Arithmetic/Complex

Arithmetic/Complex
You are encouraged to solve this task according to the task description, using any language you may know.

A   complex number   is a number which can be written as: $a+b\times i$ (sometimes shown as: $b+a\times i$ where   $a$   and   $b$  are real numbers,   and   $i$   is   √ -1 

Typically, complex numbers are represented as a pair of real numbers called the "imaginary part" and "real part",   where the imaginary part is the number to be multiplied by $i$.

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 $a+bi$ is $a-bi$

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.

11l

Translation of: Python

V z1 = 1.5 + 3i
V z2 = 1.5 + 1.5i
print(z1 + z2)
print(z1 - z2)
print(z1 * z2)
print(z1 / z2)
print(-z1)
print(conjugate(z1))
print(abs(z1))
print(z1 ^ z2)
print(z1.real)
print(z1.imag)

Output:

3+4.5i
1.5i
-2.25+6.75i
1.5+0.5i
-1.5-3i
1.5-3i
3.3541
-1.10248-0.383064i
1.5
3

Action!

Library: Action! Tool Kit

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

DEFINE R_="+0"
DEFINE I_="+6"
TYPE Complex=[CARD cr1,cr2,cr3,ci1,ci2,ci3]

BYTE FUNC Positive(REAL POINTER x)
  BYTE ARRAY tmp

  tmp=x
  IF (tmp(0)&$80)=$00 THEN
    RETURN (1)
  FI
RETURN (0)

PROC PrintComplex(Complex POINTER x)
  PrintR(x R_)
  IF Positive(x I_) THEN
    Put('+)
  FI
  PrintR(x I_) Put('i)
RETURN

PROC PrintComplexXYZ(Complex POINTER x,y,z CHAR ARRAY s)
  Print("(") PrintComplex(x)
  Print(") ") Print(s)
  Print(" (") PrintComplex(y)
  Print(") = ") PrintComplex(z)
  PutE()
RETURN

PROC PrintComplexXY(Complex POINTER x,y CHAR ARRAY s)
  Print(s)
  Print("(") PrintComplex(x)
  Print(") = ") PrintComplex(y)
  PutE()
RETURN

PROC ComplexAdd(Complex POINTER x,y,res)
  RealAdd(x R_,y R_,res R_) ;res.r=x.r+y.r
  RealAdd(x I_,y I_,res I_) ;res.i=x.i+y.i
RETURN

PROC ComplexSub(Complex POINTER x,y,res)
  RealSub(x R_,y R_,res R_) ;res.r=x.r-y.r
  RealSub(x I_,y I_,res I_) ;res.i=x.i-y.i
RETURN

PROC ComplexMult(Complex POINTER x,y,res)
  REAL tmp1,tmp2

  RealMult(x R_,y R_,tmp1)  ;tmp1=x.r*y.r
  RealMult(x I_,y I_,tmp2)  ;tmp2=x.i*y.i
  RealSub(tmp1,tmp2,res R_) ;res.r=x.r*y.r-x.i*y.i

  RealMult(x R_,y I_,tmp1)  ;tmp1=x.r*y.i
  RealMult(x I_,y R_,tmp2)  ;tmp2=x.i*y.r
  RealAdd(tmp1,tmp2,res I_) ;res.i=x.r*y.i+x.i*y.r
RETURN

PROC ComplexDiv(Complex POINTER x,y,res)
  REAL tmp1,tmp2,tmp3,tmp4

  RealMult(x R_,y R_,tmp1)  ;tmp1=x.r*y.r
  RealMult(x I_,y I_,tmp2)  ;tmp2=x.i*y.i
  RealAdd(tmp1,tmp2,tmp3)   ;tmp3=x.r*y.r+x.i*y.i
  RealMult(y R_,y R_,tmp1)  ;tmp1=y.r^2
  RealMult(y I_,y I_,tmp2)  ;tmp2=y.i^2
  RealAdd(tmp1,tmp2,tmp4)   ;tmp4=y.r^2+y.i^2
  RealDiv(tmp3,tmp4,res R_) ;res.r=(x.r*y.r+x.i*y.i)/(y.r^2+y.i^2)

  RealMult(x I_,y R_,tmp1)  ;tmp1=x.i*y.r
  RealMult(x R_,y I_,tmp2)  ;tmp2=x.r*y.i
  RealSub(tmp1,tmp2,tmp3)   ;tmp3=x.i*y.r-x.r*y.i
  RealDiv(tmp3,tmp4,res I_) ;res.i=(x.i*y.r-x.r*y.i)/(y.r^2+y.i^2)
RETURN

PROC ComplexNeg(Complex POINTER x,res)
  REAL neg

  ValR("-1",neg)            ;neg=-1
  RealMult(x R_,neg,res R_) ;res.r=-x.r
  RealMult(x I_,neg,res I_) ;res.r=-x.r
RETURN

PROC ComplexInv(Complex POINTER x,res)
  REAL tmp1,tmp2,tmp3

  RealMult(x R_,x R_,tmp1)  ;tmp1=x.r^2
  RealMult(x I_,x I_,tmp2)  ;tmp2=x.i^2
  RealAdd(tmp1,tmp2,tmp3)   ;tmp3=x.r^2+x.i^2
  RealDiv(x R_,tmp3,res R_) ;res.r=x.r/(x.r^2+x.i^2)

  ValR("-1",tmp1)           ;tmp1=-1
  RealMult(x I_,tmp1,tmp2)  ;tmp2=-x.i
  RealDiv(tmp2,tmp3,res I_) ;res.i=-x.i/(x.r^2+x.i^2)
RETURN

PROC ComplexConj(Complex POINTER x,res)
  REAL neg

  ValR("-1",neg)            ;neg=-1
  RealAssign(x R_,res R_)   ;res.r=x.r
  RealMult(x I_,neg,res I_) ;res.i=-x.i
RETURN

PROC Main()
  Complex x,y,res

  IntToReal(5,x R_) IntToReal(3,x I_)
  IntToReal(4,y R_) ValR("-3",y I_)

  Put(125) PutE() ;clear screen

  ComplexAdd(x,y,res)
  PrintComplexXYZ(x,y,res,"+")

  ComplexSub(x,y,res)
  PrintComplexXYZ(x,y,res,"-")

  ComplexMult(x,y,res)
  PrintComplexXYZ(x,y,res,"*")

  ComplexDiv(x,y,res)
  PrintComplexXYZ(x,y,res,"/")

  ComplexNeg(y,res)
  PrintComplexXY(y,res,"        -")

  ComplexInv(y,res)
  PrintComplexXY(y,res,"     1 / ")

  ComplexConj(y,res)
  PrintComplexXY(y,res,"     conj")
RETURN

Output:

Screenshot from Atari 8-bit computer

(5+3i) + (4-3i) = 9+0i
(5+3i) - (4-3i) = 1+6i
(5+3i) * (4-3i) = 29-3i
(5+3i) / (4-3i) = .44+1.08i
        -(4-3i) = -4+3i
     1 / (4-3i) = .16+.12i
     conj(4-3i) = 4+3i

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
   Put("Conjugate(-A) = ");
   C := Conjugate (C); Put(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                  ⍝addition
6.25J2.5
   x×y                  ⍝multiplication
3.75J6.75
    ⌹x                  ⍝inversion
0.5j_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...

Arturo

a: to :complex [1 1]
b: to :complex @[pi 1.2]

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

print ["a + b:" a + b]
print ["a * b:" a * b]
print ["1 / a:" 1 / a]
print ["neg a:" neg a]
print ["conj a:" conj a]

Output:

a: 1.0+1.0i 
b: 3.141592653589793+1.2i 
a + b: 4.141592653589793+2.2i 
a * b: 1.941592653589793+4.341592653589793i 
1 / a: 0.5-0.5i 
neg a: -1.0-1.0i 
conj a: 1.0-1.0i

AutoHotkey

contributed by Laszlo on the ahk forum

Cset(C,1,1)
MsgBox % Cstr(C)  ; 1 + i*1
Cneg(C,C)
MsgBox % Cstr(C)  ; -1 - i*1
Cadd(C,C,C)
MsgBox % Cstr(C)  ; -2 - i*2
Cinv(D,C)
MsgBox % Cstr(D)  ; -0.25 + 0.25*i
Cmul(C,C,D)
MsgBox % Cstr(C)  ; 1 + i*0

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

AWK

contributed by af

# simulate a struct using associative arrays
function complex(arr, re, im) {
    arr["re"] = re
    arr["im"] = im
}

function re(cmplx) {
    return cmplx["re"]
}

function im(cmplx) {
    return cmplx["im"]
}

function printComplex(cmplx) {
    print re(cmplx), im(cmplx)
}

function abs2(cmplx) {
    return re(cmplx) * re(cmplx) + im(cmplx) * im(cmplx)
}

function abs(cmplx) {
    return sqrt(abs2(cmplx))
}

function add(res, cmplx1, cmplx2) {
    complex(res, re(cmplx1) + re(cmplx2), im(cmplx1) + im(cmplx2)) 
}

function mult(res, cmplx1, cmplx2) {
    complex(res, re(cmplx1) * re(cmplx2) - im(cmplx1) * im(cmplx2), re(cmplx1) * im(cmplx2) + im(cmplx1) * re(cmplx2))
}

function scale(res, cmplx, scalar) {
    complex(res, re(cmplx) * scalar, im(cmplx) * scalar)
}

function negate(res, cmplx) {
    scale(res, cmplx, -1)
}

function conjugate(res, cmplx) {
    complex(res, re(cmplx), -im(cmplx))
}

function invert(res, cmplx) {
    conjugate(res, cmplx)
    scale(res, res, 1 / abs(cmplx))
}

BEGIN {
    complex(i, 0, 1)
    mult(i, i, i)
    printComplex(i)
}

BASIC

Works with: QuickBasic version 4.5

TYPE complex
        real AS DOUBLE
        imag AS DOUBLE
END TYPE

DECLARE SUB suma (a AS complex, b AS complex, c AS complex)
DECLARE SUB rest (a AS complex, b AS complex, c AS complex)
DECLARE SUB mult (a AS complex, b AS complex, c AS complex)
DECLARE SUB divi (a AS complex, b AS complex, c AS complex)
DECLARE SUB neg (a AS complex, b AS complex)
DECLARE SUB inv (a AS complex, b AS complex)
DECLARE SUB conj (a AS complex, b AS complex)

CLS
DIM x AS complex
DIM y AS complex
DIM z AS complex
x.real = 1
x.imag = 1
y.real = 2
y.imag = 2

PRINT "Siendo x = "; x.real; "+"; x.imag; "i"
PRINT "     e y = "; y.real; "+"; y.imag; "i"
PRINT
CALL suma(x, y, z)
PRINT "x + y = "; z.real; "+"; z.imag; "i"
CALL rest(x, y, z)
PRINT "x - y = "; z.real; "+"; z.imag; "i"
CALL mult(x, y, z)
PRINT "x * y = "; z.real; "+"; z.imag; "i"
CALL divi(x, y, z)
PRINT "x / y = "; z.real; "+"; z.imag; "i"
CALL neg(x, z)
PRINT "   -x = "; z.real; "+"; z.imag; "i"
CALL inv(x, z)
PRINT "1 / x = "; z.real; "+"; z.imag; "i"
CALL conj(x, z)
PRINT "   x* = "; z.real; "+"; z.imag; "i"
END

SUB suma (a AS complex, b AS complex, c AS complex)
        c.real = a.real + b.real
        c.imag = a.imag + b.imag
END SUB

SUB inv (a AS complex, b AS complex)
        denom = a.real ^ 2 + a.imag ^ 2
        b.real = a.real / denom
        b.imag = -a.imag / denom
END SUB

SUB mult (a AS complex, b AS complex, c AS complex)
        c.real = a.real * b.real - a.imag * b.imag
        c.imag = a.real * b.imag + a.imag * b.real
END SUB

SUB neg (a AS complex, b AS complex)
        b.real = -a.real
        b.imag = -a.imag
END SUB

SUB conj (a AS complex, b AS complex)
    b.real =  a.real
    b.imag = -a.imag
END SUB

SUB divi (a AS complex, b AS complex, c AS complex)
    c.real = ((a.real * b.real + b.imag * a.imag) / (b.real ^ 2 + b.imag ^ 2))
    c.imag = ((a.imag * b.real - a.real * b.imag) / (b.real ^ 2 + b.imag ^ 2))
END SUB

SUB rest (a AS complex, b AS complex, c AS complex)
    c.real = a.real - b.real
    c.imag = a.imag - b.imag
END SUB

Output:

Siendo x =  1+ 3i
     e y =  5+ 2i

x + y =  6 + 5 i
x - y = -4 + 1 i
x * y = -1 + 17 i
x / y =  .3793103448275862 + .4482758620689655 i
   -x = -1 +-3 i
1 / x =  .1 +-.3 i
   x* =  1 +-3 i

BBC BASIC

Works with: BBC BASIC for Windows

      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# 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# 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#

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

Implementation

      $SET SOURCEFORMAT "FREE"
class-id Prog.
method-id. Main static.
procedure division.
    declare a as type Complex = new Complex(1, 1)
    declare b as type Complex = new Complex(3.14159, 1.25)
    
    display "a = " a
    display "b = " b
    display space
    
    declare result as type Complex = a + b
    display "a + b = " result
    move (a - b) to result
    display "a - b = " result
    move (a * b) to result
    display "a * b = " result
    move (a / b) to result
    display "a / b = " result
    move (- b) to result
    display "-b = " result
    display space
    
    display "Inverse of b: " type Complex::Inverse(b)
    display "Conjugate of b: " type Complex::Conjugate(b)
end method.
end class.      
      
class-id Complex.

01  Real                               float-long property.
01  Imag                               float-long property.

method-id new.
    set Real, Imag to 0
end method.

method-id new.
procedure division using value real-val as float-long, imag-val as float-long.
    set Real to real-val
    set Imag to imag-val
end method.

method-id Norm static.
procedure division using value a as type Complex returning ret as float-long.
    compute ret = a::Real ** 2 + a::Imag ** 2
end method.

method-id Inverse static.
procedure division using value a as type Complex returning ret as type Complex.
    declare norm as float-long = type Complex::Norm(a)
    set ret to new Complex(a::Real / norm, (0 - a::Imag) / norm)
end method.

method-id Conjugate static.
procedure division using value a as type Complex returning c as type Complex.
    set c to new Complex(a::Real, 0 - a::Imag)
end method.

method-id ToString override.
procedure division returning str as string.
    set str to type String::Format("{0}{1:+#0;-#}i", Real, Imag)
end method.

operator-id + .
procedure division using value a as type Complex, b as type Complex
        returning c as type Complex.
    set c to new Complex(a::Real + b::Real, a::Imag + b::Imag)
end operator.

operator-id - .
procedure division using value a as type Complex, b as type Complex
        returning c as type Complex.
    set c to new Complex(a::Real - b::Real, a::Imag - b::Imag)
end operator.

operator-id * .
procedure division using value a as type Complex, b as type Complex
        returning c as type Complex.
    set c to new Complex(a::Real * b::Real - a::Imag * b::Imag,
        a::Real * b::Imag + a::Imag * b::Real)
end operator.

operator-id / .
procedure division using value a as type Complex, b as type Complex
        returning c as type Complex.
    set c to new Complex()        
    declare b-norm as float-long = type Complex::Norm(b)
    compute c::Real = (a::Real * b::Real + a::Imag * b::Imag) / b-norm
    compute c::Imag = (a::Imag * b::Real - a::Real * b::Imag) / b-norm
end operator.

operator-id - .
procedure division using value a as type Complex returning ret as type Complex.
    set ret to new Complex(- a::Real, 0 - a::Imag)
end operator.

end class.

CoffeeScript

# create an immutable Complex type
class Complex
  constructor: (@r=0, @i=0) ->
    @magnitude = @r*@r + @i*@i

  plus: (c2) ->
    new Complex(
      @r + c2.r,
      @i + c2.i
    )

  times: (c2) ->
    new Complex(
      @r*c2.r - @i*c2.i,
      @r*c2.i + @i*c2.r
    )

  negation: ->
    new Complex(
      -1 * @r,
      -1 * @i
    )

  inverse: ->
    throw Error "no inverse" if @magnitude is 0
    new Complex(
      @r / @magnitude,
      -1 * @i / @magnitude
    )

  toString: ->
    return "#{@r}" if @i == 0
    return "#{@i}i" if @r == 0
    if @i > 0
      "#{@r} + #{@i}i"
    else
      "#{@r} - #{-1 * @i}i"
      
# test
do ->
  a = new Complex(5, 3)
  b = new Complex(4, -3)
  
  sum = a.plus b
  console.log "(#{a}) + (#{b}) = #{sum}"
  
  product = a.times b
  console.log "(#{a}) * (#{b}) = #{product}"
  
  negation = b.negation()
  console.log "-1 * (#{b}) = #{negation}"
  
  diff = a.plus negation
  console.log "(#{a}) - (#{b}) = #{diff}"
  
  inverse = b.inverse()
  console.log "1 / (#{b}) = #{inverse}"
  
  quotient = product.times inverse
  console.log "(#{product}) / (#{b}) = #{quotient}"

Output:

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

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 r
END New;

PROCEDURE (x: Complex) Add*(y: Complex): Complex,NEW;
BEGIN
        RETURN New(x.r + y.r,x.i + y.i)
END Add;
 
PROCEDURE ( x: Complex) Sub*( y: Complex): Complex, NEW;
BEGIN
        RETURN New(x.r - y.r,x.i - y.i)
END Sub;
 
PROCEDURE ( x: Complex) Mul*( y: Complex): Complex, NEW;
BEGIN
        RETURN New(x.r*y.r - x.i*y.i,x.r*y.i + x.i*y.r)
END Mul;
 
PROCEDURE ( x: Complex) Div*( y: Complex): Complex, NEW;
VAR
        d: REAL;
BEGIN
        d := y.r * y.r + y.i * y.i;
        RETURN New((x.r*y.r + x.i*y.i)/d,(x.i*y.r - x.r*y.i)/d)
END Div;
 
(* Reciprocal *)
PROCEDURE (x: Complex) Rec*(): Complex,NEW;
VAR
        d: REAL;
BEGIN
        d := x.r * x.r + x.i * x.i;
        RETURN New(x.r/d,(-1.0 * x.i)/d);
END Rec;
 
(* Conjugate *)
PROCEDURE (x: Complex) Con*(): Complex,NEW;
BEGIN
        RETURN New(x.r, (-1.0) * x.i);
END Con;
 
PROCEDURE (x: Complex) Out(),NEW;
BEGIN
	   StdLog.String("Complex(");
	   StdLog.Real(x.r);StdLog.String(',');StdLog.Real(x.i);
	   StdLog.String("i );")
END Out;

PROCEDURE Do*;
BEGIN
        x := New(1.5,3);
        y := New(1.0,1.0);
 
        StdLog.String("x: ");x.Out();StdLog.Ln;
        StdLog.String("y: ");y.Out();StdLog.Ln;        
        r := x.Add(y);
        StdLog.String("x + y: ");r.Out();StdLog.Ln;
        r := x.Sub(y);
        StdLog.String("x - y: ");r.Out();StdLog.Ln;
        r := x.Mul(y);
        StdLog.String("x * y: ");r.Out();StdLog.Ln;
        r := x.Div(y);
        StdLog.String("x / y: ");r.Out();StdLog.Ln;
        r := y.Rec();
        StdLog.String("1 / y: ");r.Out();StdLog.Ln;
        r := x.Con();
        StdLog.String("x': ");r.Out();StdLog.Ln;
END Do;

END Complex.

Execute: ^Q Complex.Do

Output:

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

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());

}

Delphi

Library: System.SysUtils

Library: System.VarCmplx

program Arithmetic_Complex;

{$APPTYPE CONSOLE}

uses
  System.SysUtils,
  System.VarCmplx;

var
  a, b: Variant;

begin
  a := VarComplexCreate(5, 3);
  b := VarComplexCreate(0.5, 6.0);

  writeln(format('(%s) + (%s) = %s',[a,b, a+b]));

  writeln(format('(%s) * (%s) = %s',[a,b, a*b]));

  writeln(format('-(%s) = %s',[a,- a]));

  writeln(format('1/(%s) = %s',[a,1/a]));

  writeln(format('conj(%s) = %s',[a,VarComplexConjugate(a)]));

  Readln;
end.

Output:

(5 + 3i) + (0,5 + 6i) = 5,5 + 9i
(5 + 3i) * (0,5 + 6i) = -15,5 + 31,5i
-(5 + 3i) = -5 - 3i
1/(5 + 3i) = 0,147058823529412 - 0,0882352941176471i
conj(5 + 3i) = 5 - 3i

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)}"
  end
end

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"
  end
end

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

ERRE

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 = 2
type complex(sequence s)
    return length(s) = 2 and atom(s[REAL]) and atom(s[IMAG])
end type

function add(complex a, complex b)
    return a + b
end function

function mult(complex a, complex b)
    return {a[REAL] * b[REAL] - a[IMAG] * b[IMAG],
        a[REAL] * b[IMAG] + a[IMAG] * b[REAL]}
end function

function inv(complex a)
    atom denom
    denom = a[REAL] * a[REAL] + a[IMAG] * a[IMAG]
    return {a[REAL] / denom, -a[IMAG] / denom}
end function

function neg(complex a)
    return -a
end function

function scomplex(complex a)
    sequence s
    if a[REAL] != 0 then
        s = sprintf("%g",a)
    else
        s = {}
    end if
    
    if a[IMAG] != 0 then
        if a[IMAG] = 1 then
            s &= "+i"
        elsif a[IMAG] = -1 then
            s &= "-i"
        else
            s &= sprintf("%+gi",a[IMAG])
        end if
    end if
    
    if length(s) = 0 then
        return "0"
    else
        return s
    end if
end function

complex a, b
a = { 1.0,     1.0 }
b = { 3.14159, 1.2 }
printf(1,"a = %s\n",{scomplex(a)})
printf(1,"b = %s\n",{scomplex(b)})
printf(1,"a+b = %s\n",{scomplex(add(a,b))})
printf(1,"a*b = %s\n",{scomplex(mult(a,b))})
printf(1,"1/a = %s\n",{scomplex(inv(a))})
printf(1,"-a = %s\n",{scomplex(neg(a))})

Output:

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

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
    [ recip . ]         ! multiplicative inverse
    [ conjugate . ]     ! complex conjugate
    [ sin . ]           ! sine
    [ log . ]           ! natural logarithm
    [ sqrt . ]          ! square root
} cleave

Forth

Library: Forth Scientific Library

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" REQUIRED
S" complex.fs" REQUIRED

zvariable x
zvariable y
1e 1e   x z!
pi 1.2e y z!

x z@ y z@ z+ z.
x z@ y z@ z* z.
1e 0e zconstant 1+0i
1+0i x z@ z/ z.
x z@ 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 / a
end program cdemo

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

program cdemo2
    complex :: a = (5,3), b = (0.5, 6)        ! complex initializer
    real, parameter :: pi = 3.141592653589793 ! The constant "pi"
    complex, parameter :: i = (0, 1)          ! the imaginary unit "i" (sqrt(-1))
    complex :: abdiff, abquot, abpow, aconj, p2cart, newc
    real :: areal, aimag, anorm, rho = 10, theta = pi / 3.0, x = 2.3, y = 3.0
    integer, parameter :: n = 50
    integer :: j
    complex, dimension(0:n-1) :: unit_circle
    
    abdiff = a - b
    abquot = a / b
    abpow  = a ** b
    areal = real(a)               ! Real part
    aimag = imag(a)               ! Imaginary part. 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 = imag
End 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     = "; x
Print "y     = "; y
Print "x + y = "; x + y
Print "x - y = "; x - y
Print "x * y = "; x * y
Print "x / y = "; x / y
Print "-x    = "; -x
Print "1 / x = "; x.invert
Print "x*    = "; x.conjugate
Print
Print "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

Free Pascal

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.

Frink

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

add[x,y] := x + y
multiply[x,y] := x * y
negate[x] := -x
invert[x] := 1/x  // Could also use inv[x] or recip[x]
conjugate[x] := Re[x] - Im[x] i

a = 3 + 2.5i
b = 7.3 - 10i
println["$a + $b = " + add[a,b]]
println["$a * $b = " + multiply[a,b]]
println["-$a = " + negate[a]]
println["1/$a = " + invert[a]]
println["conjugate[$a] = " + conjugate[a]]

Output:

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

Futhark

This example is incorrect. Please fix the code and remove this message. Details: Futhark's syntax has changed, so "fun" should be "let"

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 1
i := Sqrt(-1);
# E(4)
(3 + 2*i)*(5 - 7*i);
# 29-11*E(4)
1/i;
# -E(4)
Sqrt(-3);
# E(3)-E(3)^2

i in GaussianIntegers;
# true
i/2 in GaussianIntegers;
# false
i/2 in GaussianRationals;
# true
Sqrt(-3) in Cyclotomics;
# true

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) {
        def zero = [0] as Complex
        (this == zero && c != zero)  \
                ?  zero  \
                :  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 <N extends Number,T> T asType (N a, Class<T> type) {
        type == Complex \
            ? [a as Number] 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
Integer.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 Complex
def a1 = [real:5, imag:3] as Complex
def a2 = 5 + 3.i
def a3 = 5 + 3*i
assert a == a1 && a == a2 && a == a3
println 'a == ' + a
def b = [0.5,6] as Complex
println 'b == ' + b
 
println "a + b == (${a}) + (${b}) == " + (a + b)
println "a * b == (${a}) * (${b}) == " + (a * b)
assert a + (-a) == 0
println "-a == -(${a}) == " + (-a)
assert (a * a.recip - 1).abs < ε
println "1/a == (${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 = 10
def d = 10 as Complex
assert d instanceof Complex && c instanceof Number && d == c
assert a + c == c + a
println "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 * c
println "10 * a == a * 10 == " + (c * a)
assert (c / a - (a / c).recip).abs < ε
println "10 / a == 1 / (a / 10) == " + (c / a)
println "a / b == (${a}) / (${b}) == " + (a / b)
assert (a ** 2 - a * a).abs < ε
println "a ** 2 == a * a == " + (a ** 2)
println "0.9 ** b == " + (0.9 ** b)
println "a ** b == (${a}) ** (${b}) == " + (a ** b)
println 'a.real == ' + a.real
println 'a.imag == ' + a.imag
println '|a| == ' + a.abs
println 'a.rho == ' + a.rho
println 'a.ρ == ' + a.ρ
println 'a.theta == ' + a.theta
println 'a.θ == ' + a.θ
println '~a (conjugate) == ' + ~a
 
def ρ = 10
def π = Math.PI
def n = 3
def θ = π / n
 
def fromPolar1 = fromPolar(ρ, θ)    // direct polar-to-cartesian conversion
def fromPolar2 = exp(θ.i) * ρ       // Euler's equation
println "ρ*cos(θ) + i*ρ*sin(θ) == ${ρ}*cos(π/${n}) + i*${ρ}*sin(π/${n})"
println "                      == 10*0.5      + i*10*√(3/4)    == " + fromPolar1
println "ρ*exp(i*θ)            == ${ρ}*exp(i*π/${n})                == " + fromPolar2
assert (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) == -0.013750112198456853 - 0.09332524760169052i
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

Hare

use fmt;
use math::complex::{c128,addc128,mulc128,divc128,negc128,conjc128};

export fn main() void = {
	let x: c128 = (1.0, 1.0);
	let y: c128 = (3.14159265, 1.2);

	// addition
	let (re, im) = addc128(x, y);
	fmt::printfln("{} + {}i", re, im)!;
	// multiplication
	let (re, im) = mulc128(x, y);
	fmt::printfln("{} + {}i", re, im)!;
	// inversion
	let (re, im) = divc128((1.0, 0.0), x);
	fmt::printfln("{} + {}i", re, im)!;
	// negation
	let (re, im) = negc128(x);
	fmt::printfln("{} + {}i", re, im)!;
	// conjugate
	let (re, im) = conjc128(x);
	fmt::printfln("{} + {}i", re, im)!;
};

Haskell

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

import Data.Complex

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

Output:

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

Icon and Unicon

Icon doesn't provide native support for complex numbers. Support is included in the IPL. Note: see the Unicon section below for a Unicon-specific solution.

procedure main()

SetupComplex()
a := complex(1,2)   
b := complex(3,4)

c := complex(&pi,1.5)
d := complex(1)
e := complex(,1)

every v := !"abcde" do write(v," := ",cpxstr(variable(v)))

write("a+b := ", cpxstr(cpxadd(a,b)))
write("a-b := ", cpxstr(cpxsub(a,b)))
write("a*b := ", cpxstr(cpxmul(a,b)))
write("a/b := ", cpxstr(cpxdiv(a,b)))
write("neg(a) := ", cpxstr(cpxneg(a)))
write("inv(a) := ", cpxstr(cpxinv(a)))
write("conj(a) := ", cpxstr(cpxconj(a)))
write("abs(a) := ", cpxabs(a))
write("neg(1) := ", cpxstr(cpxneg(1)))
end

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

Library: Icon Programming Library

provides complex number support supplemented by the code below.

link complex                            # for complex number support

procedure SetupComplex()                #: used to setup safe complex
COMPLEX()				#  replace complex record constructor	
SetupComplex := 1                       #  never call here again
return
end

procedure COMPLEX(rpart,ipart)          #: new safe record constructor and coercion
initial complex :=: COMPLEX             # get in front of record constructor
return if /ipart & (type(rpart) == "complex") 
   then rpart                           #                  already complex
   else COMPLEX( real(\rpart | 0.0), real(\ipart|0) )    # create a new complex number
end        

procedure cpxneg(z)                     #: negate z
   z := complex(z)                      # coerce
   return complex( -z.rpart, -z.ipart)
end

procedure cpxinv(z)                     #: inverse of z
   local denom
   z := complex(z)                      # coerce

   denom := z.rpart ^ 2 + z.ipart ^ 2
   return complex(z.rpart / denom, z.ipart / denom)
end

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

Output:

#complexdemo.exe

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

IDL

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

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

J

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

   x=: 1j1
   y=: 3.14159j1.2
   x+y            NB. addition
4.14159j2.2
   x*y            NB. multiplication
1.94159j4.34159
   %x             NB. inversion
0.5j_0.5
   -x             NB. negation
_1j_1
   +x             NB. (complex) conjugation
1j_1

Java

public class Complex {
    public final double real;
    public final double imag;

    public Complex() {
        this(0, 0);
    }

    public Complex(double r, double i) {
        real = r;
        imag = i;
    }

    public Complex add(Complex b) {
        return new Complex(this.real + b.real, this.imag + b.imag);
    }

    public Complex mult(Complex b) {
        // FOIL of (a+bi)(c+di) with i*i = -1
        return new Complex(this.real * b.real - this.imag * b.imag,
                this.real * b.imag + this.imag * b.real);
    }

    public Complex inv() {
        // 1/(a+bi) * (a-bi)/(a-bi) = 1/(a+bi) but it's more workable
        double denom = real * real + imag * imag;
        return new Complex(real / denom, -imag / denom);
    }

    public Complex neg() {
        return new Complex(-real, -imag);
    }

    public Complex conj() {
        return new Complex(real, -imag);
    }

    @Override
    public String toString() {
        return real + " + " + imag + " * i";
    }

    public static void main(String[] args) {
        Complex a = new Complex(Math.PI, -5); //just some numbers
        Complex b = new Complex(-1, 2.5);
        System.out.println(a.neg());
        System.out.println(a.add(b));
        System.out.println(a.inv());
        System.out.println(a.mult(b));
        System.out.println(a.conj());
    }
}

JavaScript

function Complex(r, i) {
	this.r = r;
	this.i = i;
}

Complex.add = function() {
	var num = arguments[0];
	
	for(var i = 1, ilim = arguments.length; i < ilim; i += 1){
		num.r += arguments[i].r;
		num.i += arguments[i].i;
	}
	
	return num;
}

Complex.multiply = function() {
	var num = arguments[0];
	
	for(var i = 1, ilim = arguments.length; i < ilim; i += 1){
		num.r = (num.r * arguments[i].r) - (num.i * arguments[i].i);
		num.i = (num.i * arguments[i].r) - (num.r * arguments[i].i);
	}
	
	return num;
}

Complex.negate = function (z) {
	return new Complex(-1*z.r, -1*z.i);
}

Complex.invert = function(z) {
	var denom = Math.pow(z.r,2) + Math.pow(z.i,2);
	return new Complex(z.r/denom, -1*z.i/denom);
}

Complex.conjugate = function(z) {
	return new Complex(z.r, -1*z.i);
}

// BONUSES!


Complex.prototype.toString = function() {
	return this.r === 0 && this.i === 0
          ? "0"
          : (this.r !== 0 ? this.r : "") 
          + ((this.r !== 0 || this.i < 0) && this.i !== 0 
              ? (this.i > 0 ? "+" : "-") 
              : "" ) + ( this.i !== 0 ? Math.abs(this.i) + "i" : "" ); 
}

Complex.prototype.getMod = function() {
	return Math.sqrt( Math.pow(this.r,2) , Math.pow(this.i,2) )
}

jq

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

Recent versions of jq support modules, so these functions could all be placed in a module to avoid name conflicts, and thus no special prefix is used here.

def real(z): if (z|type) == "number" then z else z[0] end;

def imag(z): if (z|type) == "number" then 0 else z[1] end;

def plus(x; y):
    if (x|type) == "number" then
       if  (y|type) == "number" then [ x+y, 0 ]
       else [ x + y[0], y[1]]
       end
    elif (y|type) == "number" then plus(y;x)
    else [ x[0] + y[0], x[1] + y[1] ]
    end;
 
def multiply(x; y):
    if (x|type) == "number" then
       if  (y|type) == "number" then [ x*y, 0 ]
       else [x * y[0], x * y[1]]
       end
    elif (y|type) == "number" then multiply(y;x)
    else [ x[0] * y[0] - x[1] * y[1],
           x[0] * y[1] + x[1] * y[0]]
    end;

def multiply: reduce .[] as $x (1; multiply(.; $x));

def negate(x): multiply(-1; x);

def minus(x; y): plus(x; multiply(-1; y));
 
def conjugate(z):
  if (z|type) == "number" then [z, 0]
  else  [z[0], -(z[1]) ]
  end;

def invert(z):
  if (z|type) == "number" then [1/z, 0]
  else
    ( (z[0] * z[0]) + (z[1] * z[1]) ) as $d
   # use "0 + ." to convert -0 back to 0
    | [ z[0]/$d, (0 + -(z[1]) / $d)]
  end;

def divide(x;y): multiply(x; invert(y));

def exp(z):
  def expi(x): [ (x|cos), (x|sin) ];
  if (z|type) == "number" then z|exp
  elif z[0] == 0 then expi(z[1])  # for efficiency
  else multiply( (z[0]|exp); expi(z[1]) )
  end ;
 
def test(x;y):
  "x =      \( x )",
  "y =      \( y )",
  "x+y:     \( plus(x;y))",
  "x*y:     \( multiply(x;y))",
  "-x:      \( negate(x))",
  "1/x:     \( invert(x))",
  "conj(x): \( conjugate(x))",
  "(x/y)*y: \( multiply( divide(x;y) ; y) )",
  "e^iπ:    \( exp( [0, 4 * (1|atan)  ] ) )"
;

test( [1,1]; [0,1] )

Output:

$ jq -n -f complex.jq
"x =      [1,1]"
"y =      [0,1]"
"x+y:     [1,2]"
"x*y:     [-1,1]"
"-x:      [-1,-1]"
"1/x:     [0.5,-0.5]"
"conj(x): [1,-1]"
"(x/y)*y: [1,1]"
"e^iπ:    [-1,1.2246467991473532e-16]"

Julia

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

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

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

Lambdatalk

{require lib_complex}

{def z1 {C.new 1 1}} 
-> z1 = (1 1)

{C.x {z1}}      -> 1
{C.y {z1}}      -> 1
{C.mod {z1}}    -> 1.4142135623730951
{C.arg {z1}}    -> 0.7853981633974483  // 45°
{C.conj {z1}}   -> (1 -1)
{C.negat {z1}}  -> (-1 -1)
{C.invert {z1}} -> (0.5 -0.4999999999999999)
{C.sqrt {z1}}   -> (1.0986841134678098 0.45508986056222733)
{C.exp {z1}}    -> (1.4686939399158851 2.2873552871788423)
{C.log {z1}}    -> (0.3465735902799727 0.7853981633974483) 

{def z2 {C.new 1.5 1.5}}
-> z2 = (1.5 1.5)

{C.add {z1} {z2}} -> (2.5 2.5)
{C.sub {z1} {z2}} -> (-0.5 -0.5)
{C.mul {z1} {z2}} -> (0 3)
{C.div {z1} {z2}} -> (0.6666666666666667 0)

Lang

fp.cprint = ($z) -> fn.printf(%.3f%+.3fi%n, fn.creal($z), fn.cimag($z))

$a = fn.complex(1.5, 3)
$b = fn.complex(1.5, 1.5)

fn.print(a =\s)
fp.cprint($a)

fn.print(b =\s)
fp.cprint($b)

# Addition
fn.print(a + b =\s)
fp.cprint(fn.cadd($a, $b))

# Multiplication
fn.print(a * b =\s)
fp.cprint(fn.cmul($a, $b))

# Inversion
fn.print(1/a =\s)
fp.cprint(fn.cdiv(fn.complex(1, 0), $a))

# Negation
fn.print(-a =\s)
fp.cprint(fn.cinv($a))

# Conjugate
fn.print(conj(a) =\s)
fp.cprint(fn.conj($a))

Output:

a = 1.500+3.000i
b = 1.500+1.500i
a + b = 3.000+4.500i
a * b = -2.250+6.750i
1/a = 0.133-0.267i
-a = -1.500-3.000i
conj(a) = 1.500-3.000i

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

Liberty BASIC

mainwin 50 10

print " Adding"
call cprint cadd$(   complex$( 1, 1), complex$( 3.14159265, 1.2))
print " Multiplying"
call cprint cmulti$( complex$( 1, 1), complex$( 3.14159265, 1.2))
print " Inverting"
call cprint cinv$(   complex$( 1, 1))
print " Negating"
call cprint cneg$(   complex$( 1, 1))

end

sub cprint cx$
    print "( "; word$( cx$, 1); " + i *"; word$( cx$, 2); ")"
end sub

function complex$( a , bj )
''complex number string-object constructor
  complex$ = str$( a ) ; " " ; str$( bj )
end function

function cadd$( a$ , b$ )
  ar = val( word$( a$ , 1 ) )
  ai = val( word$( a$ , 2 ) )
  br = val( word$( b$ , 1 ) )
  bi = val( word$( b$ , 2 ) )
  cadd$ = complex$( ar + br , ai + bi )
end function

function cmulti$( a$ , b$ )
  ar = val( word$( a$ , 1 ) )
  ai = val( word$( a$ , 2 ) )
  br = val( word$( b$ , 1 ) )
  bi = val( word$( b$ , 2 ) )
  cmulti$ = complex$( ar * br - ai * bi _
                    , ar * bi + ai * br )
end function

function cneg$( a$)
  ar = val( word$( a$ , 1 ) )
  ai = val( word$( a$ , 2 ) )
  cneg$ =complex$( 0 -ar, 0 -ai)
end function

function cinv$( a$)
  ar = val( word$( a$ , 1 ) )
  ai = val( word$( a$ , 2 ) )
  D =ar^2 +ai^2
  cinv$ =complex$( ar /D , 0 -ai /D )
end function

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+2I
y=3+4I

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

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

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

and many many more. The documentation states:

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

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

Modula-2

MODULE complex;

IMPORT  InOut;

TYPE    Complex         = RECORD   R, Im    : REAL    END;

VAR     z               : ARRAY [0..3] OF Complex;

PROCEDURE ShowComplex (str  : ARRAY OF CHAR;  p  : Complex);

BEGIN
  InOut.WriteString (str);              InOut.WriteString (" = ");
  InOut.WriteReal (p.R, 6, 2);
  IF  p.Im >= 0.0  THEN  InOut.WriteString (" + ")  ELSE  InOut.WriteString (" - ")  END;
  InOut.WriteReal (ABS (p.Im), 6, 2);   InOut.WriteString (" i ");
  InOut.WriteLn;                        InOut.WriteBf
END ShowComplex;

PROCEDURE AddComplex (x1, x2 : Complex; VAR x3  : Complex);

BEGIN
  x3.R  := x1.R  + x2.R;
  x3.Im := x1.Im + x2.Im
END AddComplex;

PROCEDURE SubComplex (x1, x2 : Complex; VAR x3  : Complex);

BEGIN
  x3.R := x1.R - x2.R;
  x3.Im := x1.Im - x2.Im
END SubComplex;

PROCEDURE MulComplex (x1, x2  : Complex; VAR x3  : Complex);

BEGIN
  x3.R := x1.R * x2.R - x1.Im * x2.Im;
  x3.Im := x1.R * x2.Im + x1.Im * x2.R
END MulComplex;

PROCEDURE InvComplex (x1 : Complex; VAR x2  : Complex);

BEGIN
  x2.R := x1.R / (x1.R * x1.R + x1.Im * x1.Im);
  x2.Im := -1.0 * x1.Im / (x1.R * x1.R + x1.Im * x1.Im)
END InvComplex;

PROCEDURE NegComplex (x1 : Complex; VAR x2  : Complex);

BEGIN
  x2.R := - x1.R;       x2.Im := - x1.Im
END NegComplex;

BEGIN
  InOut.WriteString ("Enter two complex numbers : ");
  InOut.WriteBf;
  InOut.ReadReal (z[0].R);              InOut.ReadReal (z[0].Im);
  InOut.ReadReal (z[1].R);              InOut.ReadReal (z[1].Im);
  ShowComplex ("z1", z[0]);             ShowComplex ("z2", z[1]);
  InOut.WriteLn;
  AddComplex (z[0], z[1], z[2]);        ShowComplex ("z1 + z2", z[2]);
  SubComplex (z[0], z[1], z[2]);        ShowComplex ("z1 - z2", z[2]);
  MulComplex (z[0], z[1], z[2]);        ShowComplex ("z1 * z2", z[2]);
  InvComplex (z[0], z[2]);              ShowComplex ("1  / z1", z[2]);
  NegComplex (z[0], z[2]);              ShowComplex ("   - z1", z[2]);
  InOut.WriteLn
END complex.

Output:

Enter two complex numbers : 5 3 0.5 6
z1 =   5.00 +   3.00 i
z2 =   0.50 +   6.00 i

z1 + z2 =   5.50 +   9.00 i
z1 - z2 =   4.50 -   3.00 i
z1 * z2 = -15.50 +  31.50 i
1  / z1 =   0.15 -   0.09 i
   - z1 =  -5.00 -   3.00 i

Nanoquery

Translation of: Java

This is a translation of the Java version, but it uses operator redefinition where possible.

import math

class Complex
        declare real
        declare imag

        def Complex()
                real = 0.0
                imag = 0.0
        end

        def Complex(r, i)
                real = double(r)
                imag = double(i)
        end

        def operator-(b)
                return new(Complex, this.real - b.real, this.imag - b.imag)
        end

        def operator+(b)
                return new(Complex, this.real + b.real, this.imag + b.imag)
        end

        def operator*(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)
        end

        def inv()
                // 1/(a+bi) * (a-bi)/(a-bi) = 1/(a+bi) but it's more workable
                denom = this.real * this.real + this.imag * this.imag
                return new(Complex, real/denom, -imag/denom)
        end

        def neg()
                return new(Complex, -this.real, -this.imag)
        end

        def conj()
                return new(Complex, this.real, -this.imag)
        end

        def toString()
                return this.real + " + " + this.imag + " * i"
        end
end

a = new(Complex, math.pi, -5)
b = new(Complex, -1, 2.5)
println a.neg()
println a + b
println a.inv()
println a * b
println a.conj()

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 complex
var 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 r
END New;

VAR
        r,x,y: Complex;
BEGIN
        x := New(1.5,3);
        y := New(1.0,1.0);
       
        Out.String("x: ");x.Out(Files.stdout);Out.Ln;
        Out.String("y: ");y.Out(Files.stdout);Out.Ln;        
        r := x.Add(y);
        Out.String("x + y: ");r.Out(Files.stdout);Out.Ln;
        r := x.Sub(y);
        Out.String("x - y: ");r.Out(Files.stdout);Out.Ln;
        r := x.Mul(y);
        Out.String("x * y: ");r.Out(Files.stdout);Out.Ln;
        r := x.Div(y);
        Out.String("x / y: ");r.Out(Files.stdout);Out.Ln;
        r := y.Rec();
        Out.String("1 / y: ");r.Out(Files.stdout);Out.Ln;
        r := x.Con();
        Out.String("x': ");r.Out(Files.stdout);Out.Ln;
       
END Complex.

Output:

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

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

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 + .cr
2 3 Complex new  1.2 >complex + .cr
2 3 Complex new  1.2 >complex * .cr
2 >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           =" c1
say "c2           =" c2
say "r            =" r
say "-c1          =" (-c1)
say "c1 + r       =" c1 + r
say "c1 + c2      =" c1 + c2
say "c1 - r       =" c1 - r
say "c1 - c2      =" c1 - c2
say "c1 * r       =" c1 * r
say "c1 * c2      =" c1 * c2
say "inv(c1)      =" c1~inv
say "conj(c1)     =" c1~conjugate
say "c1 / r       =" c1 / r
say "c1 / c2      =" c1 / c2
say "c1 == c1     =" (c1 == c1)
say "c1 == c2     =" (c1 == c2)


::class complex
::method init
  expose r i
  use strict arg r, i = 0

-- complex instances are immutable, so these are
-- read only attributes
::attribute r GET
::attribute i GET

::method negative
  expose r i
  return self~class~new(-r, -i)

::method add
  expose r i
  use strict arg other
  if other~isa(.complex) then
     return self~class~new(r + other~r, i + other~i)
  else return self~class~new(r + other, i)

::method subtract
  expose r i
  use strict arg other
  if other~isa(.complex) then
     return self~class~new(r - other~r, i - other~i)
  else return self~class~new(r - other, i)

::method times
  expose r i
  use strict arg other
  if other~isa(.complex) then
     return self~class~new(r * other~r - i * other~i, r * other~i + i * other~r)
  else return self~class~new(r * other, i * other)

::method inv
  expose r i
  denom = r * r + i * i
  return self~class~new(r/denom,-i/denom)

::method conjugate
  expose r i
  return self~class~new(r, -i)

::method divide
  use strict arg other
  -- this is easier if everything is a complex number
  if \other~isA(.complex) then other = .complex~new(other)
  -- division is multiplication with the inversion
  return self * other~inv

::method "=="
  expose r i
  use strict arg other

  if \other~isa(.complex) then return .false
  -- Note:  these are numeric comparisons, so we're using the "="
  -- method so those are handled correctly
  return r = other~r & i = other~i

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

::method "="
  -- this is equivalent of "=="
  forward message("==")

::method "\="
  -- this is equivalent of "\=="
  forward message("\==")

::method "<>"
  -- this is equivalent of "\=="
  forward message("\==")

::method "><"
  -- this is equivalent of "\=="
  forward message("\==")

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

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

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

::method "+"
  -- need to check if this is a prefix plus or an addition
  if arg() == 0 then
      return self  -- we can return this copy since it is 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
  @a=@accum+i
  if pp then
    @b=@a+sizeof accum
    pp=0
  else
    @b=@this
  end if
end def

method "load"()
  operands
  a.x=b.x
  a.y=b.y
end method

method "push"()
  i+=sizeof accum
end method

method "pop"()
  pp=1
  i-=sizeof accum
end method

method "="()
  operands
  b.x=a.x
  b.y=a.y
end method

method "+"()
  operands
  a.x+=b.x
  a.y+=b.y
end method

method "-"()
  operands
  a.x-=b.x
  a.y-=b.y
end method

method "*"()
  operands
  double d
  d=a.x
  a.x = a.x * b.x - a.y * b.y
  a.y = a.y * b.x + d   * b.y
end method

method "/"()
  operands
  double d,v
  v=1/(b.x * b.x + b.y * b.y)
  d=a.x
  a.x = (a.x * b.x + a.y * b.y) * v
  a.y = (a.y * b.x - d   * b.y) * v
end method

method power(double n)
  operands
  'Using DeMoivre theorem
  double r,an,mg
  r = hypot(b.x,b.y)
  mg = r^n
  if b.x=0 then
    ay=.5*pi
    if b.y<0 then ay=-ay
  else
    an = atan(b.y,b.x)
  end if
  an  *= n 
  a.x  = mg * cos(an)
  a.y  = mg * sin(an)
end method

method show() as string
  return str(x,14) ", " str(y,14)
end method

end class

'#recordof complexop  

'====
'TEST
'====

complex z1,z2,z3,z4,z5

'ENTER VALUES

z1 <=  0, 0
z2 <=  2, 1
z3 <= -2, 1
z4 <=  2, 4
z5 <=  1, 1

'EVALUATE COMPLEX EXPRESSIONS

z1 =  z2 * z3
print "Z1 = "+z1.show 'RESULT  -5.0, 0

z1 = z3+(z2.power(2))
print "Z1 = "+z1.show  'RESULT  1.0, 5.0


z1 = z5/z4
print "Z1 = "+z1.show  'RESULT 0.3, 0.1

z1 = z5/z1
print "Z1 = "+z1.show  'RESULT 2.0, 4.0

z1 = z2/z4
print "Z1 = "+z1.show  'RESULT  -0.4, -0.3

z1 = z1*z4
print "Z1 = "+z1.show  'RESULT  2.0, 1.0

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

Works with: Extended Pascal

The simple data type complex is part of Extended Pascal, ISO standard 10206.

program complexDemo(output);

const
	{ I experienced some hiccups with -1.0 using GPC (GNU Pascal Compiler) }
	negativeOne = -1.0;

type
	line = string(80);

{ as per task requirements wrap arithmetic operations into separate functions }
function sum(protected x, y: complex): complex;
begin
	sum := x + y
end;

function product(protected x, y: complex): complex;
begin
	product := x * y
end;

function negative(protected x: complex): complex;
begin
	negative := -x
end;

function inverse(protected x: complex): complex;
begin
	inverse := x ** negativeOne
end;

{ only this function is not covered by Extended Pascal, ISO 10206 }
function conjugation(protected x: complex): complex;
begin
	conjugation := cmplx(re(x), im(x) * negativeOne)
end;

{ --- test suite ------------------------------------------------------------- }
function asString(protected x: complex): line;
const
	totalWidth = 5;
	fractionDigits = 2;
var
	result: line;
begin
	writeStr(result, '(', re(x):totalWidth:fractionDigits, ', ',
		im(x):totalWidth:fractionDigits, ')');
	asString := result
end;

{ === MAIN =================================================================== }
var
	x: complex;
	{ for demonstration purposes: how to initialize complex variables }
	y: complex value cmplx(1.0, 4.0);
	z: complex value polar(exp(1.0), 3.14159265358979);
begin
	x := cmplx(-3, 2);
	
	writeLn(asString(x), ' + ', asString(y), ' = ', asString(sum(x, y)));
	writeLn(asString(x), ' * ', asString(z), ' = ', asString(product(x, z)));
	
	writeLn;
	
	writeLn('               −', asString(z), '  = ', asString(negative(z)));
	writeLn('        inverse(', asString(z), ') = ', asString(inverse(z)));
	writeLn('    conjugation(', asString(y), ') = ', asString(conjugation(y)));
end.

Output:

(-3.00,  2.00) + ( 1.00,  4.00) = (-2.00,  6.00)
(-3.00,  2.00) * (-2.72,  0.00) = ( 8.15, -5.44)

               −(-2.72,  0.00)  = ( 2.72, -0.00)
        inverse((-2.72,  0.00)) = (-0.37, -0.00)
    conjugation(( 1.00,  4.00)) = ( 1.00, -4.00)

The GPC, GNU Pascal Compiler, supports Extended Pascal’s complex data type and operations as shown. Furthermore, the GPC defines a function conjugate so there is no need for writing such a custom function. The PXSC, Pascal eXtensions for scientific computing, define a standard data type similar to Free Pascal’s ucomplex data type.

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

Phix

-- demo\rosetta\ArithComplex.exw
with javascript_semantics
include complex.e
complex a = complex_new(1,1),  -- (or just {1,1})
        b = complex_new(3.14159,1.25),
        c = complex_new(1,0),
        d = complex_new(0,1)
printf(1,"a = %s\n",{complex_sprint(a)})
printf(1,"b = %s\n",{complex_sprint(b)})
printf(1,"c = %s\n",{complex_sprint(c)})
printf(1,"d = %s\n",{complex_sprint(d)})
printf(1,"a+b = %s\n",{complex_sprint(complex_add(a,b))})
printf(1,"a*b = %s\n",{complex_sprint(complex_mul(a,b))})
printf(1,"1/a = %s\n",{complex_sprint(complex_inv(a))})
printf(1,"c/a = %s\n",{complex_sprint(complex_div(c,a))})
printf(1,"c-a = %s\n",{complex_sprint(complex_sub(c,a))})
printf(1,"d-a = %s\n",{complex_sprint(complex_sub(d,a))})
printf(1,"-a = %s\n",{complex_sprint(complex_neg(a))})
printf(1,"conj a = %s\n",{complex_sprint(complex_conjugate(a))})

Output:

a = 1+i
b = 3.14159+1.25i
c = 1
d = i
a+b = 4.14159+2.25i
a*b = 1.89159+4.39159i
1/a = 0.5-0.5i
c/a = 0.5-0.5i
c-a = -i
d-a = -1
-a = -1-i
conj 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 values
1 +: 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] def
z 0 x 0 get y 0 get add put 
z 1 x 1 get y 1 get add put
z pstack
}def 

%Subtracting one complex number from another
/subcomp{
/x exch def
/y exch def
/z [0 0] def
z 0 x 0 get y 0 get sub put 
z 1 x 1 get y 1 get sub put
z pstack
}def

%Multiplying two complex numbers
/mulcomp{
/x exch def
/y exch def
/z [0 0] def
z 0 x 0 get y 0 get mul x 1 get y 1 get mul sub  put 
z 1 x 1 get y 0 get mul x 0 get y 1 get mul add put
z pstack
}def

%Negating a complex number
/negcomp{
/x exch def
/z [0 0] def
z 0 x 0 get neg put
z 1 x 1 get neg put
z pstack
}def

%Inverting a complex number
/invcomp{
/x exch def
/z [0 0] def
z 0 x 0 get x 0 get 2 exp x 1 get 2 exp add div put
z 0 x 1 get neg x 0 get 2 exp x 1 get 2 exp add div put
z pstack
}def

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

PureBasic

Structure Complex
  real.d
  imag.d
EndStructure

Procedure Add_Complex(*A.Complex, *B.Complex) 
  Protected *R.Complex=AllocateMemory(SizeOf(Complex))
  If *R
    *R\real=*A\real+*B\real
    *R\imag=*A\imag+*B\imag
  EndIf
  ProcedureReturn *R
EndProcedure

Procedure Inv_Complex(*A.Complex)
  Protected *R.Complex=AllocateMemory(SizeOf(Complex)), denom.d
  If *R
    denom  = *A\real * *A\real + *A\imag * *A\imag
    *R\real= *A\real / denom
    *R\imag=-*A\imag / denom
  EndIf
  ProcedureReturn *R
EndProcedure

Procedure Mul_Complex(*A.Complex, *B.Complex)
  Protected *R.Complex=AllocateMemory(SizeOf(Complex))
  If *R
    *R\real=*A\real * *B\real - *A\imag * *B\imag
    *R\imag=*A\real * *B\imag + *A\imag * *B\real
  EndIf
  ProcedureReturn *R
EndProcedure

Procedure Neg_Complex(*A.Complex)
  Protected *R.Complex=AllocateMemory(SizeOf(Complex))
  If *R
    *R\real=-*A\real
    *R\imag=-*A\imag
  EndIf
  ProcedureReturn *R
EndProcedure

Procedure ShowAndFree(Header$, *Complex.Complex)
  If *Complex
    Protected.d i=*Complex\imag, r=*Complex\real 
    Print(LSet(Header$,7))
    Print("= "+StrD(r,3))
    If i>=0:  Print(" + ")
    Else:     Print(" - ")
    EndIf
    PrintN(StrD(Abs(i),3)+"i")
    FreeMemory(*Complex)
  EndIf
EndProcedure

If OpenConsole()
  Define.Complex a, b, *c
  a\real=1.0: a\imag=1.0
  b\real=#PI: b\imag=1.2
  *c=Add_Complex(a,b):  ShowAndFree("a+b",    *c)
  *c=Mul_Complex(a,b):  ShowAndFree("a*b",    *c)
  *c=Inv_Complex(a):    ShowAndFree("Inv(a)", *c)
  *c=Neg_Complex(a):    ShowAndFree("-a",     *c)
  Print(#CRLF$+"Press ENTER to exit"):Input()
EndIf

Python

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

R

Translation of: Octave

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

Racket

#lang racket

(define a 3+4i)
(define b 8+0i)

(+ a b)       ; addition
(- a b)       ; subtraction
(/ a b)       ; division
(* a b)       ; multiplication
(- a)         ; negation
(/ 1 a)       ; reciprocal
(conjugate a) ; conjugation

Raku

(formerly 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

REXX

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

/*REXX program  demonstrates how to support some  math functions  for  complex numbers. */
x = '(5,3i)'                                     /*define  X    ─── can use  I i J or j */
y = "( .5,  6j)"                                 /*define  Y         "   "   " " "  " " */

say '      addition:   '        x        " + "         y         ' = '          Cadd(x, y)
say '   subtraction:   '        x        " - "         y         ' = '          Csub(x, y)
say 'multiplication:   '        x        " * "         y         ' = '          Cmul(x, y)
say '      division:   '        x        " ÷ "         y         ' = '          Cdiv(x, y)
say '       inverse:   '        x        "                         = "          Cinv(x, y)
say '  conjugate of:   '        x        "                         = "          Conj(x, y)
say '   negation of:   '        x        "                         = "          Cneg(x, y)
exit                                             /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
Conj: procedure; parse arg a ',' b,c ',' d;   call C#;    return C$(  a      ,  -b    )
Cadd: procedure; parse arg a ',' b,c ',' d;   call C#;    return C$(  a+c    ,   b+d  )
Csub: procedure; parse arg a ',' b,c ',' d;   call C#;    return C$(  a-c    ,   b-d  )
Cmul: procedure; parse arg a ',' b,c ',' d;   call C#;    return C$( ac-bd   ,   bc+ad)
Cdiv: procedure; parse arg a ',' b,c ',' d;   call C#;    return C$((ac+bd)/s,  (bc-ad)/s)
Cinv: return  Cdiv(1,  arg(1))
Cneg: return  Cmul(arg(1), -1)
C_:   return  word(translate(arg(1), , '{[(JjIi)]}')  0,  1)                /*get # or 0*/
C#:   a=C_(a); b=C_(b); c=C_(c); d=C_(d); ac=a*c; ad=a*d; bc=b*c; bd=b*d;s=c*c+d*d; return
C$:   parse arg r,c;    _='['r;   if c\=0  then _=_","c'j';   return _"]"   /*uses  j   */

output

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

RLaB

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

RPL

Input:

(1.5,3) 'Z1' STO
(1.5,1.5) 'Z2' STO
Z1 Z2 +
Z1 Z2 -
Z1 Z2 *
Z1 Z2 /
Z1 NEG
Z1 CONJ
Z1 ABS
Z1 RE
Z1 IM

Output:

(3,4.5)
(0,1.5)
(-2.25,6.75)
(1.5,.5)
(-1.5,-3)
(1.5,-3)
63.4349488229
1.5
3

Ruby

# Four ways to write complex numbers:
a = Complex(1, 1)       # 1. call Kernel#Complex
i = Complex::I          # 2. use Complex::I
b = 3.14159 + 1.25 * i
c = '1/2+3/4i'.to_c     # 3. Use the .to_c method from String, result ((1/2)+(3/4)*i)
c =  1.0/2+3/4i         # (0.5-(3/4)*i) 

# Operations:
puts a + b              # addition
puts a * b              # multiplication
puts -a                 # negation
puts 1.quo a            # multiplicative inverse
puts a.conjugate        # complex conjugate
puts a.conj             # alias for complex conjugate

Notes:

• All of these operations are safe with other numeric types. For example, 42.conjugate returns 42.

# Other ways to find the multiplicative inverse:
puts 1.quo a            # always works
puts 1.0 / a            # works, but forces floating-point math
puts 1 / a              # might truncate to integer

Rust

extern crate num;
use num::complex::Complex;

fn main() {
    // two valid forms of definition
    let a = Complex {re:-4.0, im: 5.0};
    let b = Complex::new(1.0, 1.0);

    println!("   a    = {}", a);
    println!("   b    = {}", b);
    println!(" a + b  = {}", a + b);
    println!(" a * b  = {}", a * b);
    println!(" 1 / a  = {}", a.inv());
    println!("  -a    = {}", -a);
    println!("conj(a) = {}", a.conj());
}

Scala

Works with: Scala version 2.8

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

package org.rosettacode

package object ArithmeticComplex {
  val i = Complex(0, 1)
  
  implicit def fromDouble(d: Double) = Complex(d)
  implicit def fromInt(i: Int) = Complex(i.toDouble)
}  

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

Usage example:

scala> import org.rosettacode.ArithmeticComplex._
import org.rosettacode.ArithmeticComplex._

scala> 1 + i
res0: org.rosettacode.ArithmeticComplex.Complex = 1.0 + 1.0i

scala> 1 + 2 * i
res1: org.rosettacode.ArithmeticComplex.Complex = 1.0 + 2.0i

scala> 2 + 1.i
res2: org.rosettacode.ArithmeticComplex.Complex = 2.0 + 1.0i

scala> res0 + res1
res3: org.rosettacode.ArithmeticComplex.Complex = 2.0 + 3.0i

scala> res1 * res2
res4: org.rosettacode.ArithmeticComplex.Complex = 0.0 + 5.0i

scala> res2 / res0
res5: org.rosettacode.ArithmeticComplex.Complex = 1.5 + -0.5i

scala> res1.inverse
res6: org.rosettacode.ArithmeticComplex.Complex = 0.2 + -0.4i

scala> -res6
res7: org.rosettacode.ArithmeticComplex.Complex = -0.2 + 0.4i

Scheme

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

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

(define a 1+i)
(define b 3.14159+1.25i)

(define c (+ a b))
(define c (* a b))
(define c (/ 1 a))
(define c (- a))

Seed7

$ include "seed7_05.s7i";
  include "float.s7i";
  include "complex.s7i";

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

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.

Works with: Smalltalk/X

Complex is already in the basic class library. Multiples of imaginary are created by sending an "i" message to a number, which can be added to another number. Thus 5i => (0+5i), 1+(1/3)I => (1+1/3i) and (1.0+2i) => (1.0+2i). Notice that the real and imaginary components can be arbitrary integers, fractions or floating point numbers. And the results will be exact (i.e. have fractions or integer) if possible.

|a b|
a := 1 + 1i.
b := 3.14159 + 1.2i.
Transcript show:'a => '; showCR:a.
Transcript show:'b => '; showCR:b.
Transcript show:'a+b => '; showCR:(a + b).
Transcript show:'a-b => '; showCR:(a - b).
Transcript show:'a*b => '; showCR:(a * b).
Transcript show:'a/b => '; showCR:(a / b).
Transcript show:'a reciprocal => '; showCR:a reciprocal.
Transcript show:'a conjugated => '; showCR:a conjugated.
Transcript show:'a abs => '; showCR:a abs.
Transcript show:'a real => '; showCR:a real.
Transcript show:'a imaginary => '; showCR:a imaginary.
Transcript show:'a negated => '; showCR:a negated.
Transcript show:'a sqrt => '; showCR:a sqrt.
a2 := (1/2) + 1i.
b2 := (2/3) + 2i.
Transcript show:'a2+b2 => '; showCR:(a2 + b2).
Transcript show:'a2-b2 => '; showCR:(a2 - b2).
Transcript show:'a2*b2 => '; showCR:(a2 * b2).
Transcript show:'a2/b2 => '; showCR:(a2 / b2).
Transcript show:'a2 reciprocal => '; showCR:a2 reciprocal.

Output:

a => (1+1i)
b => (3.14159+1.2i)
a+b => (4.14159+2.2i)
a-b => (-2.14159-0.2i)
a*b => (1.94159+4.34159i)
a/b => (0.383885788269082+0.171676461306887i)
a reciprocal => ((1/2)-(1/2)i)
a conjugated => (1-1i)
a abs => 1.4142135623731
a real => 1
a imaginary => 1
a negated => (-1-1i)
a sqrt => (1.09868411346781+0.455089860562227i)
a2+b2 => ((7/6)+3i)
a2-b2 => ((-1/6)-1i)
a2*b2 => ((-5/3)+(5/3)i)
a2/b2 => ((21/40)-(3/40)i)
a2 reciprocal => ((2/5)-(4/5)i)

smart BASIC

Original author unknown {:o(

' complex numbers are native for "smart BASIC"
A=1+2i
B=3-5i

' all math operations and functions work with complex numbers
C=A*B
PRINT SQR(-4)

' example of solving quadratic equation with complex roots
' x^2+2x+5=0
a=1 ! b=2 ! c=5
x1=(-b+sqr(b^2-4*a*c))/(2*a)
x2=(-b-sqr(b^2-4*a*c))/(2*a)
print x1,x2

' gives output
-1+2i    -1-2i

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 -> unit
end;

(* Actual implementation *)
structure Complex :> COMPLEX = struct
  type num = real * real
  
  fun complex (a, b) = (a, b)
  
  fun negative (a, b) = (Real.~a, Real.~b)
  fun plus (a1, b1) (a2, b2) = (Real.+ (a1, a2), Real.+(b1, b2))
  fun minus i1 i2 = plus i1 (negative i2)
  fun times (a1, b1) (a2, b2)= (Real.*(a1, a2) - Real.*(b1, b2), Real.*(a1, b2) + Real.*(a2, b1))
  fun invert (a, b) =
    let
      val denom = a * a + b * b
    in
      (a / denom, ~b / denom)
    end
  
  fun print_number (a, b) =
    print (Real.toString(a) ^ " + " ^ Real.toString(b) ^ "i\n")
end;

val i1 = Complex.complex(1.0,2.0); (* 1 + 2i *)
val i2 = Complex.complex(3.0,4.0); (* 3 + 4i *)

Complex.print_number(Complex.negative(i1)); (* -1 - 2i *)
Complex.print_number(Complex.plus i1 i2); (* 4 + 6i *)
Complex.print_number(Complex.minus i2 i1); (* 2 + 2i *)
Complex.print_number(Complex.times i1 i2); (* -5 + 10i *)
Complex.print_number(Complex.invert i1); (* 1/5 - 2i/5 *)

Stata

mata
C(2,3)
2 + 3i

a=2+3i
b=1-2*i


a+b
-5 + 3i

a-b
9 + 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 useful
extension 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::complexnumbers
namespace import math::complexnumbers::*

set a [complex 1 1]
set b [complex 3.14159 1.2]
puts [tostring [+ $a $b]] ;# ==> 4.14159+2.2i
puts [tostring [* $a $b]] ;# ==> 1.94159+4.34159i
puts [tostring [pow $a [complex -1 0]]] ;# ==> 0.5-0.4999999999999999i
puts [tostring [- $a]] ;# ==> -1.0-i

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.

Unicon

Takes advantage of Unicon's operator overloading extension and Unicon's Complex class. Negation is not supported by the Complex class.

import math

procedure main()
   write("c1: ",(c1 := Complex(1.5,3)).toString())
   write("c2: ",(c2 := Complex(1.5,1.5)).toString())
   write("+: ",(c1+c2).toString())
   write("-: ",(c1-c2).toString())
   write("*: ",(c1*c2).toString())
   write("/: ",(c1/c2).toString())
   write("additive inverse: ",c1.addInverse().toString())
   write("multiplicative inverse: ",c1.multInverse().toString())
   write("conjugate of (4,-3i): ",Complex(4,-3).conjugate().toString())
end

Output:

c1: (1.5,3i)
c2: (1.5,1.5i)
+: (3.0,4.5i)
-: (0.0,1.5i)
*: (-2.25,6.75i)
/: (1.5,0.5i)
additive inverse: (-1.5,-3i)
multiplicative inverse: (0.1333333333333333,-0.2666666666666667i)
conjugate of (4,-3i): (4,3i)

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 c
c.add a b
c.to_s        # 4.14159 + 2.2 i

c.negate
c.to_s        # -4.14159 + -2.2 i

c.conjugate
c.to_s        # -4.14159 + 2.2 i

c.dup a
c.multiply b
c.to_s        # 1.94159 + 4.34159 i

Complex_t d=(real=2 imag=1)
d.inverse
d.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+00i
v = 9.545e-01-3.305e+00j

#cast %jL

examples = 

<
   complex..add (u,v),
   complex..mul (u,v),
   complex..sub (0.,u),
   complex..div (1.,v)>

Output:

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

VBA

Public Type Complex
    re As Double
    im As Double
End Type

Function CAdd(a As Complex, b As Complex) As Complex
    CAdd.re = a.re + b.re
    CAdd.im = a.im + b.im
End Function

Function CSub(a As Complex, b As Complex) As Complex
    CSub.re = a.re - b.re
    CSub.im = a.im - b.im
End Function

Function CMult(a As Complex, b As Complex) As Complex
    CMult.re = (a.re * b.re) - (a.im * b.im)
    CMult.im = (a.re * b.im) + (a.im * b.re)
End Function

Function CConj(a As Complex) As Complex
    CConj.re = a.re
    CConj.im = -a.im
End Function

Function CNeg(a As Complex) As Complex
    CNeg.re = -a.re
    CNeg.im = -a.im
End Function

Function CInv(a As Complex) As Complex
    CInv.re = a.re / (a.re * a.re + a.im * a.im)
    CInv.im = -a.im / (a.re * a.re + a.im * a.im)
End Function

Function CDiv(a As Complex, b As Complex) As Complex
    CDiv = CMult(a, CInv(b))
End Function

Function CAbs(a As Complex) As Double
    CAbs = Math.Sqr(a.re * a.re + a.im * a.im)
End Function

Function CSqr(a As Complex) As Complex
    CSqr.re = Math.Sqr((a.re + Math.Sqr(a.re * a.re + a.im * a.im)) / 2)
    CSqr.im = Math.Sgn(a.im) * Math.Sqr((-a.re + Math.Sqr(a.re * a.re + a.im * a.im)) / 2)
End Function

Function CPrint(a As Complex) As String
    If a.im > 0 Then
        Sep = "+"
    Else
        Sep = ""
    End If
    CPrint = a.re & Sep & a.im & "i"
End Function

Sub ShowComplexCalc()
Dim a As Complex
Dim b As Complex
Dim c As Complex

a.re = 1.5
a.im = 3
b.re = 1.5
b.im = 1.5

Debug.Print "a = " & CPrint(a)
Debug.Print "b = " & CPrint(b)

c = CAdd(a, b)
Debug.Print "a + b = " & CPrint(c)
c = CSub(a, b)
Debug.Print "a - b = " & CPrint(c)
c = CMult(a, b)
Debug.Print "a * b = " & CPrint(c)
c = CConj(a)
Debug.Print "Conj(a) = " & CPrint(c)
c = CNeg(a)
Debug.Print "-a = " & CPrint(c)
c = CInv(a)
Debug.Print "Inv(a) = " & CPrint(c)
c = CDiv(a, b)
Debug.Print "a / b = " & CPrint(c)
Debug.Print "Abs(a) = " & CAbs(a)
c = CSqr(a)
Debug.Print "Sqrt(a) = " & CPrint(c)
End Sub

Output:

a = 1.5+3i
b = 1.5+1.5i
a + b = 3+4.5i
a - b = 0+1.5i
a * b = -2.25+6.75i
Conj(a) = 1.5-3i
-a = -1.5-3i
Inv(a) = 0.133333333333333-0.266666666666667i
a / b = 1.5+0.5i
Abs(a) = 3.35410196624968
Sqrt(a) = 1.55789954205168+0.962834868045836i

V (Vlang)

import math.complex
fn main() {
    a := complex.complex(1, 1)
    b := complex.complex(3.14159, 1.25)
    println("a:      $a")
    println("b:      $b")
    println("a + b:  ${a+b}")
    println("a * b:  ${a*b}")
    println("-a:     ${a.addinv()}")
    println("1 / a:  ${complex.complex(1,0)/a}")
    println("a̅:      ${a.conjugate()}")
}

Output:

a:      1.000000+1.000000i
b:      3.141590+1.250000i
a + b:  4.141590+2.250000i
a * b:  1.891590+4.391590i
-a:     -1.000000-1.000000i
1 / a:  0.500000-0.500000i
a̅:      1.000000-1.000000i

Wortel

Translation of: CoffeeScript

@class Complex {
  &[r i] @: {
    ^r || r 0
    ^i || i 0
    ^m +@sq^r @sq^i
  }
  add &o @new Complex[+ ^r o.r + ^i o.i]
  mul &o @new Complex[-* ^r o.r * ^i o.i +* ^r o.i * ^i o.r]
  neg &^ @new Complex[@-^r @-^i]
  inv &^ @new Complex[/ ^r ^m / @-^i ^m]
  toString &^?{
    =^i 0 "{^r}"
    =^r 0 "{^i}i"
    >^i 0 "{^r} + {^i}i"
    "{^r} - {@-^i}i"
  }
}

@vars {
  a @new Complex[5 3]
  b @new Complex[4 3N]
}
@each &x !console.log x [
  "({a}) + ({b}) = {!a.add b}"
  "({a}) * ({b}) = {!a.mul b}"
  "-1 * ({b}) = {b.neg.}"
  "({a}) - ({b}) = {!a.add b.neg.}"
  "1 / ({b}) = {b.inv.}"
  "({!a.mul b}) / ({b}) = {`!.mul b.inv. !a.mul b}"
]

Output:

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

Wren

Library: Wren-complex

import "./complex" for Complex

var x = Complex.new(1, 3)
var y = Complex.new(5, 2)
System.print("x     =  %(x)")
System.print("y     =  %(y)")
System.print("x + y =  %(x + y)")
System.print("x - y =  %(x - y)")
System.print("x * y =  %(x * y)")
System.print("x / y =  %(x / y)")
System.print("-x    =  %(-x)")
System.print("1 / x =  %(x.inverse)")
System.print("x*    =  %(x.conj)")

Output:

x     =  1 + 3i
y     =  5 + 2i
x + y =  6 + 5i
x - y =  -4 + 1i
x * y =  -1 + 17i
x / y =  0.37931034482759 + 0.44827586206897i
-x    =  -1 - 3i
1 / x =  0.1 - 0.3i
x*    =  1 - 3i

XPL0

include c:\cxpl\codes;

func real CAdd(A, B, C);        \Return complex sum of two complex numbers
real A, B, C;
[C(0):= A(0) + B(0);
 C(1):= A(1) + B(1);
return C;
];

func real CMul(A, B, C);        \Return complex product of two complex numbers
real A, B, C;
[C(0):= A(0)*B(0) - A(1)*B(1);
 C(1):= A(1)*B(0) + A(0)*B(1);
return C;
];

func real CNeg(A, C);           \Return negative of a complex number
real A, C;
[C(0):= -A(0);
 C(1):= -A(1);
return C;
];

func real CInv(A, C);           \Return inversion (reciprical) of complex number
real A, C;
real D;
[D:= sq(A(0)) + sq(A(1));
C(0):= A(0)/D;
C(1):=-A(1)/D;
return C;
];

func real Conj(A, C);           \Return conjugate of a complex number
real A, C;
[C(0):= A(0);
 C(1):=-A(1);
return C;
];

proc COut(D, A);                \Output a complex number to specified device
int D; real A;
[RlOut(D, A(0));
  Text(D, if A(1)>=0.0 then " +" else " -");
 RlOut(D, abs(A(1)));
ChOut(D, ^i);
];

real U, V, W(2);
[Format(2,2);
U:= [1.0,  1.0];
V:= [3.14, 1.2];
COut(0, CAdd(U,V,W)); CrLf(0);
COut(0, CMul(U,V,W)); CrLf(0);
COut(0, CNeg(U,W));   CrLf(0);
COut(0, CInv(U,W));   CrLf(0);
COut(0, Conj(U,W));   CrLf(0);
]

Output:

 4.14 + 2.20i
 1.94 + 4.34i
-1.00 - 1.00i
 0.50 - 0.50i
 1.00 - 1.00i

Yabasic

rem ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
rem 	CADDI/CADDR  addition of complex numbers  Z1 + Z2    with Z1 = a1 + b1 *i   Z2 = a2 + b2*i
rem                CADDI returns imaginary part and CADDR the real part
rem ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
export sub caddi( a1 , b1 , a2 , b2)
    return (b1 + b2)
end sub
export 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*i
rem                CDIVI returns imaginary part and CDIVR the real part
rem ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
export sub cdivi(r,s,t,u)
    return ((s*t- u*r) / (t^2 + u^2))
end sub
export 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*i
rem                CMULI returns imaginary part and CMULR the real part
rem ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
export sub cmuli( r , s , t , u)
    return (r * u + s * t)
end sub
export 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*i
rem                CSUBI returns imaginary part and CSUBR the real part
rem ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
export sub csubi( a1 , b1 , a2 , b2)
    return (b1 - b2)
end sub
export 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/-1
end 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 r
end "+";

operator {public} "-" (a,b: Complex): Complex;
var
	r: Complex;
begin
	r.re := a.re - b.re;
	r.im := a.im - b.im;
	return r
end "-";

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 r
end "*";

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 r
end "/";

operator {public} "-" (a: Complex): Complex;
begin
	a.im := -1 * a.im;
	return a
end "-";

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 c
end "~";

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=2
10 DIM a(complex): LET a(r)=1.0: LET a(i)=1.0
20 DIM b(complex): LET b(r)=PI: LET b(i)=1.2
30 DIM o(complex)
40 REM add
50 LET o(r)=a(r)+b(r)
60 LET o(i)=a(i)+b(i)
70 PRINT "Result of addition is:": GO SUB 1000
80 REM mult
90 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 1000
120 REM neg
130 LET o(r)=-a(r)
140 LET o(i)=-a(i)
150 PRINT "Result of negation is:": GO SUB 1000
160 LET denom=a(r)^2+a(i)^2
170 LET o(r)=a(r)/denom
180 LET o(i)=-a(i)/denom
190 PRINT "Result of inversion is:": GO SUB 1000
200 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