
{
return Exp(0.5 * Log(a));
}
 
public static ComplexNumber Sin(ComplexNumber a)
{
return Sinh(ComplexNumber.i * a) / ComplexNumber.i;
}
 
public static ComplexNumber Cos(ComplexNumber a)
{
return Cosh(ComplexNumber.i * a);
}
 
public static ComplexNumber Sinh(ComplexNumber a)
{
return 0.5 * (Exp(a) - Exp(-a));
}
 
public static ComplexNumber Cosh(ComplexNumber a)
{
return 0.5 * (Exp(a) + Exp(-a));
}
 
}
 
class Program
{
static void Main(string[] args)
{
// usage
ComplexNumber i = 2;
ComplexNumber j = new ComplexNumber(1, -2);
Console.WriteLine(i * j);
Console.WriteLine(ComplexMath.Power(j, 2));
Console.WriteLine((double)ComplexMath.Sin(i) + " vs " + Math.Sin(2));
Console.WriteLine(ComplexMath.Power(j, 0) == 1.0);
}
}

[edit] C++

#include <iostream>
#include <complex>
using std::complex;
 
void complex_operations() {
complex<double> a(1.0, 1.0);
complex<double> b(3.14159, 1.25);
 
// addition
std::cout << a + b << std::endl;
// multiplication
std::cout << a * b << std::endl;
// inversion
std::cout << 1.0 / a << std::endl;
// negation
std::cout << -a << std::endl;
// conjugate
std::cout << std::conj(a) << std::endl;
}

[edit] Clojure

Clojure on the JVM has no native support for Complex numbers. Therefore, we use defrecord and the multimethods in clojure.algo.generic.arithmetic to make a Complex number type.

(ns rosettacode.arithmetic.cmplx
(:require [clojure.algo.generic.arithmetic :as ga])
(:import [java.lang Number]))
 
(defrecord Complex [^Number r ^Number i]
Object
(toString [{:keys [r i]}]
(apply str
(cond
(zero? r) [(if (= i 1) "" i) "i"]
(zero? i) [r]
 :else [r (if (neg? i) "-" "+") i "i"]))))
 
(defmethod ga/+ [Complex Complex]
[x y] (map->Complex (merge-with + x y)))
 
(defmethod ga/+ [Complex Number] ; reals become y + 0i
[{:keys [r i]} y] (->Complex (+ r y) i))
 
(defmethod ga/- Complex
[x] (->> x vals (map -) (apply ->Complex)))
 
(defmethod ga/* [Complex Complex]
[x y] (map->Complex (merge-with * x y)))
 
(defmethod ga/* [Complex Number]
[{:keys [r i]} y] (->Complex (* r y) (* i y)))
 
(ga/defmethod* ga / Complex
[x] (->> x vals (map /) (apply ->Complex)))
 
(defn conj [^Complex {:keys [r i]}]
(->Complex r (- i)))
 
(defn inv [^Complex {:keys [r i]}]
(let [m (+ (* r r) (* i i))]
(->Complex (/ r m) (- (/ i m)))))
 

[edit] COBOL

The following is in the Managed COBOL dialect.

Works with: Visual COBOL

[edit] .NET Complex class

Translation of: C#
      $SET SOURCEFORMAT "FREE"
$SET ILUSING "System"
$SET ILUSING "System.Numerics"
class-id Prog.
method-id. Main static.
procedure division.
declare num as type Complex = type Complex::ImaginaryOne()
declare results as type Complex occurs any
set content of results to ((num + num), (num * num), (- num), (1 / num), type Complex::Conjugate(num))
perform varying result as type Complex thru results
display result
end-perform
end method.
end class.

[edit] Implementation

      $SET SOURCEFORMAT "FREE"
class-id Prog.
method-id. Main static.
procedure division.
declare a as type Complex = new Complex(1, 1)
declare b as type Complex = new Complex(3.14159, 1.25)
 
display "a = " a
display "b = " b
display space
 
declare result as type Complex = a + b
display "a + b = " result
move (a - b) to result
display "a - b = " result
move (a * b) to result
display "a * b = " result
move (a / b) to result
display "a / b = " result
move (- b) to result
display "-b = " result
display space
 
display "Inverse of b: " type Complex::Inverse(b)
display "Conjugate of b: " type Complex::Conjugate(b)
end method.
end class.
 
class-id Complex.
 
01 Real float-long property.
01 Imag float-long property.
 
method-id new.
set Real, Imag to 0
end method.
 
method-id new.
procedure division using value real-val as float-long, imag-val as float-long.
set Real to real-val
set Imag to imag-val
end method.
 
method-id Norm static.
procedure division using value a as type Complex returning ret as float-long.
compute ret = a::Real ** 2 + a::Imag ** 2
end method.
 
method-id Inverse static.
procedure division using value a as type Complex returning ret as type Complex.
declare norm as float-long = type Complex::Norm(a)
set ret to new Complex(a::Real / norm, (0 - a::Imag) / norm)
end method.
 
method-id Conjugate static.
procedure division using value a as type Complex returning c as type Complex.
set c to new Complex(a::Real, 0 - a::Imag)
end method.
 
method-id ToString override.
procedure division returning str as string.
set str to type String::Format("{0}{1:+#0;-#}i", Real, Imag)
end method.
 
operator-id + .
procedure division using value a as type Complex, b as type Complex
returning c as type Complex.
set c to new Complex(a::Real + b::Real, a::Imag + b::Imag)
end operator.
 
operator-id - .
procedure division using value a as type Complex, b as type Complex
returning c as type Complex.
set c to new Complex(a::Real - b::Real, a::Imag - b::Imag)
end operator.
 
operator-id * .
procedure division using value a as type Complex, b as type Complex
returning c as type Complex.
set c to new Complex(a::Real * b::Real - a::Imag * b::Imag,
a::Real * b::Imag + a::Imag * b::Real)
end operator.
 
operator-id / .
procedure division using value a as type Complex, b as type Complex
returning c as type Complex.
set c to new Complex()
declare b-norm as float-long = type Complex::Norm(b)
compute c::Real = (a::Real * b::Real + a::Imag * b::Imag) / b-norm
compute c::Imag = (a::Imag * b::Real - a::Real * b::Imag) / b-norm
end operator.
 
operator-id - .
procedure division using value a as type Complex returning ret as type Complex.
set ret to new Complex(- a::Real, 0 - a::Imag)
end operator.
 
end class.

[edit] CoffeeScript

 
# create an immutable Complex type
class Complex
constructor: (@r=0, @i=0) ->
@magnitude = @r*@r + @i*@i
 
plus: (c2) ->
new Complex(
@r + c2.r,
@i + c2.i
)
 
times: (c2) ->
new Complex(
@r*c2.r - @i*c2.i,
@r*c2.i + @i*c2.r
)
 
negation: ->
new Complex(
-1 * @r,
-1 * @i
)
 
inverse: ->
throw Error "no inverse" if @magnitude is 0
new Complex(
@r / @magnitude,
-1 * @i / @magnitude
)
 
toString: ->
return "#{@r}" if @i == 0
return "#{@i}i" if @r == 0
if @i > 0
"#{@r} + #{@i}i"
else
"#{@r} - #{-1 * @i}i"
 
# test
do ->
a = new Complex(5, 3)
b = new Complex(4, -3)
 
sum = a.plus b
console.log "(#{a}) + (#{b}) = #{sum}"
 
product = a.times b
console.log "(#{a}) * (#{b}) = #{product}"
 
negation = b.negation()
console.log "-1 * (#{b}) = #{negation}"
 
diff = a.plus negation
console.log "(#{a}) - (#{b}) = #{diff}"
 
inverse = b.inverse()
console.log "1 / (#{b}) = #{inverse}"
 
quotient = product.times inverse
console.log "(#{product}) / (#{b}) = #{quotient}"
 

output

 
> coffee complex.coffee
(5 + 3i) + (4 - 3i) = 9
(5 + 3i) * (4 - 3i) = 29 - 3i
-1 * (4 - 3i) = -4 + 3i
(5 + 3i) - (4 - 3i) = 1 + 6i
1 / (4 - 3i) = 0.16 + 0.12i
(29 - 3i) / (4 - 3i) = 5 + 3i
 

[edit] Common Lisp

Complex numbers are a built-in numeric type in Common Lisp. The literal syntax for a complex number is #C(real imaginary). The components of a complex number may be integers, ratios, or floating-point. Arithmetic operations automatically return complex (or real) numbers when appropriate:

> (sqrt -1)
#C(0.0 1.0)
 
> (expt #c(0 1) 2)
-1

Here are some arithmetic operations on complex numbers:

> (+ #c(0 1) #c(1 0))
#C(1 1)
 
> (* #c(1 1) 2)
#C(2 2)
 
> (* #c(1 1) #c(0 2))
#C(-2 2)
 
> (- #c(1 1))
#C(-1 -1)
 
> (/ #c(0 2))
#C(0 -1/2)
 
> (conjugate #c(1 1))
#C(1 -1)

Complex numbers can be constructed from real and imaginary parts using the complex function, and taken apart using the realpart and imagpart functions.

> (complex 64 (/ 3 4))
#C(64 3/4)
 
> (realpart #c(5 5))
5
 
> (imagpart (complex 0 pi))
3.141592653589793d0

[edit] Component Pascal

BlackBox Component Builder

 
MODULE Complex;
IMPORT StdLog;
TYPE
Complex* = POINTER TO ComplexDesc;
ComplexDesc = RECORD
r-,i-: REAL;
END;
 
VAR
r,x,y: Complex;
 
PROCEDURE New(x,y: REAL): Complex;
VAR
r: Complex;
BEGIN
NEW(r);r.r := x;r.i := y;
RETURN r
END New;
 
PROCEDURE (x: Complex) Add*(y: Complex): Complex,NEW;
BEGIN
RETURN New(x.r + y.r,x.i + y.i)
END Add;
 
PROCEDURE ( x: Complex) Sub*( y: Complex): Complex, NEW;
BEGIN
RETURN New(x.r - y.r,x.i - y.i)
END Sub;
 
PROCEDURE ( x: Complex) Mul*( y: Complex): Complex, NEW;
BEGIN
RETURN New(x.r*y.r - x.i*y.i,x.r*y.i + x.i*y.r)
END Mul;
 
PROCEDURE ( x: Complex) Div*( y: Complex): Complex, NEW;
VAR
d: REAL;
BEGIN
d := y.r * y.r + y.i * y.i;
RETURN New((x.r*y.r + x.i*y.i)/d,(x.i*y.r - x.r*y.i)/d)
END Div;
 
(* Reciprocal *)
PROCEDURE (x: Complex) Rec*(): Complex,NEW;
VAR
d: REAL;
BEGIN
d := x.r * x.r + x.i * x.i;
RETURN New(x.r/d,(-1.0 * x.i)/d);
END Rec;
 
(* Conjugate *)
PROCEDURE (x: Complex) Con*(): Complex,NEW;
BEGIN
RETURN New(x.r, (-1.0) * x.i);
END Con;
 
PROCEDURE (x: Complex) Out(),NEW;
BEGIN
StdLog.String("Complex(");
StdLog.Real(x.r);StdLog.String(',');StdLog.Real(x.i);
StdLog.String("i );")
END Out;
 
PROCEDURE Do*;
BEGIN
x := New(1.5,3);
y := New(1.0,1.0);
 
StdLog.String("x: ");x.Out();StdLog.Ln;
StdLog.String("y: ");y.Out();StdLog.Ln;
r := x.Add(y);
StdLog.String("x + y: ");r.Out();StdLog.Ln;
r := x.Sub(y);
StdLog.String("x - y: ");r.Out();StdLog.Ln;
r := x.Mul(y);
StdLog.String("x * y: ");r.Out();StdLog.Ln;
r := x.Div(y);
StdLog.String("x / y: ");r.Out();StdLog.Ln;
r := y.Rec();
StdLog.String("1 / y: ");r.Out();StdLog.Ln;
r := x.Con();
StdLog.String("x': ");r.Out();StdLog.Ln;
END Do;
 
END Complex.
 

Execute: ^Q Complex.Do
Output:

x: Complex( 1.5, 3.0i );
y: Complex( 1.0, 1.0i );
x + y: Complex( 2.5, 4.0i );
x - y: Complex( 0.5, 2.0i );
x * y: Complex( -1.5, 4.5i );
x / y: Complex( 2.25, 0.75i );
1 / y: Complex( 0.5, -0.5i );
x': Complex( 1.5, -3.0i );

[edit] D

Built-in complex numbers are now deprecated in D, to simplify the language.

import std.stdio, std.complex;
 
void main() {
auto x = complex(1, 1); // complex of doubles on default
auto y = complex(3.14159, 1.2);
 
writeln(x + y); // addition
writeln(x * y); // multiplication
writeln(1.0 / x); // inversion
writeln(-x); // negation
}

Output:

4.14159+2.2i
1.94159+4.34159i
0.5-0.5i
-1-1i

[edit] Dart

 
 
class complex {
 
num real=0;
num imag=0;
 
complex(num r,num i){
this.real=r;
this.imag=i;
}
 
 
complex add(complex b){
return new complex(this.real + b.real, this.imag + b.imag);
}
 
complex mult(complex b){
//FOIL of (a+bi)(c+di) with i*i = -1
return new complex(this.real * b.real - this.imag * b.imag, this.real * b.imag + this.imag * b.real);
}
 
complex inv(){
//1/(a+bi) * (a-bi)/(a-bi) = 1/(a+bi) but it's more workable
num denom = real * real + imag * imag;
double r =real/denom;
double i= -imag/denom;
return new complex( r,-i);
}
 
complex neg(){
return new complex(-real, -imag);
}
 
complex conj(){
return new complex(real, -imag);
}
 
 
String toString(){
return this.real.toString()+' + '+ this.imag.toString()+'*i';
}
}
void main() {
var cl= new complex(1,2);
var cl2= new complex(3,-1);
print(cl.toString());
print(cl2.toString());
print(cl.inv().toString());
print(cl2.mult(cl).toString());
 
}
 

[edit] Erlang

%% Task: Complex Arithmetic
%% Author: Abhay Jain
 
-module(complex_number).
-export([calculate/0]).
 
-record(complex, {real, img}).
 
calculate() ->
A = #complex{real=1, img=3},
B = #complex{real=5, img=2},
 
Sum = add (A, B),
print (Sum),
 
Product = multiply (A, B),
print (Product),
 
Negation = negation (A),
print (Negation),
 
Inversion = inverse (A),
print (Inversion),
 
Conjugate = conjugate (A),
print (Conjugate).
 
add (A, B) ->
RealPart = A#complex.real + B#complex.real,
ImgPart = A#complex.img + B#complex.img,
#complex{real=RealPart, img=ImgPart}.
 
multiply (A, B) ->
RealPart = (A#complex.real * B#complex.real) - (A#complex.img * B#complex.img),
ImgPart = (A#complex.real * B#complex.img) + (B#complex.real * A#complex.img),
#complex{real=RealPart, img=ImgPart}.
 
negation (A) ->
#complex{real=-A#complex.real, img=-A#complex.img}.
 
inverse (A) ->
C = conjugate (A),
Mod = (A#complex.real * A#complex.real) + (A#complex.img * A#complex.img),
RealPart = C#complex.real / Mod,
ImgPart = C#complex.img / Mod,
#complex{real=RealPart, img=ImgPart}.
 
conjugate (A) ->
RealPart = A#complex.real,
ImgPart = -A#complex.img,
#complex{real=RealPart, img=ImgPart}.
 
print (A) ->
if A#complex.img < 0 ->
io:format("Ans = ~p~pi~n", [A#complex.real, A#complex.img]);
true ->
io:format("Ans = ~p+~pi~n", [A#complex.real, A#complex.img])
end.

Output:

Ans = 6+5i
Ans = -1+17i
Ans = -1-3i
Ans = 0.1-0.3i
Ans = 1-3i

[edit] Euler Math Toolbox

 
>a=1+4i; b=5-3i;
>a+b
6+1i
>a-b
-4+7i
>a*b
17+17i
>a/b
-0.205882352941+0.676470588235i
>fraction a/b
-7/34+23/34i
>conj(a)
1-4i
 


[edit] Euphoria

constant REAL = 1, IMAG = 2
type complex(sequence s)
return length(s) = 2 and atom(s[REAL]) and atom(s[IMAG])
end type
 
function add(complex a, complex b)
return a + b
end function
 
function mult(complex a, complex b)
return {a[REAL] * b[REAL] - a[IMAG] * b[IMAG],
a[REAL] * b[IMAG] + a[IMAG] * b[REAL]}
end function
 
function inv(complex a)
atom denom
denom = a[REAL] * a[REAL] + a[IMAG] * a[IMAG]
return {a[REAL] / denom, -a[IMAG] / denom}
end function
 
function neg(complex a)
return -a
end function
 
function scomplex(complex a)
sequence s
if a[REAL] != 0 then
s = sprintf("%g",a)
else
s = {}
end if
 
if a[IMAG] != 0 then
if a[IMAG] = 1 then
s &= "+i"
elsif a[IMAG] = -1 then
s &= "-i"
else
s &= sprintf("%+gi",a[IMAG])
end if
end if
 
if length(s) = 0 then
return "0"
else
return s
end if
end function
 
complex a, b
a = { 1.0, 1.0 }
b = { 3.14159, 1.2 }
printf(1,"a = %s\n",{scomplex(a)})
printf(1,"b = %s\n",{scomplex(b)})
printf(1,"a+b = %s\n",{scomplex(add(a,b))})
printf(1,"a*b = %s\n",{scomplex(mult(a,b))})
printf(1,"1/a = %s\n",{scomplex(inv(a))})
printf(1,"-a = %s\n",{scomplex(neg(a))})

Output:

a = 1+i
b = 3.14159+1.2i
a+b = 4.14159+2.2i
a*b = 1.94159+4.34159i
1/a = 0.5-0.5i
-a = -1-i

[edit] Excel

Take 7 cells, say A1 to G1. Type in :

C1:

 
=IMSUM(A1;B1)
 

D1:

 
=IMPRODUCT(A1;B1)
 

E1:

 
=IMSUB(0;D1)
 

F1:

 
=IMDIV(1;E28)
 

G1:

 
=IMCONJUGATE(C28)
 

E1 will have the negation of D1's value

 
1+2i 3+5i 4+7i -7+11i 7-11i 0,0411764705882353+0,0647058823529412i 4-7i
 

[edit] F#

Entered into an interactive session to show the results:

 
> open Microsoft.FSharp.Math;;
 
> let a = complex 1.0 1.0;;
val a : complex = 1r+1i
 
> let b = complex 3.14159 1.25;;
val b : complex = 3.14159r+1.25i
 
> a + b;;
val it : Complex = 4.14159r+2.25i {Conjugate = 4.14159r-2.25i;
ImaginaryPart = 2.25;
Magnitude = 4.713307515;
Phase = 0.497661247;
RealPart = 4.14159;
i = 2.25;
r = 4.14159;}
 
> a * b;;
val it : Complex = 1.89159r+4.39159i {Conjugate = 1.89159r-4.39159i;
ImaginaryPart = 4.39159;
Magnitude = 4.781649868;
Phase = 1.164082262;
RealPart = 1.89159;
i = 4.39159;
r = 1.89159;}
 
> a / b;;
val it : Complex =
0.384145932435901r+0.165463215905043i
{Conjugate = 0.384145932435901r-0.165463215905043i;
ImaginaryPart = 0.1654632159;
Magnitude = 0.418265673;
Phase = 0.4067140652;
RealPart = 0.3841459324;
i = 0.1654632159;
r = 0.3841459324;}
 
> -a;;
val it : complex = -1r-1i {Conjugate = -1r+1i;
ImaginaryPart = -1.0;
Magnitude = 1.414213562;
Phase = -2.35619449;
RealPart = -1.0;
i = -1.0;
r = -1.0;}
 

[edit] Factor

USING: combinators kernel math math.functions prettyprint ;
 
C{ 1 2 } C{ 0.9 -2.78 } {
[ + . ]  ! addition
[ - . ]  ! subtraction
[ * . ]  ! multiplication
[ / . ]  ! division
[ ^ . ]  ! power
} 2cleave
 
C{ 1 2 } {
[ neg . ]  ! negation
[ 1 swap / . ]  ! multiplicative inverse
[ conjugate . ]  ! complex conjugate
[ sin . ]  ! sine
[ log . ]  ! natural logarithm
[ sqrt . ]  ! square root
} cleave

[edit] Forth

There is no standard syntax or mechanism for complex numbers. The FSL provides several implementations suitable for different uses. This example uses the existing floating point stack, but other libraries define a separate complex stack and/or a fixed-point implementation suitable for microcontrollers and DSPs.

include complex.seq
 
: ZNEGATE ( r i -- -r -i ) fswap fnegate fswap fnegate ;
 
zvariable x
zvariable y
1e 1e x z!
pi 1.2e y z!
 
x z@ y z@ z+ z.
x z@ y z@ z* z.
1+0i x z@ z/ z.
x z@ znegate z.

[edit] Fortran

In ANSI FORTRAN 66 or later, COMPLEX is a built-in data type with full access to intrinsic arithmetic operations. Putting each native operation in a function is horribly inefficient, so I will simply demonstrate the operations. This example shows usage for Fortran 90 or later:

program cdemo
complex :: a = (5,3), b = (0.5, 6.0) ! complex initializer
complex :: absum, abprod, aneg, ainv
 
absum = a + b
abprod = a * b
aneg = -a
ainv = 1.0 / a
end program cdemo

And, although you did not ask, here are demonstrations of some other common complex number operations

program cdemo2
complex :: a = (5,3), b = (0.5, 6) ! complex initializer
real, parameter :: pi = 3.141592653589793 ! The constant "pi"
complex, parameter :: i = (0, 1) ! the imaginary unit "i" (sqrt(-1))
complex :: abdiff, abquot, abpow, aconj, p2cart, newc
real :: areal, aimag, anorm, rho = 10, theta = pi / 3.0, x = 2.3, y = 3.0
integer, parameter :: n = 50
integer :: j
complex, dimension(0:n-1) :: unit_circle
 
abdiff = a - b
abquot = a / b
abpow = a ** b
areal = real(a) ! Real part
aimag = imag(a) ! Imaginary part
newc = cmplx(x,y) ! Creating a complex on the fly from two reals intrinsically
! (initializer only works in declarations)
newc = x + y*i ! Creating a complex on the fly from two reals arithmetically
anorm = abs(a) ! Complex norm (or "modulus" or "absolute value")
! (use CABS before Fortran 90)
aconj = conjg(a) ! Complex conjugate (same as real(a) - i*imag(a))
p2cart = rho * exp(i * theta) ! Euler's polar complex notation to cartesian complex notation
! conversion (use CEXP before Fortran 90)
 
! The following creates an array of N evenly spaced points around the complex unit circle
! useful for FFT calculations, among other things
unit_circle = exp(2*i*pi/n * (/ (j, j=0, n-1) /) )
end program cdemo2

[edit] Frink

Frink's operations handle complex numbers naturally. The real and imaginary parts of complex numbers can be arbitrary-sized integers, arbitrary-sized rational numbers, or arbitrary-precision floating-point numbers.

 
add[x,y] := x + y
multiply[x,y] := x * y
negate[x] := -x
invert[x] := 1/x // Could also use inv[x] or recip[x]
conjugate[x] := Re[x] - Im[x] i
 
a = 3 + 2.5i
b = 7.3 - 10i
println["$a + $b = " + add[a,b]]
println["$a * $b = " + multiply[a,b]]
println["-$a = " + negate[a]]
println["1/$a = " + invert[a]]
println["conjugate[$a] = " + conjugate[a]]
 

Output:

( 3 + 2.5 i ) + ( 7.3 - 10 i ) = ( 10.3 - 7.5 i )
( 3 + 2.5 i ) * ( 7.3 - 10 i ) = ( 46.9 - 11.75 i )
-( 3 + 2.5 i ) = ( -3 - 2.5 i )
1/( 3 + 2.5 i ) = ( 0.19672131147540983607 - 0.16393442622950819672 i )
conjugate[( 3 + 2.5 i )] = ( 3 - 2.5 i )

[edit] GAP

# GAP knows gaussian integers, gaussian rationals (i.e. Q[i]), and cyclotomic fields. Here are some examples.
# E(n) is an nth primitive root of 1
i := Sqrt(-1);
# E(4)
(3 + 2*i)*(5 - 7*i);
# 29-11*E(4)
1/i;
# -E(4)
Sqrt(-3);
# E(3)-E(3)^2
 
i in GaussianIntegers;
# true
i/2 in GaussianIntegers;
# false
i/2 in GaussianRationals;
# true
Sqrt(-3) in Cyclotomics;
# true

[edit] Go

Go has complex numbers built in, with the complex conjugate in the standard library.

package main
 
import (
"fmt"
"math/cmplx"
)
 
func main() {
a := 1 + 1i
b := 3.14159 + 1.25i
fmt.Println("a: ", a)
fmt.Println("b: ", b)
fmt.Println("a + b: ", a+b)
fmt.Println("a * b: ", a*b)
fmt.Println("-a: ", -a)
fmt.Println("1 / a: ", 1/a)
fmt.Println("a̅: ", cmplx.Conj(a))
}

Output:

a:       (1+1i)
b:       (3.14159+1.25i)
a + b:   (4.14159+2.25i)
a * b:   (1.8915899999999999+4.39159i)
-a:      (-1-1i)
1 / a:   (0.5-0.5i)
a̅:       (1-1i)

[edit] Groovy

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

class Complex {
final Number real, imag
 
static final Complex I = [0,1] as Complex
 
Complex(Number real) { this(real, 0) }
 
Complex(real, imag) { this.real = real; this.imag = imag }
 
Complex plus (Complex c) { [real + c.real, imag + c.imag] as Complex }
 
Complex plus (Number n) { [real + n, imag] as Complex }
 
Complex minus (Complex c) { [real - c.real, imag - c.imag] as Complex }
 
Complex minus (Number n) { [real - n, imag] as Complex }
 
Complex multiply (Complex c) { [real*c.real - imag*c.imag , imag*c.real + real*c.imag] as Complex }
 
Complex multiply (Number n) { [real*n , imag*n] as Complex }
 
Complex div (Complex c) { this * c.recip() }
 
Complex div (Number n) { this * (1/n) }
 
Complex negative () { [-real, -imag] as Complex }
 
/** the complex conjugate of this complex number.
* Overloads the bitwise complement (~) operator. */

Complex bitwiseNegate () { [real, -imag] as Complex }
 
/** the magnitude of this complex number. */
// could also use Math.sqrt( (this * (~this)).real )
Number abs () { Math.sqrt( real*real + imag*imag ) }
 
/** the complex reciprocal of this complex number. */
Complex recip() { (~this) / ((this * (~this)).real) }
 
/** derived angle &#x03B8; (theta) for polar form.
* Normalized to 0 &#x2264; &#x03B8; < 2&#x03C0;. */

Number getTheta() {
def theta = Math.atan2(imag,real)
theta = theta < 0 ? theta + 2 * Math.PI : theta
}
 
/** derived magnitude &#x03C1; (rho) for polar form. */
Number getRho() { this.abs() }
 
/** Runs Euler's polar-to-Cartesian complex conversion,
* converting [&#x03C1;, &#x03B8;] inputs into a [real, imag]-based complex number */

static Complex fromPolar(Number rho, Number theta) {
[rho * Math.cos(theta), rho * Math.sin(theta)] as Complex
}
 
/** Creates new complex with same magnitude &#x03C1;, but different angle &#x03B8; */
Complex withTheta(Number theta) { fromPolar(this.rho, theta) }
 
/** Creates new complex with same angle &#x03B8;, but different magnitude &#x03C1; */
Complex withRho(Number rho) { fromPolar(rho, this.theta) }
 
static Complex exp(Complex c) { fromPolar(Math.exp(c.real), c.imag) }
 
static Complex log(Complex c) { [Math.log(c.rho), c.theta] as Complex }
 
Complex power(Complex c) {
this == 0 && c != 0 \
? [0] as Complex \
 : c == 1 \
? this \
 : exp( log(this) * c )
}
 
Complex power(Number n) { this ** ([n, 0] as Complex) }
 
boolean equals(other) {
other != null && (other instanceof Complex \
? [real, imag] == [other.real, other.imag] \
 : other instanceof Number && [real, imag] == [other, 0])
}
 
int hashCode() { [real, imag].hashCode() }
 
String toString() {
def realPart = "${real}"
def imagPart = imag.abs() == 1 ? "i" : "${imag.abs()}i"
real == 0 && imag == 0 \
? "0" \
 : real == 0 \
? (imag > 0 ? '' : "-") + imagPart \
 : imag == 0 \
? realPart \
 : realPart + (imag > 0 ? " + " : " - ") + imagPart
}
}

Javadoc on the polar-related methods uses the following Greek alphabet encoded entities: &#x03C1;: ρ (rho), &#x03B8;: θ (theta), and &#x03C0;: π (pi).

Test Program (patterned after the Fortran example):

def tol = 0.000000001  // tolerance: acceptable "wrongness" to account for rounding error
 
println 'Demo 1: functionality as requested'
def a = [5,3] as Complex
println 'a == ' + a
def b = [0.5,6] as Complex
println 'b == ' + b
 
println "a + b == (${a}) + (${b}) == " + (a + b)
println "a * b == (${a}) * (${b}) == " + (a * b)
assert a + (-a) == 0
println "-a == -(${a}) == " + (-a)
assert (a * a.recip() - 1).abs() < tol
println "1/a == (${a}).recip() == " + (a.recip())
println()
 
println 'Demo 2: other functionality not requested, but important for completeness'
println "a - b == (${a}) - (${b}) == " + (a - b)
println "a / b == (${a}) / (${b}) == " + (a / b)
println "a ** b == (${a}) ** (${b}) == " + (a ** b)
println 'a.real == ' + a.real
println 'a.imag == ' + a.imag
println 'a.rho == ' + a.rho
println 'a.theta == ' + a.theta
println '|a| == ' + a.abs()
println 'a_bar == ' + ~a
 
def rho = 10
def piOverTheta = 3
def theta = Math.PI / piOverTheta
def fromPolar1 = Complex.fromPolar(rho, theta) // direct polar-to-cartesian conversion
def fromPolar2 = Complex.exp(Complex.I * theta) * rho // Euler's equation
println "rho*cos(theta) + rho*i*sin(theta) == ${rho}*cos(pi/${piOverTheta}) + ${rho}*i*sin(pi/${piOverTheta}) == " + fromPolar1
println "rho * exp(i * theta) == ${rho} * exp(i * pi/${piOverTheta}) == " + fromPolar2
assert (fromPolar1 - fromPolar2).abs() < tol
println()

Output:

Demo 1: functionality as requested
a == 5 + 3i
b == 0.5 + 6i
a + b == (5 + 3i) + (0.5 + 6i) == 5.5 + 9i
a * b == (5 + 3i) * (0.5 + 6i) == -15.5 + 31.5i
-a == -(5 + 3i) == -5 - 3i
1/a == (5 + 3i).recip() == 0.1470588235 - 0.0882352941i

Demo 2: other functionality not requested, but important for completeness
a - b == (5 + 3i) - (0.5 + 6i) == 4.5 - 3i
a / b == (5 + 3i) / (0.5 + 6i) == 0.56551724145 - 0.78620689665i
a ** b == (5 + 3i) ** (0.5 + 6i) == -0.013750112198456855 - 0.09332524760169053i
a.real == 5
a.imag == 3
a.rho == 5.830951894845301
a.theta == 0.5404195002705842
|a| == 5.830951894845301
a_bar == 5 - 3i
rho*cos(theta) +  rho*i*sin(theta) == 10*cos(pi/3) +  10*i*sin(pi/3) == 5.000000000000001 + 8.660254037844386i
rho * exp(i * theta) == 10 * exp(i * pi/3) == 5.000000000000001 + 8.660254037844386i

A Groovy equivalent to the "unit circle" part of the Fortran demo is shown in the Roots of unity task.

[edit] Haskell

Complex numbers are parameterized in their base type, so you can have Complex Integer for the Gaussian Integers, Complex Float, Complex Double, etc. The operations are just the usual overloaded numeric operations.

import Data.Complex
 
main = do
let a = 1.0 :+ 2.0 -- complex number 1+2i
let b = 4 -- complex number 4+0i
-- 'b' is inferred to be complex because it's used in
-- arithmetic with 'a' below.
putStrLn $ "Add: " ++ show (a + b)
putStrLn $ "Subtract: " ++ show (a - b)
putStrLn $ "Multiply: " ++ show (a * b)
putStrLn $ "Divide: " ++ show (a / b)
putStrLn $ "Negate: " ++ show (-a)
putStrLn $ "Inverse: " ++ show (recip a)
putStrLn $ "Conjugate:" ++ show (conjugate a)

Output:

*Main> main
Add: 5.0 :+ 2.0
Subtract: (-3.0) :+ 2.0
Multiply: 4.0 :+ 8.0
Divide: 0.25 :+ 0.5
Negate: (-1.0) :+ (-2.0)
Inverse: 0.2 :+ (-0.4)
Conjugate:1.0 :+ (-2.0)

[edit] IDL

complex (and dcomplex for double-precision) is a built-in data type in IDL:

x=complex(1,1)
y=complex(!pi,1.2)
print,x+y
( 4.14159, 2.20000)
print,x*y
( 1.94159, 4.34159)
print,-x
( -1.00000, -1.00000)
print,1/x
( 0.500000, -0.500000)

[edit] Icon and Unicon

Icon doesn't provide native support for complex numbers. Support is included in the IPL.

procedure main()
 
SetupComplex()
a := complex(1,2)
b := complex(3,4)
 
c := complex(&pi,1.5)
d := complex(1)
e := complex(,1)
 
every v := !"abcde" do write(v," := ",cpxstr(variable(v)))
 
write("a+b := ", cpxstr(cpxadd(a,b)))
write("a-b := ", cpxstr(cpxsub(a,b)))
write("a*b := ", cpxstr(cpxmul(a,b)))
write("a/b := ", cpxstr(cpxdiv(a,b)))
write("neg(a) := ", cpxstr(cpxneg(a)))
write("inv(a) := ", cpxstr(cpxinv(a)))
write("conj(a) := ", cpxstr(cpxconj(a)))
write("abs(a) := ", cpxabs(a))
write("neg(1) := ", cpxstr(cpxneg(1)))
end

Icon doesn't allow for operator overloading but procedures can be overloaded as was done here to allow 'complex' to behave more robustly.

provides complex number support supplemented by the code below.

 
link complex # for complex number support
 
procedure SetupComplex() #: used to setup safe complex
COMPLEX() # replace complex record constructor
SetupComplex := 1 # never call here again
return
end
 
procedure COMPLEX(rpart,ipart) #: new safe record constructor and coercion
initial complex :=: COMPLEX # get in front of record constructor
return if /ipart & (type(rpart) == "complex")
then rpart # already complex
else COMPLEX( real(\rpart | 0.0), real(\ipart|0) ) # create a new complex number
end
 
procedure cpxneg(z) #: negate z
z := complex(z) # coerce
return complex( -z.rpart, -z.ipart)
end
 
procedure cpxinv(z) #: inverse of z
local denom
z := complex(z) # coerce
 
denom := z.rpart ^ 2 + z.ipart ^ 2
return complex(z.rpart / denom, z.ipart / denom)
end

To take full advantage of the overloaded 'complex' procedure, the other cpxxxx procedures would need to be rewritten or overloaded.

Sample output:

#complexdemo.exe

a := (1.0+2.0i)
b := (3.0+4.0i)
c := (3.141592653589793+1.5i)
d := (1.0+0.0i)
e := (0.0+1.0i)
a+b := (4.0+6.0i)
a-b := (-2.0-2.0i)
a*b := (-5.0+10.0i)
a/b := (0.44+0.08i)
neg(a) := (-1.0-2.0i)
inv(a) := (0.2+0.4i)
conj(a) := (1.0-2.0i)
abs(a) := 2.23606797749979
neg(1) := (-1.0+0.0i)


This example is in need of improvement:
This could be better implemented as an object i n Unicon. Note, however, that Unicon doesn't allow for operator overloading at the current time.

[edit] J

Complex numbers are a native numeric data type in J. Although the examples shown here are performed on scalars, all numeric operations naturally apply to arrays of complex numbers.

   x=: 1j1
y=: 3.14159j1.2
x+y
4.14159j2.2
x*y
1.94159j4.34159
 %x
0.5j_0.5
-x
_1j_1

[edit] Java

public class Complex{
public final double real;
public final double imag;
 
public Complex(){this(0,0)}//default values to 0...force of habit
public Complex(double r, double i){real = r; imag = i;}
 
public Complex add(Complex b){
return new Complex(this.real + b.real, this.imag + b.imag);
}
 
public Complex mult(Complex b){
//FOIL of (a+bi)(c+di) with i*i = -1
return new Complex(this.real * b.real - this.imag * b.imag, this.real * b.imag + this.imag * b.real);
}
 
public Complex inv(){
//1/(a+bi) * (a-bi)/(a-bi) = 1/(a+bi) but it's more workable
double denom = real * real + imag * imag;
return new Complex(real/denom,-imag/denom);
}
 
public Complex neg(){
return new Complex(-real, -imag);
}
 
public Complex conj(){
return new Complex(real, -imag);
}
 
public String toString(){ //override Object's toString
return real + " + " + imag + " * i";
}
 
public static void main(String[] args){
Complex a = new Complex(Math.PI, -5) //just some numbers
Complex b = new Complex(-1, 2.5);
System.out.println(a.neg());
System.out.println(a.add(b));
System.out.println(a.inv());
System.out.println(a.mult(b));
System.out.println(a.conj());
}
}

[edit] JavaScript

function Complex(r, i) {
this.r = r;
this.i = i;
}
 
Complex.add = function() {
var num = arguments[0];
 
for(var i = 1, ilim = arguments.length; i < ilim; i += 1){
num.r += arguments[i].r;
num.i += arguments[i].i;
}
 
return num;
}
 
Complex.multiply = function() {
var num = arguments[0];
 
for(var i = 1, ilim = arguments.length; i < ilim; i += 1){
num.r = (num.r * arguments[i].r) - (num.i * arguments[i].i);
num.i = (num.i * arguments[i].r) - (num.r * arguments[i].i);
}
 
return num;
}
 
Complex.negate = function (z) {
return new Complex(-1*z.r, -1*z.i);
}
 
Complex.invert = function(z) {
var denom = Math.pow(z.r,2) + Math.pow(z.i,2);
return new Complex(z.r/denom, -1*z.i/denom);
}
 
Complex.conjugate = function(z) {
return new Complex(z.r, -1*z.i);
}
 
// BONUSES!
 
 
Complex.prototype.toString = function() {
return this.r === 0 && this.i === 0
? "0"
: (this.r !== 0 ? this.r : "")
+ ((this.r !== 0 || this.i < 0) && this.i !== 0
? (this.i > 0 ? "+" : "-")
: "" ) + ( this.i !== 0 ? Math.abs(this.i) + "i" : "" );
}
 
Complex.prototype.getMod = function() {
return Math.sqrt( Math.pow(this.r,2) , Math.pow(this.i,2) )
}

[edit] Julia

Julia has built-in support for complex arithmetic with arbitrary real types.

julia> z1 = 1.5 + 3im
julia> z2 = 1.5 + 1.5im
julia> z1 + z2
3.0 + 4.5im
julia> z1 - z2
0.0 + 1.5im
julia> z1 * z2
-2.25 + 6.75im
julia> z1 / z2
1.5 + 0.5im
julia> - z1
-1.5 - 3.0im
julia> conj(z1), z1' # two ways to conjugate
(1.5 - 3.0im,1.5 - 3.0im)
julia> abs(z1)
3.3541019662496847
julia> z1^z2
-1.102482955327779 - 0.38306415117199305im
julia> real(z1)
1.5
julia> imag(z1)
3.0

[edit] jq

For speed and for conformance with the complex plane interpretation, x+iy is represented as [x,y]; for flexibility, all the functions defined here will accept both real and complex numbers; and for uniformity, they are implemented as functions that ignore their input.

Recent versions of jq support modules, so these functions could all be placed in a module to avoid name conflicts, and thus no special prefix is used here.
def real(z): if (z|type) == "number" then z else z[0] end;
 
def imag(z): if (z|type) == "number" then 0 else z[1] end;
 
def plus(x; y):
if (x|type) == "number" then
if (y|type) == "number" then [ x+y, 0 ]
else [ x + y[0], y[1]]
end
elif (y|type) == "number" then plus(y;x)
else [ x[0] + y[0], x[1] + y[1] ]
end;
 
def multiply(x; y):
if (x|type) == "number" then
if (y|type) == "number" then [ x*y, 0 ]
else [x * y[0], x * y[1]]
end
elif (y|type) == "number" then multiply(y;x)
else [ x[0] * y[0] - x[1] * y[1],
x[0] * y[1] + x[1] * y[0]]
end;
 
def negate(x): multiply(-1; x);
 
def minus(x; y): plus(x; multiply(-1; y));
 
def conjugate(z):
if (z|type) == "number" then [z, 0]
else [z[0], -(z[1]) ]
end;
 
def invert(z):
if (z|type) == "number" then [1/z, 0]
else
( (z[0] * z[0]) + (z[1] * z[1]) ) as $d
# use "0 + ." to convert -0 back to 0
| [ z[0]/$d, (0 + -(z[1]) / $d)]
end;
 
def divide(x;y): multiply(x; invert(y));
 
def exp(z):
def expi(x): [ (x|cos), (x|sin) ];
if (z|type) == "number" then z|exp
elif z[0] == 0 then expi(z[1]) # for efficiency
else multiply( (z[0]|exp); expi(z[1]) )
end ;
 
def test(x;y):
"x = \( x )",
"y = \( y )",
"x+y: \( plus(x;y))",
"x*y: \( multiply(x;y))",
"-x: \( negate(x))",
"1/x: \( invert(x))",
"conj(x): \( conjugate(x))",
"(x/y)*y: \( multiply( divide(x;y) ; y) )",
"e^iπ: \( exp( [0, 4 * (1|atan) ] ) )"
;
 
test( [1,1]; [0,1] )
Output:
$ jq -n -f complex.jq
"x = [1,1]"
"y = [0,1]"
"x+y: [1,2]"
"x*y: [-1,1]"
"-x: [-1,-1]"
"1/x: [0.5,-0.5]"
"conj(x): [1,-1]"
"(x/y)*y: [1,1]"
"e^iπ: [-1,1.2246467991473532e-16]"

[edit] Liberty BASIC

mainwin 50 10
 
print " Adding"
call cprint cadd$( complex$( 1, 1), complex$( 3.14159265, 1.2))
print " Multiplying"
call cprint cmulti$( complex$( 1, 1), complex$( 3.14159265, 1.2))
print " Inverting"
call cprint cinv$( complex$( 1, 1))
print " Negating"
call cprint cneg$( complex$( 1, 1))
 
end
 
sub cprint cx$
print "( "; word$( cx$, 1); " + i *"; word$( cx$, 2); ")"
end sub
 
function complex$( a , bj )
''complex number string-object constructor
complex$ = str$( a ) ; " " ; str$( bj )
end function
 
function cadd$( a$ , b$ )
ar = val( word$( a$ , 1 ) )
ai = val( word$( a$ , 2 ) )
br = val( word$( b$ , 1 ) )
bi = val( word$( b$ , 2 ) )
cadd$ = complex$( ar + br , ai + bi )
end function
 
function cmulti$( a$ , b$ )
ar = val( word$( a$ , 1 ) )
ai = val( word$( a$ , 2 ) )
br = val( word$( b$ , 1 ) )
bi = val( word$( b$ , 2 ) )
cmulti$ = complex$( ar * br - ai * bi _
, ar * bi + ai * br )
end function
 
function cneg$( a$)
ar = val( word$( a$ , 1 ) )
ai = val( word$( a$ , 2 ) )
cneg$ =complex$( 0 -ar, 0 -ai)
end function
 
function cinv$( a$)
ar = val( word$( a$ , 1 ) )
ai = val( word$( a$ , 2 ) )
D =ar^2 +ai^2
cinv$ =complex$( ar /D , 0 -ai /D )
end function

[edit] Lua

 
 
--defines addition, subtraction, negation, multiplication, division, conjugation, norms, and a conversion to strgs.
complex = setmetatable({
__add = function(u, v) return complex(u.real + v.real, u.imag + v.imag) end,
__sub = function(u, v) return complex(u.real - v.real, u.imag - v.imag) end,
__mul = function(u, v) return complex(u.real * v.real - u.imag * v.imag, u.real * v.imag + u.imag * v.real) end,
__div = function(u, v) return u * complex(v.real / v.norm, -v.imag / v.norm) end,
__unm = function(u) return complex(-u.real, -u.imag) end,
__concat = function(u, v)
if type(u) == "table" then return u.real .. " + " .. u.imag .. "i" .. v
elseif type(u) == "string" or type(u) == "number" then return u .. v.real .. " + " .. v.imag .. "i"
end end,
__index = function(u, index)
local operations = {
norm = function(u) return u.real ^ 2 + u.imag ^ 2 end,
conj = function(u) return complex(u.real, -u.imag) end,
}
return operations[index] and operations[index](u)
end,
__newindex = function() error() end
}, {
__call = function(z, realpart, imagpart) return setmetatable({real = realpart, imag = imagpart}, complex) end
} )
 
local i, j = complex(2, 3), complex(1, 1)
 
print(i .. " + " .. j .. " = " .. (i+j))
print(i .. " - " .. j .. " = " .. (i-j))
print(i .. " * " .. j .. " = " .. (i*j))
print(i .. " / " .. j .. " = " .. (i/j))
print("|" .. i .. "| = " .. math.sqrt(i.norm))
print(i .. "* = " .. i.conj)
 

[edit] Maple

Maple has I (the square root of -1) built-in. Thus:

x := 1+I;
y := Pi+I*1.2;

By itself, it will perform mathematical operations symbolically, i.e. it will not try to perform computational evaluation unless specifically asked to do so. Thus:

x*y;
==> (1 + I) (Pi + 1.2 I)
simplify(x*y);
==> 1.941592654 + 4.341592654 I

Other than that, the task merely asks for

x+y;
x*y;
-x;
1/x;

[edit] Mathematica

Mathematica has fully implemented support for complex numbers throughout the software. Addition, subtraction, division, multiplications and powering need no further syntax than for real numbers:

x=1+2I
y=3+4I
 
x+y => 4 + 6 I
x-y => -2 - 2 I
y x => -5 + 10 I
y/x => 11/5 - (2 I)/5
x^3 => -11 - 2 I
y^4 => -527 - 336 I
x^y => (1 + 2 I)^(3 + 4 I)
N[x^y] => 0.12901 + 0.0339241 I

Powering to a complex power can in general not be written shorter, so Mathematica leaves it unevaluated if the numbers are exact. An approximation can be acquired using the function N. However Mathematica goes much further, basically all functions can handle complex numbers to arbitrary precision, including (but not limited to!):

Exp  Log
Sin Cos Tan Csc Sec Cot
ArcSin ArcCos ArcTan ArcCsc ArcSec ArcCot
Sinh Cosh Tanh Csch Sech Coth
ArcSinh ArcCosh ArcTanh ArcCsch ArcSech ArcCoth
Sinc
Haversine InverseHaversine
Factorial Gamma PolyGamma LogGamma
Erf BarnesG Hyperfactorial Zeta ProductLog RamanujanTauL

and many many more. The documentation states:

Mathematica has fundamental support for both explicit complex numbers and symbolic complex variables. All applicable mathematical functions support arbitrary-precision evaluation for complex values of all parameters, and symbolic operations automatically treat complex variables with full generality.

[edit] MATLAB

Complex numbers are a primitive data type in MATLAB. All the typical complex operations can be performed. There are two keywords that specify a number as complex: "i" and "j".

>> a = 1+i
 
a =
 
1.000000000000000 + 1.000000000000000i
 
>> b = 3+7i
 
b =
 
3.000000000000000 + 7.000000000000000i
 
>> a+b
 
ans =
 
4.000000000000000 + 8.000000000000000i
 
>> a-b
 
ans =
 
-2.000000000000000 - 6.000000000000000i
 
>> a*b
 
ans =
 
-4.000000000000000 +10.000000000000000i
 
>> a/b
 
ans =
 
0.172413793103448 - 0.068965517241379i
 
>> -a
 
ans =
 
-1.000000000000000 - 1.000000000000000i
 
>> a'
 
ans =
 
1.000000000000000 - 1.000000000000000i
 
>> a^b
 
ans =
 
0.000808197112874 - 0.011556516327187i
 
>> norm(a)
 
ans =
 
1.414213562373095

[edit] Maxima

z1: 5 + 2 * %i;
2*%i+5
 
z2: 3 - 7 * %i;
3-7*%i
 
carg(z1);
atan(2/5)
 
cabs(z1);
sqrt(29)
 
rectform(z1 * z2);
29-29*%i
 
polarform(z1);
sqrt(29)*%e^(%i*atan(2/5))
 
conjugate(z1);
5-2*%i
 
z1 + z2;
8-5*%i
 
z1 - z2;
9*%i+2
 
z1 * z2;
(3-7*%i)*(2*%i+5)
 
z1 * z2, rectform;
29-29*%i
 
z1 / z2;
(2*%i+5)/(3-7*%i)
 
z1 / z2, rectform;
(41*%i)/58+1/58
 
realpart(z1);
5
 
imagpart(z1);
2

[edit] МК-61/52

Instrustion:

Z1 = a + ib; Z2 = c + id;

a С/П b С/П c С/П d С/П

Division: С/П; multiplication: БП 36 С/П; addition: БП 54 С/П; subtraction: БП 63 С/П.

ПA	С/П	ПB	С/П	ПC	С/П	ПD	С/П	ИПC	x^2
ИПD x^2 + П3 ИПA ИПC * ИПB ИПD *
+ ИП3 / П1 ИПB ИПC * ИПA ИПD *
- ИП3 / П2 ИП1 С/П ИПA ИПC * ИПB
ИПD * - П1 ИПB ИПC * ИПA ИПD *
+ П2 ИП1 С/П ИПB ИПD + П2 ИПA ИПC
+ ИП1 С/П ИПB ИПD - П2 ИПA ИПC -
П1 С/П

[edit] Modula-2

MODULE complex;
 
IMPORT InOut;
 
TYPE Complex = RECORD R, Im : REAL END;
 
VAR z : ARRAY [0..3] OF Complex;
 
PROCEDURE ShowComplex (str : ARRAY OF CHAR; p : Complex);
 
BEGIN
InOut.WriteString (str); InOut.WriteString (" = ");
InOut.WriteReal (p.R, 6, 2);
IF p.Im >= 0.0 THEN InOut.WriteString (" + ") ELSE InOut.WriteString (" - ") END;
InOut.WriteReal (ABS (p.Im), 6, 2); InOut.WriteString (" i ");
InOut.WriteLn; InOut.WriteBf
END ShowComplex;
 
PROCEDURE AddComplex (x1, x2 : Complex; VAR x3 : Complex);
 
BEGIN
x3.R := x1.R + x2.R;
x3.Im := x1.Im + x2.Im
END AddComplex;
 
PROCEDURE SubComplex (x1, x2 : Complex; VAR x3 : Complex);
 
BEGIN
x3.R := x1.R - x2.R;
x3.Im := x1.Im - x2.Im
END SubComplex;
 
PROCEDURE MulComplex (x1, x2 : Complex; VAR x3 : Complex);
 
BEGIN
x3.R := x1.R * x2.R - x1.Im * x2.Im;
x3.Im := x1.R * x2.Im + x1.Im * x2.R
END MulComplex;
 
PROCEDURE InvComplex (x1 : Complex; VAR x2 : Complex);
 
BEGIN
x2.R := x1.R / (x1.R * x1.R + x1.Im * x1.Im);
x2.Im := -1.0 * x1.Im / (x1.R * x1.R + x1.Im * x1.Im)
END InvComplex;
 
PROCEDURE NegComplex (x1 : Complex; VAR x2 : Complex);
 
BEGIN
x2.R := - x1.R; x2.Im := - x1.Im
END NegComplex;
 
BEGIN
InOut.WriteString ("Enter two complex numbers : ");
InOut.WriteBf;
InOut.ReadReal (z[0].R); InOut.ReadReal (z[0].Im);
InOut.ReadReal (z[1].R); InOut.ReadReal (z[1].Im);
ShowComplex ("z1", z[0]); ShowComplex ("z2", z[1]);
InOut.WriteLn;
AddComplex (z[0], z[1], z[2]); ShowComplex ("z1 + z2", z[2]);
SubComplex (z[0], z[1], z[2]); ShowComplex ("z1 - z2", z[2]);
MulComplex (z[0], z[1], z[2]); ShowComplex ("z1 * z2", z[2]);
InvComplex (z[0], z[2]); ShowComplex ("1 / z1", z[2]);
NegComplex (z[0], z[2]); ShowComplex (" - z1", z[2]);
InOut.WriteLn
END complex.
Output :
Enter two complex numbers : 5 3 0.5 6

z1 = 5.00 + 3.00 i z2 = 0.50 + 6.00 i

z1 + z2 = 5.50 + 9.00 i z1 - z2 = 4.50 - 3.00 i z1 * z2 = -15.50 + 31.50 i 1 / z1 = 0.15 - 0.09 i

- z1 = -5.00 - 3.00 i

[edit] Nemerle

using System;
using System.Console;
using System.Numerics;
using System.Numerics.Complex;
 
module RCComplex
{
PrettyPrint(this c : Complex) : string
{
mutable sign = '+';
when (c.Imaginary < 0) sign = '-';
$"$(c.Real) $sign $(Math.Abs(c.Imaginary))i"
}
 
Main() : void
{
def complex1 = Complex(1.0, 1.0);
def complex2 = Complex(3.14159, 1.2);
 
WriteLine(Add(complex1, complex2).PrettyPrint());
WriteLine(Multiply(complex1, complex2).PrettyPrint());
WriteLine(Negate(complex2).PrettyPrint());
WriteLine(Reciprocal(complex2).PrettyPrint());
WriteLine(Conjugate(complex2).PrettyPrint());
}
}

Output:

4.14159 + 2.2i
1.94159 + 4.34159i
-3.14159 - 1.2i
0.277781124787984 - 0.106104663481097i
3.14159 - 1.2i

[edit] Nimrod

 
import complex
var a: TComplex = (1.0,1.0)
var b: TComplex = (3.1415,1.2)
 
echo ("a  : " & $a)
echo ("b  : " & $b)
echo ("a + b: " & $(a + b))
echo ("a * b: " & $(a * b))
echo ("1/a  : " & $(1/a))
echo ("-a  : " & $(-a))
 
 

Output:

a    : (1.0000000000000000e+00, 1.0000000000000000e+00)
b    : (3.1415000000000002e+00, 1.2000000000000000e+00)
a + b: (4.1415000000000006e+00, 2.2000000000000002e+00)
a * b: (1.9415000000000002e+00, 4.3414999999999999e+00)
1/a  : (5.0000000000000000e-01, -5.0000000000000000e-01)
-a   : (-1.0000000000000000e+00, -1.0000000000000000e+00)

[edit] Oberon-2

Oxford Oberon Compiler

 
MODULE Complex;
IMPORT Files,Out;
TYPE
Complex* = POINTER TO ComplexDesc;
ComplexDesc = RECORD
r-,i-: REAL;
END;
 
PROCEDURE (CONST x: Complex) Add*(CONST y: Complex): Complex;
BEGIN
RETURN New(x.r + y.r,x.i + y.i)
END Add;
 
PROCEDURE (CONST x: Complex) Sub*(CONST y: Complex): Complex;
BEGIN
RETURN New(x.r - y.r,x.i - y.i)
END Sub;
 
PROCEDURE (CONST x: Complex) Mul*(CONST y: Complex): Complex;
BEGIN
RETURN New(x.r*y.r - x.i*y.i,x.r*y.i + x.i*y.r)
END Mul;
 
PROCEDURE (CONST x: Complex) Div*(CONST y: Complex): Complex;
VAR
d: REAL;
BEGIN
d := y.r * y.r + y.i * y.i;
RETURN New((x.r*y.r + x.i*y.i)/d,(x.i*y.r - x.r*y.i)/d)
END Div;
 
(* Reciprocal *)
PROCEDURE (CONST x: Complex) Rec*(): Complex;
VAR
d: REAL;
BEGIN
d := x.r * x.r + y.i * y.i;
RETURN New(x.r/d,(-1.0 * x.i)/d);
END Rec;
 
(* Conjugate *)
PROCEDURE (x: Complex) Con*(): Complex;
BEGIN
RETURN New(x.r, (-1.0) * x.i);
END Con;
 
PROCEDURE (x: Complex) Out(out : Files.File);
BEGIN
Files.WriteString(out,"(");
Files.WriteReal(out,x.r);
Files.WriteString(out,",");
Files.WriteReal(out,x.i);
Files.WriteString(out,"i)")
END Out;
 
PROCEDURE New(x,y: REAL): Complex;
VAR
r: Complex;
BEGIN
NEW(r);r.r := x;r.i := y;
RETURN r
END New;
 
VAR
r,x,y: Complex;
BEGIN
x := New(1.5,3);
y := New(1.0,1.0);
 
Out.String("x: ");x.Out(Files.stdout);Out.Ln;
Out.String("y: ");y.Out(Files.stdout);Out.Ln;
r := x.Add(y);
Out.String("x + y: ");r.Out(Files.stdout);Out.Ln;
r := x.Sub(y);
Out.String("x - y: ");r.Out(Files.stdout);Out.Ln;
r := x.Mul(y);
Out.String("x * y: ");r.Out(Files.stdout);Out.Ln;
r := x.Div(y);
Out.String("x / y: ");r.Out(Files.stdout);Out.Ln;
r := y.Rec();
Out.String("1 / y: ");r.Out(Files.stdout);Out.Ln;
r := x.Con();
Out.String("x': ");r.Out(Files.stdout);Out.Ln;
 
END Complex.
 

Output:

x: (1.50000,3.00000i)
y: (1.00000,1.00000i)
x + y: (2.50000,4.00000i)
x - y: (0.500000,2.00000i)
x * y: (-1.50000,4.50000i)
x / y: (2.25000,0.750000i)
1 / y: (0.500000,-0.500000i)
x': (1.50000,-3.00000i)

[edit] OCaml

The Complex module from the standard library provides the functionality of complex numbers:

open Complex
 
let print_complex z =
Printf.printf "%f + %f i\n" z.re z.im
 
let () =
let a = { re = 1.0; im = 1.0 }
and b = { re = 3.14159; im = 1.25 } in
print_complex (add a b);
print_complex (mul a b);
print_complex (inv a);
print_complex (neg a);
print_complex (conj a)

Using Delimited Overloading, the syntax can be made closer to the usual one:

let () =
Complex.(
let print txt z = Printf.printf "%s = %s\n" txt (to_string z) in
let a = 1 + I
and b = 3 + 7I in
print "a + b" (a + b);
print "a - b" (a - b);
print "a * b" (a * b);
print "a / b" (a / b);
print "-a" (- a);
print "conj a" (conj a);
print "a^b" (a**b);
Printf.printf "norm a = %g\n" (float(abs a));
)

[edit] Octave

GNU Octave handles naturally complex numbers:

z1 = 1.5 + 3i;
z2 = 1.5 + 1.5i;
disp(z1 + z2); % 3.0 + 4.5i
disp(z1 - z2); % 0.0 + 1.5i
disp(z1 * z2); % -2.25 + 6.75i
disp(z1 / z2); % 1.5 + 0.5i
disp(-z1); % -1.5 - 3i
disp(z1'); % 1.5 - 3i
disp(abs(z1)); % 3.3541 = sqrt(z1*z1')
disp(z1 ^ z2); % -1.10248 - 0.38306i
disp( exp(z1) ); % -4.43684 + 0.63246i
disp( imag(z1) ); % 3
disp( real(z2) ); % 1.5
%...

[edit] ooRexx

 
c1 = .complex~new(1, 2)
c2 = .complex~new(3, 4)
r = 7
 
say "c1 =" c1
say "c2 =" c2
say "r =" r
say "-c1 =" (-c1)
say "c1 + r =" c1 + r
say "c1 + c2 =" c1 + c2
say "c1 - r =" c1 - r
say "c1 - c2 =" c1 - c2
say "c1 * r =" c1 * r
say "c1 * c2 =" c1 * c2
say "inv(c1) =" c1~inv
say "conj(c1) =" c1~conjugate
say "c1 / r =" c1 / r
say "c1 / c2 =" c1 / c2
say "c1 == c1 =" (c1 == c1)
say "c1 == c2 =" (c1 == c2)
 
 
::class complex
::method init
expose r i
use strict arg r, i = 0
 
-- complex instances are immutable, so these are
-- read only attributes
::attribute r GET
::attribute i GET
 
::method negative
expose r i
return self~class~new(-r, -i)
 
::method add
expose r i
use strict arg other
if other~isa(.complex) then
return self~class~new(r + other~r, i + other~i)
else return self~class~new(r + other, i)
 
::method subtract
expose r i
use strict arg other
if other~isa(.complex) then
return self~class~new(r - other~r, i - other~i)
else return self~class~new(r - other, i)
 
::method times
expose r i
use strict arg other
if other~isa(.complex) then
return self~class~new(r * other~r - i * other~i, r * other~i + i * other~r)
else return self~class~new(r * other, i * other)
 
::method inv
expose r i
denom = r * r + i * i
return self~class~new(r/denom,-i/denom)
 
::method conjugate
expose r i
return self~class~new(r, -i)
 
::method divide
use strict arg other
-- this is easier if everything is a complex number
if \other~isA(.complex) then other = .complex~new(other)
-- division is multiplication with the inversion
return self * other~inv
 
::method "=="
expose r i
use strict arg other
 
if \other~isa(.complex) then return .false
-- Note: these are numeric comparisons, so we're using the "="
-- method so those are handled correctly
return r = other~r & i = other~i
 
::method "\=="
use strict arg other
return \self~"\=="(other)
 
::method "="
-- this is equivalent of "=="
forward message("==")
 
::method "\="
-- this is equivalent of "\=="
forward message("\==")
 
::method "<>"
-- this is equivalent of "\=="
forward message("\==")
 
::method "><"
-- this is equivalent of "\=="
forward message("\==")
 
-- 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 r i
return r self~formatnumber(i)"i"
 
::method formatnumber private
use arg value
if value > 0 then return "+" value
else return "-" value~abs
 
-- override hashcode for collection class hash uses
::method hashCode
expose r i
return r~hashcode~bitxor(i~hashcode)
 

Output:

c1           = 1 + 2i
c2           = 3 + 4i
r            = 7
-c1          = -1 - 2i
c1 + r       = 8 + 2i
c1 + c2      = 4 + 6i
c1 - r       = -6 + 2i
c1 - c2      = -2 - 2i
c1 * r       = 7 + 14i
c1 * c2      = -5 + 10i
inv(c1)      = 0.2 - 0.4i
conj(c1)     = 1 - 2i
c1 / r       = 0.142857143 + 0.285714286i
c1 / c2      = 0.44 + 0.08i
c1 == c1     = 1
c1 == c2     = 0  

[edit] OxygenBasic

Implementation of a complex numbers class with arithmetical operations, and powers using DeMoivre's theorem (polar conversion).

 
'COMPLEX OPERATIONS
'=================
 
type tcomplex double x,y
 
class Complex
'============
 
has tcomplex
static sys i,pp
static tcomplex accum[32]
 
def operands
tcomplex*a,*b
@a=@accum+i
if pp then
@b=@a+sizeof accum
pp=0
else
@b=@this
end if
end def
 
method "load"()
operands
a.x=b.x
a.y=b.y
end method
 
method "push"()
i+=sizeof accum
end method
 
method "pop"()
pp=1
i-=sizeof accum
end method
 
method "="()
operands
b.x=a.x
b.y=a.y
end method
 
method "+"()
operands
a.x+=b.x
a.y+=b.y
end method
 
method "-"()
operands
a.x-=b.x
a.y-=b.y
end method
 
method "*"()
operands
double d
d=a.x
a.x = a.x * b.x - a.y * b.y
a.y = a.y * b.x + d * b.y
end method
 
method "/"()
operands
double d,v
v=1/(b.x * b.x + b.y * b.y)
d=a.x
a.x = (a.x * b.x + a.y * b.y) * v
a.y = (a.y * b.x - d * b.y) * v
end method
 
method power(double n)
operands
'Using DeMoivre theorem
double r,an,mg
r = hypot(b.x,b.y)
mg = r^n
if b.x=0 then
ay=.5*pi
if b.y<0 then ay=-ay
else
an = atan(b.y,b.x)
end if
an *= n
a.x = mg * cos(an)
a.y = mg * sin(an)
end method
 
method show() as string
return str(x,14) ", " str(y,14)
end method
 
end class
 
'#recordof complexop
 
'====
'TEST
'====
 
complex z1,z2,z3,z4,z5
 
'ENTER VALUES
 
z1 <= 0, 0
z2 <= 2, 1
z3 <= -2, 1
z4 <= 2, 4
z5 <= 1, 1
 
'EVALUATE COMPLEX EXPRESSIONS
 
z1 = z2 * z3
print "Z1 = "+z1.show 'RESULT -5.0, 0
 
z1 = z3+(z2.power(2))
print "Z1 = "+z1.show 'RESULT 1.0, 5.0
 
 
z1 = z5/z4
print "Z1 = "+z1.show 'RESULT 0.3, 0.1
 
z1 = z5/z1
print "Z1 = "+z1.show 'RESULT 2.0, 4.0
 
z1 = z2/z4
print "Z1 = "+z1.show 'RESULT -0.4, -0.3
 
z1 = z1*z4
print "Z1 = "+z1.show 'RESULT 2.0, 1.0
 

[edit] PARI/GP

To use, type, e.g., inv(3 + 7*I).

add(a,b)=a+b;
mult(a,b)=a*b;
neg(a)=-a;
inv(a)=1/a;

[edit] Pascal

program showcomplex(output);
 
type
complex = record
re,im: real
end;
 
var
z1, z2, zr: complex;
 
procedure set(var result: complex; re, im: real);
begin
result.re := re;
result.im := im
end;
 
procedure print(a: complex);
begin
write('(', a.re , ',', a.im, ')')
end;
 
procedure add(var result: complex; a, b: complex);
begin
result.re := a.re + b.re;
result.im := a.im + b.im;
end;
 
procedure neg(var result: complex; a: complex);
begin
result.re := -a.re;
result.im := -a.im
end;
 
procedure mult(var result: complex; a, b: complex);
begin
result.re := a.re*b.re - a.im*b.im;
result.im := a.re*b.im + a.im*b.re
end;
 
procedure inv(var result: complex; a: complex);
var
anorm: real;
begin
anorm := a.re*a.re + a.im*a.im;
result.re := a.re/anorm;
result.im := -a.im/anorm
end;
 
begin
set(z1, 3, 4);
set(z2, 5, 6);
 
neg(zr, z1);
print(zr); { prints (-3,-4) }
writeln;
 
add(zr, z1, z2);
print(zr); { prints (8,10) }
writeln;
 
inv(zr, z1);
print(zr); { prints (0.12,-0.16) }
writeln;
 
mul(zr, z1, z2);
print(zr); { prints (-9,38) }
writeln
end.

FreePascal has a complex units. Example of usage:

Program ComplexDemo;
 
uses
ucomplex;
 
var
a, b, absum, abprod, aneg, ainv, acong: complex;
 
function complex(const re, im: real): ucomplex.complex; overload;
begin
complex.re := re;
complex.im := im;
end;
 
begin
a := complex(5, 3);
b := complex(0.5, 6.0);
absum := a + b;
writeln ('(5 + i3) + (0.5 + i6.0): ', absum.re:3:1, ' + i', absum.im:3:1);
abprod := a * b;
writeln ('(5 + i3) * (0.5 + i6.0): ', abprod.re:5:1, ' + i', abprod.im:4:1);
aneg := -a;
writeln ('-(5 + i3): ', aneg.re:3:1, ' + i', aneg.im:3:1);
ainv := 1.0 / a;
writeln ('1/(5 + i3): ', ainv.re:3:1, ' + i', ainv.im:3:1);
acong := cong(a);
writeln ('conj(5 + i3): ', acong.re:3:1, ' + i', acong.im:3:1);
end.
 

[edit] Perl

The Math::Complex module implements complex arithmetic.

use Math::Complex;
my $a = 1 + 1*i;
my $b = 3.14159 + 1.25*i;
 
print "$_\n" foreach
$a + $b, # addition
$a * $b, # multiplication
-$a, # negation
1 / $a, # multiplicative inverse
~$a; # complex conjugate

[edit] Perl 6

Works with: Rakudo version #22 "Thousand Oaks"
my $a = 1 + i;
my $b = pi + 1.25i;
 
.say for $a + $b, $a * $b, -$a, 1 / $a, $a.conj;
.say for $a.abs, $a.sqrt, $a.re, $a.im;

Output (precision varies with different implementations):

4.1415926535897931+2.25i
1.8915926535897931+4.3915926535897931i
-1-1i
0.5-0.5i
1-1i
1.4142135623730951
1.0986841134678098+0.45508986056222733i
1
1

[edit] PL/I

/* PL/I complex numbers may be integer or floating-point.  */
/* In this example, the variables are floating-pint. */
/* For integer variables, change 'float' to 'fixed binary' */
 
declare (a, b) complex float;
a = 2+5i;
b = 7-6i;
 
put skip list (a+b);
put skip list (a - b);
put skip list (a*b);
put skip list (a/b);
put skip list (a**b);
put skip list (1/a);
put skip list (conjg(a)); /* gives the conjugate of 'a'. */
 
/* Functions exist for extracting the real and imaginary parts */
/* of a complex number. */
 
/* As well, trigonometric functions may be used with complex */
/* numbers, such as SIN, COS, TAN, ATAN, and so on. */

[edit] PicoLisp

(load "@lib/math.l")
 
(de addComplex (A B)
(cons
(+ (car A) (car B)) # Real
(+ (cdr A) (cdr B)) ) ) # Imag
 
(de mulComplex (A B)
(cons
(-
(*/ (car A) (car B) 1.0)
(*/ (cdr A) (cdr B) 1.0) )
(+
(*/ (car A) (cdr B) 1.0)
(*/ (cdr A) (car B) 1.0) ) ) )
 
(de invComplex (A)
(let Denom
(+
(*/ (car A) (car A) 1.0)
(*/ (cdr A) (cdr A) 1.0) )
(cons
(*/ (car A) 1.0 Denom)
(- (*/ (cdr A) 1.0 Denom)) ) ) )
 
(de negComplex (A)
(cons (- (car A)) (- (cdr A))) )
 
(de fmtComplex (A)
(pack
(round (car A) (dec *Scl))
(and (gt0 (cdr A)) "+")
(round (cdr A) (dec *Scl))
"i" ) )
 
(let (A (1.0 . 1.0) B (cons pi 1.2))
(prinl "A = " (fmtComplex A))
(prinl "B = " (fmtComplex B))
(prinl "A+B = " (fmtComplex (addComplex A B)))
(prinl "A*B = " (fmtComplex (mulComplex A B)))
(prinl "1/A = " (fmtComplex (invComplex A)))
(prinl "-A = " (fmtComplex (negComplex A))) )

Output:

A = 1.00000+1.00000i
B = 3.14159+1.20000i
A+B = 4.14159+2.20000i
A*B = 1.94159+4.34159i
1/A = 0.50000-0.50000i
-A = -1.00000-1.00000i

[edit] Pop11

Complex numbers are a built-in data type in Pop11. Real and imaginary part of complex numbers can be floating point or exact (integer or rational) value (both part must be of the same type). Operations on floating point complex numbers always produce complex numbers. Operations on exact complex numbers give real result (integer or rational) if imaginary part of the result is 0. The '+:' and '-:' operators create complex numbers: '1 -: 3' is '1 - 3i' in mathematical notation.

lvars a = 1.0 +: 1.0, b = 2.0 +: 5.0 ;
a+b =>
a*b =>
1/a =>
a-b =>
a-a =>
a/b =>
a/a =>
 
;;; The same, but using exact values
1 +: 1 -> a;
2 +: 5 -> b;
a+b =>
a*b =>
1/a =>
a-b =>
a-a =>
a/b =>
a/a =>

[edit] PostScript

Complex numbers can be represented as 2 element vectors ( arrays ). Thus, a+bi can be written as [a b] in PostScript.

 
%Adding two complex numbers
/addcomp{
/x exch def
/y exch def
/z [0 0] def
z 0 x 0 get y 0 get add put
z 1 x 1 get y 1 get add put
z pstack
}def
 
%Subtracting one complex number from another
/subcomp{
/x exch def
/y exch def
/z [0 0] def
z 0 x 0 get y 0 get sub put
z 1 x 1 get y 1 get sub put
z pstack
}def
 
%Multiplying two complex numbers
/mulcomp{
/x exch def
/y exch def
/z [0 0] def
z 0 x 0 get y 0 get mul x 1 get y 1 get mul sub put
z 1 x 1 get y 0 get mul x 0 get y 1 get mul add put
z pstack
}def
 
%Negating a complex number
/negcomp{
/x exch def
/z [0 0] def
z 0 x 0 get neg put
z 1 x 1 get neg put
z pstack
}def
 
%Inverting a complex number
/invcomp{
/x exch def
/z [0 0] def
z 0 x 0 get x 0 get 2 exp x 1 get 2 exp add div put
z 0 x 1 get neg x 0 get 2 exp x 1 get 2 exp add div put
z pstack
}def
 
 

[edit] PureBasic

Structure Complex
real.d
imag.d
EndStructure
 
Procedure Add_Complex(*A.Complex, *B.Complex)
Protected *R.Complex=AllocateMemory(SizeOf(Complex))
If *R
*R\real=*A\real+*B\real
*R\imag=*A\imag+*B\imag
EndIf
ProcedureReturn *R
EndProcedure
 
Procedure Inv_Complex(*A.Complex)
Protected *R.Complex=AllocateMemory(SizeOf(Complex)), denom.d
If *R
denom = *A\real * *A\real + *A\imag * *A\imag
*R\real= *A\real / denom
*R\imag=-*A\imag / denom
EndIf
ProcedureReturn *R
EndProcedure
 
Procedure Mul_Complex(*A.Complex, *B.Complex)
Protected *R.Complex=AllocateMemory(SizeOf(Complex))
If *R
*R\real=*A\real * *B\real - *A\imag * *B\imag
*R\imag=*A\real * *B\imag + *A\imag * *B\real
EndIf
ProcedureReturn *R
EndProcedure
 
Procedure Neg_Complex(*A.Complex)
Protected *R.Complex=AllocateMemory(SizeOf(Complex))
If *R
*R\real=-*A\real
*R\imag=-*A\imag
EndIf
ProcedureReturn *R
EndProcedure
 
Procedure ShowAndFree(Header$, *Complex.Complex)
If *Complex
Protected.d i=*Complex\imag, r=*Complex\real
Print(LSet(Header$,7))
Print("= "+StrD(r,3))
If i>=0: Print(" + ")
Else: Print(" - ")
EndIf
PrintN(StrD(Abs(i),3)+"i")
FreeMemory(*Complex)
EndIf
EndProcedure
 
If OpenConsole()
Define.Complex a, b, *c
a\real=1.0: a\imag=1.0
b\real=#PI: b\imag=1.2
*c=Add_Complex(a,b): ShowAndFree("a+b", *c)
*c=Mul_Complex(a,b): ShowAndFree("a*b", *c)
*c=Inv_Complex(a): ShowAndFree("Inv(a)", *c)
*c=Neg_Complex(a): ShowAndFree("-a", *c)
Print(#CRLF$+"Press ENTER to exit"):Input()
EndIf

[edit] Python

>>> z1 = 1.5 + 3j
>>> z2 = 1.5 + 1.5j
>>> z1 + z2
(3+4.5j)
>>> z1 - z2
1.5j
>>> z1 * z2
(-2.25+6.75j)
>>> z1 / z2
(1.5+0.5j)
>>> - z1
(-1.5-3j)
>>> z1.conjugate()
(1.5-3j)
>>> abs(z1)
3.3541019662496847
>>> z1 ** z2
(-1.1024829553277784-0.38306415117199333j)
>>> z1.real
1.5
>>> z1.imag
3.0
>>>

[edit] R

Translation of: Octave
z1 <- 1.5 + 3i
z2 <- 1.5 + 1.5i
print(z1 + z2) # 3+4.5i
print(z1 - z2) # 0+1.5i
print(z1 * z2) # -2.25+6.75i
print(z1 / z2) # 1.5+0.5i
print(-z1) # -1.5-3i
print(Conj(z1)) # 1.5-3i
print(abs(z1)) # 3.354102
print(z1^z2) # -1.102483-0.383064i
print(exp(z1)) # -4.436839+0.632456i
print(Re(z1)) # 1.5
print(Im(z1)) # 3

[edit] Racket

 
#lang racket
 
(define a 3+4i)
(define b 8+0i)
 
(+ a b)  ; addition
(- a b)  ; subtraction
(/ a b)  ; division
(* a b)  ; multiplication
(- a)  ; negation
(/ 1 a)  ; reciprocal
(conjugate a) ; conjugation
 

[edit] REXX

The REXX language has no complex type numbers, but some complex arithmetic subroutines can easily be written.

/*REXX program to show how to support math functions for complex numbers*/
 
x = '(5,3i)' /*this little piggy uses "I" (or "i") ... */
y = '( .5, 6j)' /*this little piggy uses "J" (or "j") ... */
 
sum = Cadd(x,y); say ' addition: ' x " + " y ' = ' sum
dif = Csub(x,y); say ' subtration: ' x " + " y ' = ' dif
prod = Cmul(x,y); say 'multiplication: ' x " * " y ' = ' prod
quot = Cdiv(x,y); say ' division: ' x " ÷ " y ' = ' quot
inv = Cinv(x); say ' inverse: ' x " = " inv
cnjX = Ccnj(x); say ' conjugate of: ' x " = " cnjX
negX = Cneg(x); say ' negation of: ' x " = " negX
exit /*stick a fork in it, we're done.*/
 
/*─────────────────────────────────────one─liners───────────────────────*/
Ccnj: procedure;arg a ',' b,c ',' d;call Cg;r1=a;r2=-b;return Cr()
Cadd: procedure;arg a ',' b,c ',' d;call Cg;r1=a+c;r2=b+d;return Cr()
Csub: procedure;arg a ',' b,c ',' d;call Cg;r1=a-c;r2=b-d;return Cr()
Cmul: procedure;arg a ',' b,c ',' d;call Cg;r1=a*c-b*d; r2=b*c+a*d;return Cr()
Cdiv: procedure;arg a ',' b,c ',' d;call Cg;_=c*c+d*d;r1=(a*c+b*d)/_;r2=(b*c-a*d)/_;return Cr()
Cg: a=Cdej(a); b=Cdej(b); c=Cdej(c); d=Cdej(d); return
Cr: _='['r1; if r2\=0 then _=_','r2"j"; return _']'
Cdej: return word(translate(arg(1),,'{[(JI)]}') 0,1)
Cneg: return Cmul(arg(1),-1)
Cinv: return Cdiv(1,arg(1))

output

      addition:    (5,3i)  +  (.5,6j)  =  [5.5,9j]
    subtration:    (5,3i)  +  (.5,6j)  =  [4.5,-3j]
multiplication:    (5,3i)  *  (.5,6j)  =  [-15.5,31.5j]
      division:    (5,3i)  ÷  (.5,6j)  =  [0.565517241,-0.786206897j]
       inverse:    (5,3i)              =  [0.147058824,-0.0882352941j]
  conjugate of:    (5,3i)              =  [5,-3j]
   negation of:    (5,3i)              =  [-5,-3j]

[edit] RLaB

 
>> x = sqrt(-1)
0 + 1i
>> y = 10 + 5i
10 + 5i
>> z = 5*x-y
-10 + 0i
>> isreal(z)
1
 

[edit] Ruby

require 'complex'  # With Ruby 1.9, this line is optional.
 
# Three ways to write complex numbers:
a = Complex(1, 1) # 1. call Kernel#Complex
i = Complex::I # 2. use Complex::I
b = 3.14159 + 1.25 * i
c = '1/2+3/4i'.to_c # 3. Use the .to_c method from String, result ((1/2)+(3/4)*i)
 
#Ruby 2.1 introduced a suffix to create a complex:
c = 1.0/2+3/4i # (0.5-(3/4)*i)
 
# Operations:
puts a + b # addition
puts a * b # multiplication
puts -a # negation
puts 1.quo a # multiplicative inverse
puts a.conjugate # complex conjugate
puts a.conj # alias for complex conjugate

Notes:

  • Ruby 1.8 must require 'complex'. Ruby 1.9 moves complex numbers to core, so require 'complex' only defines a few deprecated methods.
  • Ruby 1.9 deprecates Numeric#im; code like a = 1 + 1.im or b = 3.14159 + 1.25.im would call the deprecated method.
  • All of these operations are safe with other numeric types. For example, 42.conjugate returns 42.
# Other ways to find the multiplicative inverse:
puts 1.quo a # always works
puts 1.0 / a # works, but forces floating-point math
puts 1 / a # might truncate to integer

[edit] Rust

This code works for rustc 0.10-pre.

extern crate num;
 
use num::complex::Cmplx;
 
fn main() {
let a = Cmplx::new(-4.0, 5.0);
let b = Cmplx::new(1.0, 1.0);
 
println!("a = {}", a);
println!("b = {}", b);
println!("a + b = {}", a + b);
println!("a * b = {}", a * b);
println!("1 / a = {}", Cmplx::new(1.0, 0.0) / a);
println!("-a = {}", -a);
println!("conj a = {}", a.conj());
}

[edit] Scala

Works with: Scala version 2.8

Scala doesn't come with a Complex library, but one can be made:

package org.rosettacode
 
package object ArithmeticComplex {
val i = Complex(0, 1)
 
implicit def fromDouble(d: Double) = Complex(d)
implicit def fromInt(i: Int) = Complex(i.toDouble)
}
 
package ArithmeticComplex {
case class Complex(real: Double = 0.0, imag: Double = 0.0) {
def this(s: String) =
this("[\\d.]+(?!i)".r findFirstIn s getOrElse "0" toDouble,
"[\\d.]+(?=i)".r findFirstIn s getOrElse "0" toDouble)
 
def +(b: Complex) = Complex(real + b.real, imag + b.imag)
def -(b: Complex) = Complex(real - b.real, imag - b.imag)
def *(b: Complex) = Complex(real * b.real - imag * b.imag, real * b.imag + imag * b.real)
def inverse = {
val denom = real * real + imag * imag
Complex(real / denom, -imag / denom)
}
def /(b: Complex) = this * b.inverse
def unary_- = Complex(-real, -imag)
lazy val abs = math.hypot(real, imag)
override def toString = real + " + " + imag + "i"
 
def i = { require(imag == 0.0); Complex(imag = real) }
}
 
object Complex {
def apply(s: String) = new Complex(s)
def fromPolar(rho:Double, theta:Double) = Complex(rho*math.cos(theta), rho*math.sin(theta))
}
}

Usage example:

scala> import org.rosettacode.ArithmeticComplex._
import org.rosettacode.ArithmeticComplex._
 
scala> 1 + i
res0: org.rosettacode.ArithmeticComplex.Complex = 1.0 + 1.0i
 
scala> 1 + 2 * i
res1: org.rosettacode.ArithmeticComplex.Complex = 1.0 + 2.0i
 
scala> 2 + 1.i
res2: org.rosettacode.ArithmeticComplex.Complex = 2.0 + 1.0i
 
scala> res0 + res1
res3: org.rosettacode.ArithmeticComplex.Complex = 2.0 + 3.0i
 
scala> res1 * res2
res4: org.rosettacode.ArithmeticComplex.Complex = 0.0 + 5.0i
 
scala> res2 / res0
res5: org.rosettacode.ArithmeticComplex.Complex = 1.5 + -0.5i
 
scala> res1.inverse
res6: org.rosettacode.ArithmeticComplex.Complex = 0.2 + -0.4i
 
scala> -res6
res7: org.rosettacode.ArithmeticComplex.Complex = -0.2 + 0.4i
 

[edit] Scheme

Scheme implementations are not required to support complex numbers, but if they do, they are required to support complex number literals in one of the following standard formats[3]:

  • rectangular coordinates: real+imagi (or real-imagi), where real is the real part and imag is the imaginary part. For a pure-imaginary number, the real part may be omitted but the sign of the imaginary part is mandatory (even if it is "+"): +imagi (or -imagi). If the imaginary part is 1 or -1, the imaginary part can be omitted, leaving only the +i or -i at the end.
  • polar coordinates: r@theta, where r is the absolute value (magnitude) and theta is the angle
(define a 1+i)
(define b 3.14159+1.25i)
 
(define c (+ a b))
(define c (* a b))
(define c (/ 1 a))
(define c (- a))

[edit] Seed7

$ include "seed7_05.s7i";
include "float.s7i";
include "complex.s7i";
 
const proc: main is func
local
var complex: a is complex(1.0, 1.0);
var complex: b is complex(3.14159, 1.2);
begin
writeln("a=" <& a digits 5);
writeln("b=" <& b digits 5);
# addition
writeln("a+b=" <& a + b digits 5);
# multiplication
writeln("a*b=" <& a * b digits 5);
# inversion
writeln("1/a=" <& complex(1.0) / a digits 5);
# negation
writeln("-a=" <& -a digits 5);
end func;

[edit] Slate

[| a b |
a: 1 + 1 i.
b: Pi + 1.2 i.
print: a + b.
print: a * b.
print: a / b.
print: a reciprocal.
print: a conjugated.
print: a abs.
print: a negated.
].

[edit] Smalltalk

Works with: GNU Smalltalk
PackageLoader fileInPackage: 'Complex'.
|a b|
a := 1 + 1 i.
b := 3.14159 + 1.2 i.
(a + b) displayNl.
(a * b) displayNl.
(a / b) displayNl.
a reciprocal displayNl.
a conjugate displayNl.
a abs displayNl.
a real displayNl.
a imaginary displayNl.
a negated displayNl.

[edit] smart BASIC

' complex numbers are native for "smart BASIC"
A=1+2i
B=3-5i
 
' all math operations and functions work with complex numbers
C=A*B
PRINT SQR(-4)
 
' example of solving quadratic equation with complex roots
' x^2+2x+5=0
a=1 ! b=2 ! c=5
x1=(-b+SQR(b^2-4*a*c))/(2*a)
x2=(-b-SQR(b^2-4*a*c))/(2*a)
PRINT x1,x2
 
' gives output
-1+2i -1-2i

[edit] SNOBOL4

Works with: Macro Spitbol
Works with: Snobol4+
Works with: CSnobol
*       # Define complex datatype
data('complex(r,i)')
 
* # Addition
define('addx(x1,x2)a,b,c,d') :(addx_end)
addx a = r(x1); b = i(x1); c = r(x2); d = i(x2)
addx = complex(a + c, b + d) :(return)
addx_end
 
* # Multiplication
define('multx(x1,x2)a,b,c,d') :(multx_end)
multx a = r(x1); b = i(x1); c = r(x2); d = i(x2)
multx = complex(a * c - b * d, b * c + a * d) :(return)
multx_end
 
* # Negation
define('negx(x)') :(negx_end)
negx negx = complex(-r(x), -i(x)) :(return)
negx_end
 
* # Inverse
define('invx(x)d') :(invx_end)
invx d = (r(x) * r(x)) + (i(x) * i(x))
invx = complex(1.0 * r(x) / d, 1.0 * -i(x) / d) :(return)
invx_end
 
* # Print compex number: a+bi / a-bi
define('printx(x)sign') :(printx_end)
printx sign = ge(i(x),0) '+'
printx = r(x) sign i(x) 'i' :(return)
printx_end
 
* # Test and display
a = complex(1,1)
b = complex(3.14159, 1.2)
output = printx( addx(a,b) )
output = printx( multx(a,b) )
output = printx( negx(a) ) ', ' printx( negx(b) )
output = printx( invx(a) ) ', ' printx( invx(b) )
end

Output:

4.14159+2.2i
1.94159+4.34159i
-1-1i, -3.14159-1.2i
0.5-0.5i, 0.277781125-0.106104663i

[edit] Standard ML

 
(* Signature for complex numbers *)
signature COMPLEX = sig
type num
 
val complex : real * real -> num
 
val negative : num -> num
val plus : num -> num -> num
val minus : num -> num -> num
val times : num -> num -> num
val invert : num -> num
val print_number : num -> unit
end;
 
(* Actual implementation *)
structure Complex :> COMPLEX = struct
type num = real * real
 
fun complex (a, b) = (a, b)
 
fun negative (a, b) = (Real.~a, Real.~b)
fun plus (a1, b1) (a2, b2) = (Real.+ (a1, a2), Real.+(b1, b2))
fun minus i1 i2 = plus i1 (negative i2)
fun times (a1, b1) (a2, b2)= (Real.*(a1, a2) - Real.*(b1, b2), Real.*(a1, b2) + Real.*(a2, b1))
fun invert (a, b) =
let
val denom = a * a + b * b
in
(a / denom, ~b / denom)
end
 
fun print_number (a, b) =
print (Real.toString(a) ^ " + " ^ Real.toString(b) ^ "i\n")
end;
 
val i1 = Complex.complex(1.0,2.0); (* 1 + 2i *)
val i2 = Complex.complex(3.0,4.0); (* 3 + 4i *)
 
Complex.print_number(Complex.negative(i1)); (* -1 - 2i *)
Complex.print_number(Complex.plus i1 i2); (* 4 + 6i *)
Complex.print_number(Complex.minus i2 i1); (* 2 + 2i *)
Complex.print_number(Complex.times i1 i2); (* -5 + 10i *)
Complex.print_number(Complex.invert i1); (* 1/5 - 2i/5 *)
 

[edit] Tcl

Library: Tcllib (Package: math::complexnumbers)
package require math::complexnumbers
namespace import math::complexnumbers::*
 
set a [complex 1 1]
set b [complex 3.14159 1.2]
puts [tostring [+ $a $b]] ;# ==> 4.14159+2.2i
puts [tostring [* $a $b]] ;# ==> 1.94159+4.34159i
puts [tostring [pow $a [complex -1 0]]] ;# ==> 0.5-0.4999999999999999i
puts [tostring [- $a]] ;# ==> -1.0-i

[edit] TI-83 BASIC

TI-83 BASIC has built in complex number support; the normal arithmetic operators + - * / are used.

The method complex numbers are displayed can be chosen in the "MODE" menu.
Real: Does not show complex numbers, gives an error if a number is imaginary.
a+bi: The classic display for imaginary numbers with the real and imaginary components
re^Θi: Displays imaginary numbers in Polar Coordinates.

[edit] TI-89 BASIC

TI-89 BASIC has built-in complex number support; the normal arithmetic operators + - * / are used.

Character set note: the symbol for the imaginary unit is not the normal "i" but a different character (Unicode: U+F02F "" (private use area); this character should display with the "TI Uni" font). Also, U+3013 EN DASH “”, displayed on the TI as a superscript minus, is used for the minus sign on numbers, distinct from ASCII "-" used for subtraction.
The choice of examples here is
Translation of: Common Lisp
.
■ √(–1)                    
■ ^2                     —1
■  + 1                1 + 
■ (1+) * 2          2 + 2*
■ (1+) (2)        —2 + 2*
■ —(1+)              —1 - 
■ 1/(2)              —1 - 
■ real(1 + 2)             1
■ imag(1 + 2)             2

Complex numbers can also be entered and displayed in polar form. (This example shows input in polar form while the complex display mode is rectangular and the angle mode is radians).

■ (1∠π/4)
           √(2)/2 + √(2)/2*

Note that the parentheses around ∠ notation are required. It has a related use in vectors: (1∠π/4) is a complex number, [1,∠π/4] is a vector in two dimensions in polar notation, and [(1∠π/4)] is a complex number in a vector.

[edit] Ursala

Complex numbers are a primitive type that can be parsed in fixed or exponential formats, with either i or j notation as shown. The usual complex arithmetic and transcendental functions are callable using the syntax libname..funcname or a recognizable truncation (e.g., c..add or ..csin). Real operands are promoted to complex.

u = 3.785e+00-1.969e+00i
v = 9.545e-01-3.305e+00j
 
#cast %jL
 
examples =
 
<
complex..add (u,v),
complex..mul (u,v),
complex..sub (0.,u),
complex..div (1.,v)>

output:

<
   4.740e+00-5.274e+00j,
   -2.895e+00-1.439e+01j,
   3.785e+00-1.969e+00j,
   8.066e-02+2.793e-01j>

[edit] Wortel

Translation of: CoffeeScript
@class Complex {
&[r i] @: {
^r || r 0
^i || i 0
^m +@sq^r @sq^i
}
add &o @new Complex[+ ^r o.r + ^i o.i]
mul &o @new Complex[-* ^r o.r * ^i o.i +* ^r o.i * ^i o.r]
neg &^ @new Complex[@-^r @-^i]
inv &^ @new Complex[/ ^r ^m / @-^i ^m]
toString &^?{
=^i 0 "{^r}"
=^r 0 "{^i}i"
>^i 0 "{^r} + {^i}i"
"{^r} - {@-^i}i"
}
}
 
@vars {
a @new Complex[5 3]
b @new Complex[4 3N]
}
@each &x !console.log x [
"({a}) + ({b}) = {!a.add b}"
"({a}) * ({b}) = {!a.mul b}"
"-1 * ({b}) = {b.neg.}"
"({a}) - ({b}) = {!a.add b.neg.}"
"1 / ({b}) = {b.inv.}"
"({!a.mul b}) / ({b}) = {`!.mul b.inv. !a.mul b}"
]

Output:

(5 + 3i) + (4 - 3i) = 9
(5 + 3i) * (4 - 3i) = 29 - 3i
-1 * (4 - 3i) = -4 + 3i
(5 + 3i) - (4 - 3i) = 1 + 6i
1 / (4 - 3i) = 0.16 + 0.12i
(29 - 3i) / (4 - 3i) = 5 + 3i

[edit] XPL0

include c:\cxpl\codes;
 
func real CAdd(A, B, C); \Return complex sum of two complex numbers
real A, B, C;
[C(0):= A(0) + B(0);
C(1):= A(1) + B(1);
return C;
];
 
func real CMul(A, B, C); \Return complex product of two complex numbers
real A, B, C;
[C(0):= A(0)*B(0) - A(1)*B(1);
C(1):= A(1)*B(0) + A(0)*B(1);
return C;
];
 
func real CNeg(A, C); \Return negative of a complex number
real A, C;
[C(0):= -A(0);
C(1):= -A(1);
return C;
];
 
func real CInv(A, C); \Return inversion (reciprical) of complex number
real A, C;
real D;
[D:= sq(A(0)) + sq(A(1));
C(0):= A(0)/D;
C(1):=-A(1)/D;
return C;
];
 
func real Conj(A, C); \Return conjugate of a complex number
real A, C;
[C(0):= A(0);
C(1):=-A(1);
return C;
];
 
proc COut(D, A); \Output a complex number to specified device
int D; real A;
[RlOut(D, A(0));
Text(D, if A(1)>=0.0 then " +" else " -");
RlOut(D, abs(A(1)));
ChOut(D, ^i);
];
 
real U, V, W(2);
[Format(2,2);
U:= [1.0, 1.0];
V:= [3.14, 1.2];
COut(0, CAdd(U,V,W)); CrLf(0);
COut(0, CMul(U,V,W)); CrLf(0);
COut(0, CNeg(U,W)); CrLf(0);
COut(0, CInv(U,W)); CrLf(0);
COut(0, Conj(U,W)); CrLf(0);
]

Output:

 4.14 + 2.20i
 1.94 + 4.34i
-1.00 - 1.00i
 0.50 - 0.50i
 1.00 - 1.00i
Personal tools
Namespaces

Variants
Actions
Community
Explore
Misc
Toolbox