# Arithmetic/Complex

(Redirected from Complex numbers)
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: ${\displaystyle a+b\times i}$ (sometimes shown as: ${\displaystyle b+a\times i}$ where   ${\displaystyle a}$   and   ${\displaystyle b}$  are real numbers,   and   ${\displaystyle i}$   is    -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 ${\displaystyle i}$.

Task
• 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 ${\displaystyle a+bi}$ is ${\displaystyle a-bi}$

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.

## 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;

## 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



## ALGOL W

Complex is a built-in type in Algol W.

begin    % show some complex arithmetic                                          %    % returns c + d, using the builtin complex + operator                   %    complex procedure cAdd ( complex value c, d ) ; c + d;    % returns c * d, using the builtin complex * operator                   %    complex procedure cMul ( complex value c, d ) ; c * d;    % returns the negation of c, using the builtin complex unary - operator %    complex procedure cNeg ( complex value c ) ; - c;    % returns the inverse of c, using the builtin complex / operatror       %    complex procedure cInv ( complex value c ) ; 1 / c;    % returns the conjugate of c                                            %    complex procedure cConj ( complex value c ) ; realpart( c ) - imag( imagpart( c ) );    complex c, d;    c := 1 + 2i;    d := 3 + 4i;    % set I/O format for real aand complex numbers                          %    r_format := "A"; s_w := 0; r_w := 6; r_d := 2;    write( "c      : ",        c      );    write( "d      : ",           d   );    write( "c + d  : ", cAdd(  c, d ) );    write( "c * d  : ", cMul(  c, d ) );    write( "-c     : ", cNeg(  c    ) );    write( "1/c    : ", cInv(  c    ) );    write( "conj c : ", cConj( c    ) )end.
Output:
c      :   1.00   2.00I
d      :   3.00   4.00I
c + d  :   4.00   6.00I
c * d  :  -5.00  10.00I
-c     :  -1.00  -2.00I
1/c    :   0.20  -0.40I
conj c :   1.00  -2.00I


## APL

    x←1j1                ⍝assignment   y←5.25j1.5   x+y                  ⍝addition6.25J2.5   x×y                  ⍝multiplication3.75J6.75    ⌹x                  ⍝inversion0.5j_0.5    -x                  ⍝negation¯1J¯1

## 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...

## AutoHotkey

contributed by Laszlo on the ahk forum

Cset(C,1,1)MsgBox % Cstr(C)  ; 1 + i*1Cneg(C,C)MsgBox % Cstr(C)  ; -1 - i*1Cadd(C,C,C)MsgBox % Cstr(C)  ; -2 - i*2Cinv(D,C)MsgBox % Cstr(D)  ; -0.25 + 0.25*iCmul(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")}

## BASIC

Works with: QuickBasic version 4.5
TYPE complex        real AS DOUBLE        imag AS DOUBLEEND TYPEDECLARE 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)CLSDIM x AS complexDIM y AS complexDIM z AS complexx.real = 1x.imag = 1y.real = 2y.imag = 2CALL 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.imagEND 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 / denomEND 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.realEND SUB SUB neg (a AS complex, b AS complex)        b.real = -a.real        b.imag = -a.imagEND SUB
Output:
 3 + 3 i
0 + 4 i
.5 +-.5 i
-1 +-1 i

## 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

## 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

## 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");}

## 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);    }}

## 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;}

## 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)))))

## COBOL

The following is in the Managed COBOL dialect.

Works with: Visual COBOL

### .NET Complex class

Translation of: C#

## CoffeeScript

 # create an immutable Complex typeclass 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" # testdo ->  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


## 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

## 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 rEND 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 );


## 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

## 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()); }

## EchoLisp

Complex numbers are part of the language. No special library is needed.

 (define a 42+666i) → a(define b 1+i) → b(- a) → -42-666i ; negate(+ a b) → 43+667i ; add(* a b) → -624+708i ; multiply(/ b) → 0.5-0.5i ; invert(conjugate b) → 1-i(angle b) → 0.7853981633974483 ; = PI/4(magnitude b) → 1.4142135623730951 ; = sqrt(2)(exp (* I PI)) → -1+0i ; Euler = e^(I*PI) = -1

## Elixir

defmodule Complex do  import Kernel, except: [abs: 1, div: 2]   defstruct real: 0, imag: 0   def new(real, imag) do    %__MODULE__{real: real, imag: imag}  end   def add(a, b) do    {a, b} = convert(a, b)    new(a.real + b.real, a.imag + b.imag)  end   def sub(a, b) do    {a, b} = convert(a, b)    new(a.real - b.real, a.imag - b.imag)  end   def mul(a, b) do    {a, b} = convert(a, b)    new(a.real*b.real - a.imag*b.imag, a.imag*b.real + a.real*b.imag)  end   def div(a, b) do    {a, b} = convert(a, b)    divisor = abs2(b)    new((a.real*b.real + a.imag*b.imag) / divisor,        (a.imag*b.real - a.real*b.imag) / divisor)  end   def neg(a) do    a = convert(a)    new(-a.real, -a.imag)  end   def inv(a) do    a = convert(a)    divisor = abs2(a)    new(a.real / divisor, -a.imag / divisor)  end   def conj(a) do    a = convert(a)    new(a.real, -a.imag)  end   def abs(a) do    :math.sqrt(abs2(a))  end   defp abs2(a) do    a = convert(a)    a.real*a.real + a.imag*a.imag  end   defp convert(a) when is_number(a), do: new(a, 0)  defp convert(%__MODULE__{} = a), do: a   defp convert(a, b), do: {convert(a), convert(b)}   def task do    a = new(1, 3)    b = new(5, 2)    IO.puts "a = #{a}"    IO.puts "b = #{b}"    IO.puts "add(a,b): #{add(a, b)}"    IO.puts "sub(a,b): #{sub(a, b)}"    IO.puts "mul(a,b): #{mul(a, b)}"    IO.puts "div(a,b): #{div(a, b)}"    IO.puts "div(b,a): #{div(b, a)}"    IO.puts "neg(a)  : #{neg(a)}"    IO.puts "inv(a)  : #{inv(a)}"    IO.puts "conj(a) : #{conj(a)}"  endend defimpl String.Chars, for: Complex do  def to_string(%Complex{real: real, imag: imag}) do    if imag >= 0, do: "#{real}+#{imag}j",                else: "#{real}#{imag}j"  endend Complex.task
Output:
a = 1+3j
b = 5+2j
add(a,b): 6+5j
sub(a,b): -4+1j
mul(a,b): -1+17j
div(a,b): 0.3793103448275862+0.4482758620689655j
div(b,a): 1.1-1.3j
neg(a)  : -1-3j
inv(a)  : 0.1-0.3j
conj(a) : 1-3j


## 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+5iAns = -1+17iAns = -1-3iAns = 0.1-0.3iAns = 1-3i

 PROGRAM COMPLEX_ARITH TYPE COMPLEX=(REAL#,IMAG#) DIM X:COMPLEX,Y:COMPLEX,Z:COMPLEX !! complex arithmetic routines!DIM A:COMPLEX,B:COMPLEX,C:COMPLEX PROCEDURE ADD(A.,B.->C.)    C.REAL#=A.REAL#+B.REAL#    C.IMAG#=A.IMAG#+B.IMAG#END PROCEDURE PROCEDURE INV(A.->B.)  LOCAL DENOM#    DENOM#=A.REAL#^2+A.IMAG#^2    B.REAL#=A.REAL#/DENOM#    B.IMAG#=-A.IMAG#/DENOM#END PROCEDURE PROCEDURE MULT(A.,B.->C.)    C.REAL#=A.REAL#*B.REAL#-A.IMAG#*B.IMAG#    C.IMAG#=A.REAL#*B.IMAG#+A.IMAG#*B.REAL#END PROCEDURE PROCEDURE NEG(A.->B.)    B.REAL#=-A.REAL#    B.IMAG#=-A.IMAG#END PROCEDURE BEGIN    PRINT(CHR$(12);) !CLS X.REAL#=1 X.IMAG#=1 Y.REAL#=2 Y.IMAG#=2 ADD(X.,Y.->Z.) PRINT(Z.REAL#;" + ";Z.IMAG#;"i") MULT(X.,Y.->Z.) PRINT(Z.REAL#;" + ";Z.IMAG#;"i") INV(X.->Z.) PRINT(Z.REAL#;" + ";Z.IMAG#;"i") NEG(X.->Z.) PRINT(Z.REAL#;" + ";Z.IMAG#;"i")END PROGRAM  Note: Adapted from QuickBasic source code Output:  3 + 3 i 0 + 4 i .5 +-.5 i -1 +-1 i ## 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  ## Euphoria constant REAL = 1, IMAG = 2type 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 + bend 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 -aend 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 ifend function complex a, ba = { 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 ## 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  ## 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;}  ## 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 ## Forth Historically, there was no standard syntax or mechanism for complex numbers and several implementations suitable for different uses were provided. However later a wordset was standardised as "Algorithm #60". S" fsl-util.fs" REQUIREDS" complex.fs" REQUIRED zvariable xzvariable y1e 1e x z!pi 1.2e y z! x [email protected] y [email protected] z+ z.x [email protected] y [email protected] z* z.1e 0e zconstant 1+0i1+0i x [email protected] z/ z.x [email protected] znegate z. ## 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 / aend 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. Function imag(a) is possibly not recognised. Use aimag(a) if so. 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 ## FreeBASIC ' FB 1.05.0 Win64 Type Complex As Double real, imag Declare Constructor(real As Double, imag As Double) Declare Function invert() As Complex Declare Function conjugate() As Complex Declare Operator cast() As String End Type Constructor Complex(real As Double, imag As Double) This.real = real This.imag = imagEnd Constructor Function Complex.invert() As Complex Dim denom As Double = real * real + imag * imag Return Complex(real / denom, -imag / denom)End Function Function Complex.conjugate() As Complex Return Complex(real, -imag)End Function Operator Complex.Cast() As String If imag >= 0 Then Return Str(real) + "+" + Str(imag) + "j" End If Return Str(real) + Str(imag) + "j" End Operator Operator - (c As Complex) As Complex Return Complex(-c.real, -c.imag)End Operator Operator + (c1 As Complex, c2 As Complex) As Complex Return Complex(c1.real + c2.real, c1.imag + c2.imag)End Operator Operator - (c1 As Complex, c2 As Complex) As Complex Return c1 + (-c2)End Operator Operator * (c1 As Complex, c2 As Complex) As Complex Return Complex(c1.real * c2.real - c1.imag * c2.imag, c1.real * c2.imag + c2.real * c1.imag)End Operator Operator / (c1 As Complex, c2 As Complex) As Complex Return c1 * c2.invert End Operator Var x = Complex(1, 3)Var y = Complex(5, 2)Print "x = "; xPrint "y = "; yPrint "x + y = "; x + yPrint "x - y = "; x - yPrint "x * y = "; x * yPrint "x / y = "; x / yPrint "-x = "; -xPrint "1 / x = "; x.invertPrint "x* = "; x.conjugatePrintPrint "Press any key to quit"Sleep Output: x = 1+3j y = 5+2j x + y = 6+5j x - y = -4+1j x * y = -1+17j x / y = 0.3793103448275862+0.4482758620689655j -x = -1-3j 1 / x = 0.1-0.3j x* = 1-3j  ## 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 + ymultiply[x,y] := x * ynegate[x] := -xinvert[x] := 1/x // Could also use inv[x] or recip[x]conjugate[x] := Re[x] - Im[x] i a = 3 + 2.5ib = 7.3 - 10iprintln["$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 )


## Futhark

 type complex = (f64,f64) fun complexAdd((a,b): complex) ((c,d): complex): complex =  (a + c,   b + d) fun complexMult((a,b): complex) ((c,d): complex): complex = (a*c - b * d,  a*d + b * c) fun complexInv((r,i): complex): complex =  let denom = r*r + i * i  in (r / denom,      -i / denom) fun complexNeg((r,i): complex): complex =  (-r, -i) fun complexConj((r,i): complex): complex =  (r, -i) fun main (o: int) (a: complex) (b: complex): complex =  if      o == 0 then complexAdd a b  else if o == 1 then complexMult a b  else if o == 2 then complexInv a  else if o == 3 then complexNeg a  else                complexConj a

## 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 1i := 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;# truei/2 in GaussianIntegers;# falsei/2 in GaussianRationals;# trueSqrt(-3) in Cyclotomics;# true

## 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)


## 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 r, Number i = 0) { (real, imag) = [r, i] }     Complex(Map that) { (real, imag) = [that.real ?: 0, that.imag ?: 0] }     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 getAbs() { Math.sqrt( real*real + imag*imag ) }    /** the magnitude of this complex number. */    Number abs() { this.abs }     /** the reciprocal of this complex number. */    Complex getRecip() { (~this) / (ρ**2) }    /** the reciprocal of this complex number. */    Complex recip() { this.recip }     /** derived polar angle θ (theta) for polar form. Normalized to 0 ≤ θ < 2π. */    Number getTheta() {        def θ = Math.atan2(imag,real)        θ = θ < 0 ? θ + 2 * Math.PI : θ    }    /** derived polar angle θ (theta) for polar form. Normalized to 0 ≤ θ < 2π. */    Number getΘ() { this.theta } // this is greek uppercase theta     /** derived polar magnitude ρ (rho) for polar form. */    Number getRho() { this.abs }    /** derived polar magnitude ρ (rho) for polar form. */    Number getΡ() { this.abs } // this is greek uppercase rho, not roman P     /** Runs Euler's polar-to-Cartesian complex conversion,     * converting [ρ, θ] inputs into a [real, imag]-based complex number */    static Complex fromPolar(Number ρ, Number θ) {        [ρ * Math.cos(θ), ρ * Math.sin(θ)] as Complex    }     /** Creates new complex with same magnitude ρ, but different angle θ */    Complex withTheta(Number θ) { fromPolar(this.rho, θ) }    /** Creates new complex with same magnitude ρ, but different angle θ */    Complex withΘ(Number θ) { fromPolar(this.rho, θ) }     /** Creates new complex with same angle θ, but different magnitude ρ */    Complex withRho(Number ρ) { fromPolar(ρ, this.θ) }    /** Creates new complex with same angle θ, but different magnitude ρ */    Complex withΡ(Number ρ) { fromPolar(ρ, this.θ) } // this is greek uppercase rho, not roman P     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(that) {        that != null && (that instanceof Complex \                                ? [this.real, this.imag] == [that.real, that.imag] \                                : that instanceof Number && [this.real, this.imag] == [that, 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    }}

The following ComplexCategory class allows for modification of regular Number behavior when interacting with Complex.

import org.codehaus.groovy.runtime.DefaultGroovyMethods class ComplexCategory {    static Complex getI (Number a) { [0, a] as Complex }     static Complex plus (Number a, Complex b) { b + a }    static Complex minus (Number a, Complex b) { -b + a }    static Complex multiply (Number a, Complex b) { b * a }    static Complex div (Number a, Complex b) { ([a] as Complex) / b  }    static Complex power (Number a, Complex b) { ([a] as Complex) ** b }     static <T> T asType (Number a, Class<T> type) {        type == Complex \            ? [a] as Complex            : DefaultGroovyMethods.asType(a, type)    }}

Notice also that this solution takes liberal advantage of Groovy's full Unicode support, including support for non-English alphabets used in identifiers.

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

import static Complex.* Number.metaClass.mixin ComplexCategory def ε = 0.000000001  // tolerance (epsilon): acceptable "wrongness" to account for rounding error println 'Demo 1: functionality as requested'def a = [5,3] as Complexdef a1 = [real:5, imag:3] as Complexdef a2 = 5 + 3.idef a3 = 5 + 3*iassert a == a1 && a == a2 && a == a3println 'a == ' + adef b = [0.5,6] as Complexprintln 'b == ' + b println "a + b == (${a}) + (${b}) == " + (a + b)println "a * b == (${a}) * (${b}) == " + (a * b)assert a + (-a) == 0println "-a == -(${a}) == " + (-a)assert (a * a.recip - 1).abs < εprintln "1/a == (${a}).recip == " + (a.recip)println "a * 1/a == " + (a * a.recip)println() println 'Demo 2: other functionality not requested, but important for completeness'def c = 10def d = 10 as Complexassert d instanceof Complex && c instanceof Number && d == cassert a + c == c + aprintln "a + 10 == 10 + a == " + (c + a)assert c - a == -(a - c)println "10 - a == -(a - 10) == " + (c - a)println "a - b == (${a}) - (${b}) == " + (a - b)assert c * a == a * cprintln "10 * a == a * 10 == " + (c * a)assert (c / a - (a / c).recip).abs < εprintln "10 / a == 1 / (a / 10) == " + (c / a)println "a / b == (${a}) / (${b}) == " + (a / b)assert (a ** 2 - a * a).abs < εprintln "a ** 2 == a * a == " + (a ** 2)println "0.9 ** b == " + (0.9 ** b)println "a ** b == (${a}) ** (${b}) == " + (a ** b)println 'a.real == ' + a.realprintln 'a.imag == ' + a.imagprintln '|a| == ' + a.absprintln 'a.rho == ' + a.rhoprintln 'a.ρ == ' + a.ρprintln 'a.theta == ' + a.thetaprintln 'a.θ == ' + a.θprintln '~a (conjugate) == ' + ~a def ρ = 10def π = Math.PIdef n = 3def θ = π / n def fromPolar1 = fromPolar(ρ, θ)    // direct polar-to-cartesian conversiondef fromPolar2 = exp(θ.i) * ρ       // Euler's equationprintln "ρ*cos(θ) + i*ρ*sin(θ) == ${ρ}*cos(π/${n}) + i*${ρ}*sin(π/${n})"println "                      == 10*0.5      + i*10*√(3/4)    == " + fromPolar1println "ρ*exp(i*θ)            == ${ρ}*exp(i*π/${n})                == " + fromPolar2assert (fromPolar1 - fromPolar2).abs < ε
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
a * 1/a == 0.9999999998

Demo 2: other functionality not requested, but important for completeness
a + 10 == 10 + a == 15 + 3i
10 - a == -(a - 10) == 5 - 3i
a - b == (5 + 3i) - (0.5 + 6i) == 4.5 - 3i
10 * a == a * 10 == 50 + 30i
10 / a == 1 / (a / 10) == 1.4705882350 - 0.8823529410i
a / b == (5 + 3i) / (0.5 + 6i) == 0.5655172413793104 - 0.7862068965517242i
a ** 2 == a * a == 16.000000000000004 + 30.000000000000007i
0.9 ** b == 0.7653514303676113 - 0.5605686291920475i
a ** b == (5 + 3i) ** (0.5 + 6i) == -0.013750112198456855 - 0.09332524760169053i
a.real == 5
a.imag == 3
|a| == 5.830951894845301
a.rho == 5.830951894845301
a.ρ == 5.830951894845301
a.theta == 0.5404195002705842
a.θ == 0.5404195002705842
~a (conjugate) == 5 - 3i
ρ*cos(θ) + i*ρ*sin(θ) == 10*cos(π/3) + i*10*sin(π/3)
== 10*0.5      + i*10*√(3/4)    == 5.000000000000001 + 8.660254037844386i
ρ*exp(i*θ)            == 10*exp(i*π/3)                == 5.000000000000001 + 8.660254037844386i

## 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.

Output:
$jq -n -f complex.jq"x = [1,1]""y = [0,1]""x+y: [1,2]""x*y: [-1,1]""-x: [-1,-1]""1/x: [0.5,-0.5]""conj(x): [1,-1]""(x/y)*y: [1,1]""e^iπ: [-1,1.2246467991473532e-16]" ## Julia Julia has built-in support for complex arithmetic with arbitrary real types. julia> z1 = 1.5 + 3imjulia> z2 = 1.5 + 1.5imjulia> z1 + z23.0 + 4.5imjulia> z1 - z20.0 + 1.5imjulia> z1 * z2-2.25 + 6.75imjulia> z1 / z21.5 + 0.5imjulia> - z1-1.5 - 3.0imjulia> conj(z1), z1' # two ways to conjugate(1.5 - 3.0im,1.5 - 3.0im)julia> abs(z1)3.3541019662496847julia> z1^z2-1.102482955327779 - 0.38306415117199305imjulia> real(z1)1.5julia> imag(z1)3.0 ## Kotlin class Complex(private val real: Double, private val imag: Double) { operator fun plus(other: Complex) = Complex(real + other.real, imag + other.imag) operator fun times(other: Complex) = Complex( real * other.real - imag * other.imag, real * other.imag + imag * other.real ) fun inv(): Complex { val denom = real * real + imag * imag return Complex(real / denom, -imag / denom) } operator fun unaryMinus() = Complex(-real, -imag) operator fun minus(other: Complex) = this + (-other) operator fun div(other: Complex) = this * other.inv() fun conj() = Complex(real, -imag) override fun toString() = if (imag >= 0.0) "$real + ${imag}i" else "$real - ${-imag}i"} fun main(args: Array<String>) { val x = Complex(1.0, 3.0) val y = Complex(5.0, 2.0) println("x =$x")    println("y     =  $y") println("x + y =${x + y}")    println("x - y =  ${x - y}") println("x * y =${x * y}")    println("x / y =  ${x / y}") println("-x =${-x}")    println("1 / x =  ${x.inv()}") println("x* =${x.conj()}")}
Output:
x     =  1.0 + 3.0i
y     =  5.0 + 2.0i
x + y =  6.0 + 5.0i
x - y =  -4.0 + 1.0i
x * y =  -1.0 + 17.0i
x / y =  0.3793103448275862 + 0.4482758620689655i
-x    =  -1.0 - 3.0i
1 / x =  0.1 - 0.3i
x*    =  1.0 - 3.0i


## LFE

There is no native support for complex numbers in either LFE or Erlang. As such, this example shows how to implement complex support. There is, however, an LFE library that offers a complex number data type and many mathematical functions which support this data type: complex.

A convenient data structure for a complex number is the record:

 (defrecord complex  real  img)

Here are the required functions:

 (defun add  (((match-complex real r1 img i1)    (match-complex real r2 img i2))   (new (+ r1 r2) (+ i1 i2)))) (defun mult  (((match-complex real r1 img i1)    (match-complex real r2 img i2))   (new (- (* r1 r2) (* i1 i2))              (+ (* r1 i2) (* r2 i1))))) (defun neg  (((match-complex real r img i))   (new (* -1 r) (* -1 i)))) (defun inv (cmplx)  (div (conj cmplx) (modulus cmplx)))

Bonus:

 (defun conj  (((match-complex real r img i))   (new r (* -1 i))))

The functions above are built using the following supporting functions:

 (defun new (r i)  (make-complex real r img i)) (defun modulus (cmplx)  (mult cmplx (conj cmplx))) (defun div (c1 c2)   (let* ((denom (complex-real (modulus c2)))          (c3 (mult c1 (conj c2))))     (new (/ (complex-real c3) denom)          (/ (complex-img c3) denom)))))

Finally, we have some functions for use in the conversion and display of our complex number data structure:

 (defun ->str  (((match-complex real r img i)) (when (>= i 0))   (->str r i "+"))  (((match-complex real r img i))   (->str r i ""))) (defun ->str (r i pos)  (io_lib:format "~p ~s~pi" (,r ,pos ,i))) (defun print (cmplx)  (io:format (++ (->str cmplx) "~n"))) 

Usage is as follows:

> (set ans1 (add c1 c2))
#(complex 2.5 4.0)
> (set ans2 (mult c1 c2))
#(complex -1.5 4.5)
> (set ans3 (inv c2))
#(complex 0.5 -0.5)
> (set ans4 (conj c1))
#(complex 1.5 -3.0)


These can be printed in the following manner:

> (progn (lists:map #'print/1 (,ans1 ,ans2 ,ans3 ,ans4)) 'ok)
2.5 +4.0i
-1.5 +4.5i
0.5 -0.5i
1.5 -3.0i
ok


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 ## 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)  ## 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; ## Mathematica / Wolfram Language 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+2Iy=3+4I x+y => 4 + 6 Ix-y => -2 - 2 Iy x => -5 + 10 Iy/x => 11/5 - (2 I)/5x^3 => -11 - 2 Iy^4 => -527 - 336 Ix^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 LogSin Cos Tan Csc Sec CotArcSin ArcCos ArcTan ArcCsc ArcSec ArcCotSinh Cosh Tanh Csch Sech CothArcSinh ArcCosh ArcTanh ArcCsch ArcSech ArcCothSincHaversine InverseHaversine Factorial Gamma PolyGamma LogGammaErf 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. ## 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 ## 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 ## 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^2ИПD x^2 + П3 ИПA ИПC * ИПB ИПD *+ ИП3 / П1 ИПB ИПC * ИПA ИПD *- ИП3 / П2 ИП1 С/П ИПA ИПC * ИПBИПD * - 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 ## 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.WriteBfEND ShowComplex; PROCEDURE AddComplex (x1, x2 : Complex; VAR x3 : Complex); BEGIN x3.R := x1.R + x2.R; x3.Im := x1.Im + x2.ImEND AddComplex; PROCEDURE SubComplex (x1, x2 : Complex; VAR x3 : Complex); BEGIN x3.R := x1.R - x2.R; x3.Im := x1.Im - x2.ImEND 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.REND 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.ImEND 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.WriteLnEND 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 ## 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 $(Math.Abs(c.Imaginary))i" } Main() : void { def complex1 = Complex(1.0, 1.0); def complex2 = Complex(3.14159, 1.2); WriteLine(Add(complex1, complex2).PrettyPrint()); WriteLine(Multiply(complex1, complex2).PrettyPrint()); WriteLine(Negate(complex2).PrettyPrint()); WriteLine(Reciprocal(complex2).PrettyPrint()); WriteLine(Conjugate(complex2).PrettyPrint()); }} Output: 4.14159 + 2.2i 1.94159 + 4.34159i -3.14159 - 1.2i 0.277781124787984 - 0.106104663481097i 3.14159 - 1.2i ## Nim  import complexvar a: Complex = (1.0,1.0)var b: Complex = (3.1415,1.2) echo("a : " &$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)  ## 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 rEND 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)  ## 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)); ) ## Octave GNU Octave handles naturally complex numbers: z1 = 1.5 + 3i;z2 = 1.5 + 1.5i;disp(z1 + z2); % 3.0 + 4.5idisp(z1 - z2); % 0.0 + 1.5idisp(z1 * z2); % -2.25 + 6.75idisp(z1 / z2); % 1.5 + 0.5idisp(-z1); % -1.5 - 3idisp(z1'); % 1.5 - 3idisp(abs(z1)); % 3.3541 = sqrt(z1*z1')disp(z1 ^ z2); % -1.10248 - 0.38306idisp( exp(z1) ); % -4.43684 + 0.63246idisp( imag(z1) ); % 3disp( real(z2) ); % 1.5%... ## Oforth Object Class new: Complex(re, im) Complex method: re @re ;Complex method: im @im ; Complex method: initialize := im := re ;Complex method: << '(' <<c @re << ',' <<c @im << ')' <<c ; 0 1 Complex new const: I Complex method: ==(c -- b ) c re @re == c im @im == and ; Complex method: norm -- f @re sq @im sq + sqrt ; Complex method: conj -- c @re @im neg Complex new ; Complex method: +(c -- d ) c re @re + c im @im + Complex new ; Complex method: -(c -- d ) c re @re - c im @im - Complex new ; Complex method: *(c -- d) c re @re * c im @im * - c re @im * @re c im * + Complex new ; Complex method: inv | n | @re sq @im sq + >float ->n @re n / @im neg n / Complex new; Complex method: /( c -- d ) c self inv * ; Integer method: >complex self 0 Complex new ;Float method: >complex self 0 Complex new ; Usage : 3.2 >complex I * 2 >complex + .cr2 3 Complex new 1.2 >complex + .cr2 3 Complex new 1.2 >complex * .cr2 >complex 2 3 Complex new / .cr Output: (2,3.2) (3.2,3) (2.4,3.6) (0.307692307692308,-0.461538461538462)  ## Ol Ol supports complex numbers by default. Numbers must be entered manually in form A+Bi without spaces between elements, where A and B - numbers (can be rational), i - imaginary unit; or in functional form using function complex.  (define A 0+1i) ; manually entered numbers(define B 1+0i) (print (+ A B)); <== 1+i (print (- A B)); <== -1+i (print (* A B)); <== 0+i (print (/ A B)); <== 0+i (define C (complex 2/7 -3)) ; functional way (print "real part of " C " is " (car C)); <== real part of 2/7-3i is 2/7 (print "imaginary part of " C " is " (cdr C)); <== imaginary part of 2/7-3i is -3  ## ooRexx c1 = .complex~new(1, 2)c2 = .complex~new(3, 4)r = 7 say "c1 =" c1say "c2 =" c2say "r =" rsay "-c1 =" (-c1)say "c1 + r =" c1 + rsay "c1 + c2 =" c1 + c2say "c1 - r =" c1 - rsay "c1 - c2 =" c1 - c2say "c1 * r =" c1 * rsay "c1 * c2 =" c1 * c2say "inv(c1) =" c1~invsay "conj(c1) =" c1~conjugatesay "c1 / r =" c1 / rsay "c1 / c2 =" c1 / c2say "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 immutable 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 ## 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 @[email protected]+i if pp then @[email protected]+sizeof accum pp=0 else @[email protected] end ifend def method "load"() operands a.x=b.x a.y=b.yend method method "push"() i+=sizeof accumend method method "pop"() pp=1 i-=sizeof accumend method method "="() operands b.x=a.x b.y=a.yend method method "+"() operands a.x+=b.x a.y+=b.yend method method "-"() operands a.x-=b.x a.y-=b.yend 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.yend 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) * vend 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, 0z2 <= 2, 1z3 <= -2, 1z4 <= 2, 4z5 <= 1, 1 'EVALUATE COMPLEX EXPRESSIONS z1 = z2 * z3print "Z1 = "+z1.show 'RESULT -5.0, 0 z1 = z3+(z2.power(2))print "Z1 = "+z1.show 'RESULT 1.0, 5.0 z1 = z5/z4print "Z1 = "+z1.show 'RESULT 0.3, 0.1 z1 = z5/z1print "Z1 = "+z1.show 'RESULT 2.0, 4.0 z1 = z2/z4print "Z1 = "+z1.show 'RESULT -0.4, -0.3 z1 = z1*z4print "Z1 = "+z1.show 'RESULT 2.0, 1.0  ## 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; ## 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) } writelnend. 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.  ## 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    ~$a; # complex conjugate ## Perl 6 Works with: Rakudo version 2015.12 my$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


## Phix

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 sq_add(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 sq_uminus(a)end function function scomplex(complex a)sequence s = ""atom ar, ai    {ar, ai} = a    if ar!=0 then        s = sprintf("%g",ar)    end if     if ai!=0 then        if ai=1 then            s &= "+i"        elsif ai=-1 then            s &= "-i"        else            s &= sprintf("%+gi",ai)        end if    end if     if length(s)=0 then        return "0"    end if    return send function complex a, ba = { 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


## 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

## 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.           */

## 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 values1 +: 1 -> a;2 +: 5 -> b;a+b =>a*b =>1/a =>a-b =>a-a =>a/b =>a/a =>

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

## PowerShell

### Implementation

 class Complex {  [Double]$x [Double]$y  Complex() {      $this.x = 0$this.y = 0  }  Complex([Double]$x, [Double]$y) {      $this.x =$x      $this.y =$y  }  [Double]abs2() {return $this.x*$this.x + $this.y*$this.y}  [Double]abs() {return [math]::sqrt($this.abs2())} static [Complex]add([Complex]$m,[Complex]$n) {return [Complex]::new($m.x+$n.x,$m.y+$n.y)} static [Complex]mul([Complex]$m,[Complex]$n) {return [Complex]::new($m.x*$n.x -$m.y*$n.y,$m.x*$n.y +$n.x*$m.y)} [Complex]mul([Double]$k) {return [Complex]::new($k*$this.x, $k*$this.y)}  [Complex]negate() {return $this.mul(-1)} [Complex]conjugate() {return [Complex]::new($this.x, -$this.y)} [Complex]inverse() {return$this.conjugate().mul(1/$this.abs2())} [String]show() { if(0 -ge$this.y) {        return "$($this.x)+$($this.y)i"    } else {        return "$($this.x)$($this.y)i"    }  }  static [String]show([Complex]$other) { return$other.show()  }}$m = [complex]::new(3, 4)$n = [complex]::new(7, 6)"$m:$($m.show())""$n: $($n.show())""$m + $n: $([complex]::show([complex]::add($m,$n)))""$m * $n:$([complex]::show([complex]::mul($m,$n)))""negate $m:$($m.negate().show())""1/$m: $([complex]::show($m.inverse()))""conjugate $m:$([complex]::show($m.conjugate()))"  Output: $m: 3+4i
$n: 7+6i$m + $n: 10+10i$m * $n: -3+46i negate$m: -3-4i
1/$m: 0.12-0.16i conjugate$m: 3-4i


### Library

 function show([System.Numerics.Complex]$c) { if(0 -le$c.Imaginary) {        return "$($c.Real)+$($c.Imaginary)i"    } else {        return "$($c.Real)$($c.Imaginary)i"    }  }$m = [System.Numerics.Complex]::new(3, 4)$n = [System.Numerics.Complex]::new(7, 6)"$m:$(show $m)""$n: $(show$n)""$m + $n: $(show ([System.Numerics.Complex]::Add($m,$n)))""$m * $n:$(show ([System.Numerics.Complex]::Multiply($m,$n)))""negate $m:$(show ([System.Numerics.Complex]::Negate($m)))""1/$m: $(show ([System.Numerics.Complex]::Reciprocal($m)))""conjugate $m:$(show ([System.Numerics.Complex]::Conjugate($m)))"  Output: $m: 3+4i
$n: 7+6i$m + $n: 10+10i$m * $n: -3+46i negate$m: -3-4i
1/$m: 0.12-0.16i conjugate$m: 3-4i


## Sidef

var a = 1:1                 # Complex(1, 1)var b = 3.14159:1.25        # Complex(3.14159, 1.25) [   a + b,                  # addition    a * b,                  # multiplication    -a,                     # negation    a.inv,                  # multiplicative inverse    a.conj,                 # complex conjugate    a.abs,                  # abs    a.sqrt,                 # sqrt    b.re,                   # real    b.im,                   # imaginary].each { |c| say c }
Output:
4.14159+2.25i
1.89159+4.39159i
-1-i
0.5-0.5i
1-i
1.4142135623730950488016887242097
1.09868411346780996603980119524068+0.45508986056222734130435775782247i
3.14159
1.25

## 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.].

## 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.

## smart BASIC

Original author unknown {:o(

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

## 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

## 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 -> unitend; (* 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 *)

## Stata

mataC(2,3)2 + 3i a=2+3ib=1-2*i  a+b-5 + 3i a-b9 + 3i a*b-14 - 21i a/b-.285714286 - .428571429i -a-2 - 3i 1/a.153846154 - .230769231i conj(a)2 - 3i abs(a)3.605551275 arg(a).9827937232 exp(a)-7.31511009 + 1.04274366i log(a)1.28247468 + .982793723i end

## Swift

Works with: Swift version 2.0

Use a struct to create a complex number type in Swift. Math Operations can be added using operator overloading

 public struct Complex {     public let real : Double    public let imaginary : Double     public init(real inReal:Double, imaginary inImaginary:Double) {        real = inReal        imaginary = inImaginary    }     public static var i : Complex = Complex(real:0, imaginary: 1)    public static var zero : Complex = Complex(real: 0, imaginary: 0)     public var negate : Complex {        return Complex(real: -real, imaginary: -imaginary)    }     public var invert : Complex {        let d = (real*real + imaginary*imaginary)        return Complex(real: real/d, imaginary: -imaginary/d)    }     public var conjugate : Complex {        return Complex(real: real, imaginary: -imaginary)    } }public func + (left: Complex, right: Complex) -> Complex {     return Complex(real: left.real+right.real, imaginary: left.imaginary+right.imaginary)}public func * (left: Complex, right: Complex) -> Complex {     return Complex(real: left.real*right.real - left.imaginary*right.imaginary,        imaginary: left.real*right.imaginary+left.imaginary*right.real)}public prefix func - (right:Complex) -> Complex {    return right.negate} // Checking equality is almost necessary for a struct of this type  to be usefulextension Complex : Equatable {}public func == (left:Complex, right:Complex) -> Bool {    return left.real == right.real && left.imaginary == right.imaginary}

Make the Complex Number struct printable and easier to debug by adding making it conform to CustomStringConvertible

  extension Complex : CustomStringConvertible {     public var description : String {         guard real != 0 || imaginary != 0 else { return "0" }         let rs : String = real != 0 ? "\(real)" : ""        let iS : String        let sign : String        let iSpace = real != 0 ? " " : ""        switch imaginary {        case let i where i < 0:            sign = "-"            iS = i == -1 ? "i" : "\(-i)i"        case let i where i > 0:            sign = real != 0 ? "+" : ""            iS = i == 1 ? "i" : "\(i)i"        default:            sign = ""            iS = ""        }        return "\(rs)\(iSpace)\(sign)\(iSpace)\(iS)"    }}

Explicitly support subtraction and division

 public func - (left:Complex, right:Complex) -> Complex {    return left + -right} public func / (divident:Complex, divisor:Complex) -> Complex {    let rc = divisor.conjugate    let num = divident * rc    let den = divisor * rc    return Complex(real: num.real/den.real, imaginary: num.imaginary/den.real)}

## Tcl

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

## 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.

## 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.

## UNIX Shell

Works with: ksh93
typeset -T Complex_t=(    float real=0    float imag=0     function to_s {         print -- "${_.real} +${_.imag} i"    }     function dup {        nameref other=$1 _=( real=${other.real} imag=${other.imag} ) } function add { typeset varname for varname; do nameref other=$varname            (( _.real += other.real ))            (( _.imag += other.imag ))        done    }     function negate {        (( _.real *= -1 ))        (( _.imag *= -1 ))    }     function conjugate {        (( _.imag *= -1 ))    }     function multiply {        typeset varname        for varname; do            nameref other=$varname float a=${_.real} b=${_.imag} c=${other.real} d=${other.imag} (( _.real = a*c - b*d )) (( _.imag = b*c + a*d )) done } function inverse { if (( _.real == 0 && _.imag == 0 )); then print -u2 "division by zero" return 1 fi float denom=$(( _.real*_.real + _.imag*_.imag ))        (( _.real = _.real / denom ))        (( _.imag = -1 * _.imag / denom ))    }) Complex_t a=(real=1 imag=1)a.to_s        # 1 + 1 i Complex_t b=(real=3.14159 imag=1.2)b.to_s        # 3.14159 + 1.2 i Complex_t cc.add a bc.to_s        # 4.14159 + 2.2 i c.negatec.to_s        # -4.14159 + -2.2 i c.conjugatec.to_s        # -4.14159 + 2.2 i c.dup ac.multiply bc.to_s        # 1.94159 + 4.34159 i Complex_t d=(real=2 imag=1)d.inversed.to_s        # 0.4 + -0.2 i

## 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+00iv = 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>

## Wortel

Translation of: CoffeeScript
@class Complex {  &[r i] @: {    ^r || r 0    ^i || i 0    ^m [email protected]^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

## XPL0

include c:\cxpl\codes; func real CAdd(A, B, C);        \Return complex sum of two complex numbersreal 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 numbersreal 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 numberreal A, C;[C(0):= -A(0); C(1):= -A(1);return C;]; func real CInv(A, C);           \Return inversion (reciprical) of complex numberreal 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 numberreal A, C;[C(0):= A(0); C(1):=-A(1);return C;]; proc COut(D, A);                \Output a complex number to specified deviceint 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


## Yabasic

rem ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++rem 	CADDI/CADDR  addition of complex numbers  Z1 + Z2    with Z1 = a1 + b1 *i   Z2 = a2 + b2*irem                CADDI returns imaginary part and CADDR the real partrem ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++export sub caddi( a1 , b1 , a2 , b2)    return (b1 + b2)end subexport sub caddr( a1 , b1 , a2 , b2)    return (a1 + a2)end sub rem ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++rem   CDIVI/CDIVR  division of complex numbers  Z1 / Z2        with Z1 = r + s *i   Z2 = t + u*irem                CDIVI returns imaginary part and CDIVR the real partrem ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++export sub cdivi(r,s,t,u)    return ((s*t- u*r) / (t^2 + u^2))end subexport sub cdivr( r , s , t , u)    return ((r*t- s*u) / (t^2 + u^2))end sub rem ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++rem   CMULI/CMULR  multiplication of complex numbers  Z1 * Z2, with Z1 = r + s *i   Z2 = t + u*irem                CMULI returns imaginary part and CMULR the real partrem ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++export sub cmuli( r , s , t , u)    return (r * u + s * t)end subexport sub cmulr( r , s , t , u)    return (r * t - s * u)end sub rem ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++rem   CSUBI/CSUBR  subtraction of complex numbers Z1 - Z2  with Z1 = a1 + b1 *i   Z2 = a2 + b2*irem                CSUBI returns imaginary part and CSUBR the real partrem ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++export sub csubi( a1 , b1 , a2 , b2)    return (b1 - b2)end subexport sub csubr( a1 , b1 , a2 , b2)    return (a1 - a2)end sub if (peek\$("library") = "main") then    print "Example: Z1 + Z2 with Z1 = 3 +2i , Z2 = 1-3i: Z1 + Z2 = 4 -1i"    print caddr(3,2,1,-2), "/", caddi(3,2,1,-3)   //   4/-1end if

## zkl

var [const] GSL=Import("zklGSL");    // libGSL (GNU Scientific Library)(GSL.Z(3,4) + GSL.Z(1,2)).println(); // (4.00+6.00i)(GSL.Z(3,4) - GSL.Z(1,2)).println(); // (2.00+2.00i)(GSL.Z(3,4) * GSL.Z(1,2)).println(); // (-5.00+10.00i)(GSL.Z(3,4) / GSL.Z(1,2)).println(); // (2.20-0.40i)(GSL.Z(1,0) / GSL.Z(1,1)).println(); // (0.50-0.50i) // inversion(-GSL.Z(3,4)).println(); 	     // (-3.00-4.00i)GSL.Z(3,4).conjugate().println();    // (3.00-4.00i)
Output:
(4.00+6.00i)
(2.00+2.00i)
(-5.00+10.00i)
(2.20-0.40i)
(0.50-0.50i)
(-3.00-4.00i)
(3.00-4.00i)


## zonnon

 module Numbers;type	{public,immutable} 	Complex = record		re,im: real;	end Complex; operator {public} "+" (a,b: Complex): Complex;var	r: Complex;begin	r.re := a.re + b.re;	r.im := a.im + b.im;	return rend "+"; operator {public} "-" (a,b: Complex): Complex;var	r: Complex;begin	r.re := a.re - b.re;	r.im := a.im - b.im;	return rend "-"; operator {public} "*" (a,b: Complex): Complex;var	r: Complex;begin	r.re := a.re*b.re - a.im*b.im;	r.im := a.re*b.im + a.im*b.re;	return rend "*"; operator {public} "/" (a,b: Complex): Complex;var	r: Complex;	d: real;begin	d := b.re * b.re + b.im * b.im;	r.re := (a.re * b.re + a.im * b.im)/d;	r.im := (a.im * b.re - a.re * b.im)/d;	return rend "/"; operator {public} "-" (a: Complex): Complex;begin	a.im := -1 * a.im;	return aend "-"; operator {public} "~" (a: Complex): Complex;var	d: real;	c: Complex;begin	d := a.re * a.re + a.im * a.im;	c.re := a.re/d;	c.im := (-1.0 * a.im)/d;	return cend "~"; end Numbers.  module Main;import Numbers; var	a,b,c: Numbers.Complex; 	procedure Writeln(c: Numbers.Complex);	begin		writeln("(",c.re:4:2,";",c.im:4:2,"i)");	end Writeln; 	procedure NewComplex(x,y: real): Numbers.Complex;	var		r: Numbers.Complex;	begin		r.re := x;r.im := y;		return r	end NewComplex; begin	a := NewComplex(1.5,3.0);	b := NewComplex(1.0,1.0);	Writeln(a + b);	Writeln(a - b);	Writeln(a * b);	Writeln(a / b);	Writeln(-a);	Writeln(~b);end Main. 
Output:
   ( 2,5   ;   4  i)
(  ,5   ;   2  i)
(-1,5   ; 4,5  i)
(2,25   ; ,75  i)
( 1,5   ;  -3  i)
(  ,5   ; -,5  i)


## ZX Spectrum Basic

Translation of: BBC BASIC
5 LET complex=2: LET r=1: LET i=210 DIM a(complex): LET a(r)=1.0: LET a(i)=1.020 DIM b(complex): LET b(r)=PI: LET b(i)=1.230 DIM o(complex)40 REM add50 LET o(r)=a(r)+b(r)60 LET o(i)=a(i)+b(i)70 PRINT "Result of addition is:": GO SUB 100080 REM mult90 LET o(r)=a(r)*b(r)-a(i)*b(i)100 LET o(i)=a(i)*b(r)+a(r)*b(i)110 PRINT "Result of multiplication is:": GO SUB 1000120 REM neg130 LET o(r)=-a(r)140 LET o(i)=-a(i)150 PRINT "Result of negation is:": GO SUB 1000160 LET denom=a(r)^2+a(i)^2170 LET o(r)=a(r)/denom180 LET o(i)=-a(i)/denom190 PRINT "Result of inversion is:": GO SUB 1000200 STOP 1000 IF o(i)>=0 THEN PRINT o(r);" + ";o(i);"i": RETURN 1010 PRINT o(r);" - ";-o(i);"i": RETURN  
Output:
Result of addition is:
4.1415927 + 2.2i
Result of multiplication is:
1.9415927 + 4.3415927i
Result of negation is:
-1 - 1i
Result of inversion is:
0.5 - 0.5i`