# 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: ${\displaystyle a+b\times i}$ (sometimes shown as: ${\displaystyle b+a\times i}$ where   ${\displaystyle a}$   and   ${\displaystyle b}$  are real numbers,   and   ${\displaystyle i}$   is    -1

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

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

By definition, the   complex conjugate   of ${\displaystyle a+bi}$ is ${\displaystyle a-bi}$

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

## 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!

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

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
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
RealMult(y R_,y R_,tmp1)  ;tmp1=y.r^2
RealMult(y I_,y I_,tmp2)  ;tmp2=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
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

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:
(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


with Ada.Numerics.Generic_Complex_Types;

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;

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
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));

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
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
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))
}

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

      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{}

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

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);

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 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(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);

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)

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;
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);   // 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;
}

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

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
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

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(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

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


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

provides complex number support supplemented by the code below.

link complex                            # for complex number support

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

procedure COMPLEX(rpart,ipart)          #: new safe record constructor and coercion
initial complex :=: COMPLEX             # get in front of record constructor
return if /ipart & (type(rpart) == "complex")
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
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;
}

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.inv());
System.out.println(a.mult(b));
System.out.println(a.conj());
}
}


## JavaScript

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

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)

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

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

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;
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(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)

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;
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 (mul a b);
print_complex (inv a);
print_complex (neg a);
print_complex (conj a)


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
::attribute r GET
::attribute i GET

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

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

::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

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.

## PascalABC.NET

begin
var a := Cplx(1,2);
var b := Cplx(3,4);
Println(a + b);
Println(a - b);
Println(a * b);
Println(a / b);
Println(-a);
Println(1/a);
Println(a.Real,a.Imaginary);
Println(a.Conjugate);
Println(Abs(a));
Println(a ** b);
end.

Output:
4+6i
-2-2i
-5+10i
0.44+0.08i
-1-2i
0.2-0.4i
1 2
1-2i
2.23606797749979
0.129009594074467+0.0339240929051702i


## 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_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")

(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
/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

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=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))


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} )
}

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, 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 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
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"
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)
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`