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: (sometimes shown as: where and are real numbers, and 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 .
- Task
- Show addition, multiplication, negation, and inversion of complex numbers in separate functions. (Subtraction and division operations can be made with pairs of these operations.)
- Print the results for each operation tested.
- Optional: Show complex conjugation.
By definition, the complex conjugate of
is
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
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
PROC ComplexAdd(Complex POINTER x,y,res)
RealAdd(x R_,y R_,res R_) ;res.r=x.r+y.r
RealAdd(x I_,y I_,res I_) ;res.i=x.i+y.i
RETURN
PROC ComplexSub(Complex POINTER x,y,res)
RealSub(x R_,y R_,res R_) ;res.r=x.r-y.r
RealSub(x I_,y I_,res I_) ;res.i=x.i-y.i
RETURN
PROC ComplexMult(Complex POINTER x,y,res)
REAL tmp1,tmp2
RealMult(x R_,y R_,tmp1) ;tmp1=x.r*y.r
RealMult(x I_,y I_,tmp2) ;tmp2=x.i*y.i
RealSub(tmp1,tmp2,res R_) ;res.r=x.r*y.r-x.i*y.i
RealMult(x R_,y I_,tmp1) ;tmp1=x.r*y.i
RealMult(x I_,y R_,tmp2) ;tmp2=x.i*y.r
RealAdd(tmp1,tmp2,res I_) ;res.i=x.r*y.i+x.i*y.r
RETURN
PROC ComplexDiv(Complex POINTER x,y,res)
REAL tmp1,tmp2,tmp3,tmp4
RealMult(x R_,y R_,tmp1) ;tmp1=x.r*y.r
RealMult(x I_,y I_,tmp2) ;tmp2=x.i*y.i
RealAdd(tmp1,tmp2,tmp3) ;tmp3=x.r*y.r+x.i*y.i
RealMult(y R_,y R_,tmp1) ;tmp1=y.r^2
RealMult(y I_,y I_,tmp2) ;tmp2=y.i^2
RealAdd(tmp1,tmp2,tmp4) ;tmp4=y.r^2+y.i^2
RealDiv(tmp3,tmp4,res R_) ;res.r=(x.r*y.r+x.i*y.i)/(y.r^2+y.i^2)
RealMult(x I_,y R_,tmp1) ;tmp1=x.i*y.r
RealMult(x R_,y I_,tmp2) ;tmp2=x.r*y.i
RealSub(tmp1,tmp2,tmp3) ;tmp3=x.i*y.r-x.r*y.i
RealDiv(tmp3,tmp4,res I_) ;res.i=(x.i*y.r-x.r*y.i)/(y.r^2+y.i^2)
RETURN
PROC ComplexNeg(Complex POINTER x,res)
REAL neg
ValR("-1",neg) ;neg=-1
RealMult(x R_,neg,res R_) ;res.r=-x.r
RealMult(x I_,neg,res I_) ;res.r=-x.r
RETURN
PROC ComplexInv(Complex POINTER x,res)
REAL tmp1,tmp2,tmp3
RealMult(x R_,x R_,tmp1) ;tmp1=x.r^2
RealMult(x I_,x I_,tmp2) ;tmp2=x.i^2
RealAdd(tmp1,tmp2,tmp3) ;tmp3=x.r^2+x.i^2
RealDiv(x R_,tmp3,res R_) ;res.r=x.r/(x.r^2+x.i^2)
ValR("-1",tmp1) ;tmp1=-1
RealMult(x I_,tmp1,tmp2) ;tmp2=-x.i
RealDiv(tmp2,tmp3,res I_) ;res.i=-x.i/(x.r^2+x.i^2)
RETURN
PROC ComplexConj(Complex POINTER x,res)
REAL neg
ValR("-1",neg) ;neg=-1
RealAssign(x R_,res R_) ;res.r=x.r
RealMult(x I_,neg,res I_) ;res.i=-x.i
RETURN
PROC Main()
Complex x,y,res
IntToReal(5,x R_) IntToReal(3,x I_)
IntToReal(4,y R_) ValR("-3",y I_)
Put(125) PutE() ;clear screen
ComplexAdd(x,y,res)
PrintComplexXYZ(x,y,res,"+")
ComplexSub(x,y,res)
PrintComplexXYZ(x,y,res,"-")
ComplexMult(x,y,res)
PrintComplexXYZ(x,y,res,"*")
ComplexDiv(x,y,res)
PrintComplexXYZ(x,y,res,"/")
ComplexNeg(y,res)
PrintComplexXY(y,res," -")
ComplexInv(y,res)
PrintComplexXY(y,res," 1 / ")
ComplexConj(y,res)
PrintComplexXY(y,res," conj")
RETURN- Output:
Screenshot from Atari 8-bit computer
(5+3i) + (4-3i) = 9+0i
(5+3i) - (4-3i) = 1+6i
(5+3i) * (4-3i) = 29-3i
(5+3i) / (4-3i) = .44+1.08i
-(4-3i) = -4+3i
1 / (4-3i) = .16+.12i
conj(4-3i) = 4+3i
Ada
with Ada.Numerics.Generic_Complex_Types;
with Ada.Text_IO.Complex_IO;
procedure Complex_Operations is
-- Ada provides a pre-defined generic package for complex types
-- That package contains definitions for composition,
-- negation, addition, subtraction, multiplication, division,
-- conjugation, exponentiation, and absolute value, as well as
-- basic comparison operations.
-- Ada provides a second pre-defined package for sin, cos, tan, cot,
-- arcsin, arccos, arctan, arccot, and the hyperbolic versions of
-- those trigonometric functions.
-- The package Ada.Numerics.Generic_Complex_Types requires definition
-- with the real type to be used in the complex type definition.
package Complex_Types is new Ada.Numerics.Generic_Complex_Types (Long_Float);
use Complex_Types;
package Complex_IO is new Ada.Text_IO.Complex_IO (Complex_Types);
use Complex_IO;
use Ada.Text_IO;
A : Complex := Compose_From_Cartesian (Re => 1.0, Im => 1.0);
B : Complex := Compose_From_Polar (Modulus => 1.0, Argument => 3.14159);
C : Complex;
begin
-- Addition
C := A + B;
Put("A + B = "); Put(C);
New_Line;
-- Multiplication
C := A * B;
Put("A * B = "); Put(C);
New_Line;
-- Inversion
C := 1.0 / A;
Put("1.0 / A = "); Put(C);
New_Line;
-- Negation
C := -A;
Put("-A = "); Put(C);
New_Line;
-- Conjugation
Put("Conjugate(-A) = ");
C := Conjugate (C); Put(C);
end Complex_Operations;
ALGOL 68
main:(
FORMAT compl fmt = $g(-7,5)"⊥"g(-7,5)$;
PROC compl operations = VOID: (
LONG COMPL a = 1.0 ⊥ 1.0;
LONG COMPL b = 3.14159 ⊥ 1.2;
LONG COMPL c;
printf(($x"a="f(compl fmt)l$,a));
printf(($x"b="f(compl fmt)l$,b));
# addition #
c := a + b;
printf(($x"a+b="f(compl fmt)l$,c));
# multiplication #
c := a * b;
printf(($x"a*b="f(compl fmt)l$,c));
# inversion #
c := 1.0 / a;
printf(($x"1/c="f(compl fmt)l$,c));
# negation #
c := -a;
printf(($x"-a="f(compl fmt)l$,c))
);
compl operations
)- Output:
a=1.00000⊥1.00000 b=3.14159⊥1.20000 a+b=4.14159⊥2.20000 a*b=1.94159⊥4.34159 1/c=0.50000⊥-.50000 -a=-1.0000⊥-1.0000
ALGOL W
Complex is a built-in type in Algol W.
begin
% show some complex arithmetic %
% returns c + d, using the builtin complex + operator %
complex procedure cAdd ( complex value c, d ) ; c + d;
% returns c * d, using the builtin complex * operator %
complex procedure cMul ( complex value c, d ) ; c * d;
% returns the negation of c, using the builtin complex unary - operator %
complex procedure cNeg ( complex value c ) ; - c;
% returns the inverse of c, using the builtin complex / operatror %
complex procedure cInv ( complex value c ) ; 1 / c;
% returns the conjugate of c %
complex procedure cConj ( complex value c ) ; realpart( c ) - imag( imagpart( c ) );
complex c, d;
c := 1 + 2i;
d := 3 + 4i;
% set I/O format for real aand complex numbers %
r_format := "A"; s_w := 0; r_w := 6; r_d := 2;
write( "c : ", c );
write( "d : ", d );
write( "c + d : ", cAdd( c, d ) );
write( "c * d : ", cMul( c, d ) );
write( "-c : ", cNeg( c ) );
write( "1/c : ", cInv( c ) );
write( "conj c : ", cConj( c ) )
end.- Output:
c : 1.00 2.00I d : 3.00 4.00I c + d : 4.00 6.00I c * d : -5.00 10.00I -c : -1.00 -2.00I 1/c : 0.20 -0.40I conj c : 1.00 -2.00I
APL
x←1j1 ⍝assignment
y←5.25j1.5
x+y ⍝addition
6.25J2.5
x×y ⍝multiplication
3.75J6.75
⌹x ⍝inversion
0.5j_0.5
-x ⍝negation
¯1J¯1
App Inventor
App Inventor has native support for complex numbers.
The linked image gives a few examples of complex arithmetic and a custom complex conjugate function.
View the blocks and app screen...
Arturo
a: to :complex [1 1]
b: to :complex @[pi 1.2]
print ["a:" a]
print ["b:" b]
print ["a + b:" a + b]
print ["a * b:" a * b]
print ["1 / a:" 1 / a]
print ["neg a:" neg a]
print ["conj a:" conj a]
- Output:
a: 1.0+1.0i b: 3.141592653589793+1.2i a + b: 4.141592653589793+2.2i a * b: 1.941592653589793+4.341592653589793i 1 / a: 0.5-0.5i neg a: -1.0-1.0i conj a: 1.0-1.0i
AutoHotkey
contributed by Laszlo on the ahk forum
Cset(C,1,1)
MsgBox % Cstr(C) ; 1 + i*1
Cneg(C,C)
MsgBox % Cstr(C) ; -1 - i*1
Cadd(C,C,C)
MsgBox % Cstr(C) ; -2 - i*2
Cinv(D,C)
MsgBox % Cstr(D) ; -0.25 + 0.25*i
Cmul(C,C,D)
MsgBox % Cstr(C) ; 1 + i*0
Cset(ByRef C, re, im) {
VarSetCapacity(C,16)
NumPut(re,C,0,"double")
NumPut(im,C,8,"double")
}
Cre(ByRef C) {
Return NumGet(C,0,"double")
}
Cim(ByRef C) {
Return NumGet(C,8,"double")
}
Cstr(ByRef C) {
Return Cre(C) ((i:=Cim(C))<0 ? " - i*" . -i : " + i*" . i)
}
Cadd(ByRef C, ByRef A, ByRef B) {
VarSetCapacity(C,16)
NumPut(Cre(A)+Cre(B),C,0,"double")
NumPut(Cim(A)+Cim(B),C,8,"double")
}
Cmul(ByRef C, ByRef A, ByRef B) {
VarSetCapacity(C,16)
t := Cre(A)*Cim(B)+Cim(A)*Cre(B)
NumPut(Cre(A)*Cre(B)-Cim(A)*Cim(B),C,0,"double")
NumPut(t,C,8,"double") ; A or B can be C!
}
Cneg(ByRef C, ByRef A) {
VarSetCapacity(C,16)
NumPut(-Cre(A),C,0,"double")
NumPut(-Cim(A),C,8,"double")
}
Cinv(ByRef C, ByRef A) {
VarSetCapacity(C,16)
d := Cre(A)**2 + Cim(A)**2
NumPut( Cre(A)/d,C,0,"double")
NumPut(-Cim(A)/d,C,8,"double")
}
AWK
contributed by af
# simulate a struct using associative arrays
function complex(arr, re, im) {
arr["re"] = re
arr["im"] = im
}
function re(cmplx) {
return cmplx["re"]
}
function im(cmplx) {
return cmplx["im"]
}
function printComplex(cmplx) {
print re(cmplx), im(cmplx)
}
function abs2(cmplx) {
return re(cmplx) * re(cmplx) + im(cmplx) * im(cmplx)
}
function abs(cmplx) {
return sqrt(abs2(cmplx))
}
function add(res, cmplx1, cmplx2) {
complex(res, re(cmplx1) + re(cmplx2), im(cmplx1) + im(cmplx2))
}
function mult(res, cmplx1, cmplx2) {
complex(res, re(cmplx1) * re(cmplx2) - im(cmplx1) * im(cmplx2), re(cmplx1) * im(cmplx2) + im(cmplx1) * re(cmplx2))
}
function scale(res, cmplx, scalar) {
complex(res, re(cmplx) * scalar, im(cmplx) * scalar)
}
function negate(res, cmplx) {
scale(res, cmplx, -1)
}
function conjugate(res, cmplx) {
complex(res, re(cmplx), -im(cmplx))
}
function invert(res, cmplx) {
conjugate(res, cmplx)
scale(res, res, 1 / abs(cmplx))
}
BEGIN {
complex(i, 0, 1)
mult(i, i, i)
printComplex(i)
}
BASIC
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{}
PROCcomplexadd(o{}, a{}, b{})
PRINT "Result of addition is " FNcomplexshow(o{})
PROCcomplexmul(o{}, a{}, b{})
PRINT "Result of multiplication is " ; FNcomplexshow(o{})
PROCcomplexneg(o{}, a{})
PRINT "Result of negation is " ; FNcomplexshow(o{})
PROCcomplexinv(o{}, a{})
PRINT "Result of inversion is " ; FNcomplexshow(o{})
END
DEF PROCcomplexadd(dst{}, one{}, two{})
dst.r = one.r + two.r
dst.i = one.i + two.i
ENDPROC
DEF PROCcomplexmul(dst{}, one{}, two{})
dst.r = one.r*two.r - one.i*two.i
dst.i = one.i*two.r + one.r*two.i
ENDPROC
DEF PROCcomplexneg(dst{}, src{})
dst.r = -src.r
dst.i = -src.i
ENDPROC
DEF PROCcomplexinv(dst{}, src{})
LOCAL denom : denom = src.r^2 + src.i^ 2
dst.r = src.r / denom
dst.i = -src.i / denom
ENDPROC
DEF FNcomplexshow(src{})
IF src.i >= 0 THEN = STR$(src.r) + " + " +STR$(src.i) + "i"
= STR$(src.r) + " - " + STR$(-src.i) + "i"
- Output:
Result of addition is 4.14159265 + 2.2i Result of multiplication is 1.94159265 + 4.34159265i Result of negation is -1 - 1i Result of inversion is 0.5 - 0.5i
Bracmat
Bracmat recognizes the symbol i as the square root of -1. The results of the functions below are not necessarily of the form a+b*i, but as the last example shows, Bracmat nevertheless can work out that two different representations of the same mathematical object, when subtracted from each other, give zero. You may wonder why in the functions multiply and negate there are terms 1 and -1. These terms are a trick to force Bracmat to expand the products. As it is more costly to factorize a sum than to expand a product into a sum, Bracmat retains isolated products. However, when in combination with a non-zero term, the product is expanded.
(add=a b.!arg:(?a,?b)&!a+!b)
& ( multiply
= a b.!arg:(?a,?b)&1+!a*!b+-1
)
& (negate=.1+-1*!arg+-1)
& ( conjugate
= a b
. !arg:i&-i
| !arg:-i&i
| !arg:?a_?b&(conjugate$!a)_(conjugate$!b)
| !arg
)
& ( invert
= conjugated
. conjugate$!arg:?conjugated
& multiply$(!arg,!conjugated)^-1*!conjugated
)
& out$("(a+i*b)+(a+i*b) =" add$(a+i*b,a+i*b))
& out$("(a+i*b)+(a+-i*b) =" add$(a+i*b,a+-i*b))
& out$("(a+i*b)*(a+i*b) =" multiply$(a+i*b,a+i*b))
& out$("(a+i*b)*(a+-i*b) =" multiply$(a+i*b,a+-i*b))
& out$("-1*(a+i*b) =" negate$(a+i*b))
& out$("-1*(a+-i*b) =" negate$(a+-i*b))
& out$("sin$x = " sin$x)
& out$("conjugate sin$x =" conjugate$(sin$x))
& out
$ ("sin$x minus conjugate sin$x =" sin$x+negate$(conjugate$(sin$x)))
& done;- Output:
(a+i*b)+(a+i*b) = 2*a+2*i*b (a+i*b)+(a+-i*b) = 2*a (a+i*b)*(a+i*b) = a^2+-1*b^2+2*i*a*b (a+i*b)*(a+-i*b) = a^2+b^2 -1*(a+i*b) = -1*a+-i*b -1*(a+-i*b) = -1*a+i*b sin$x = i*(-1/2*e^(i*x)+1/2*e^(-i*x)) conjugate sin$x = -i*(1/2*e^(i*x)+-1/2*e^(-i*x)) sin$x minus conjugate sin$x = 0
C
The more recent C99 standard has built-in complex number primitive types, which can be declared with float, double, or long double precision. To use these types and their associated library functions, you must include the <complex.h> header. (Note: this is a different header than the <complex> templates that are defined by C++.) [1] [2]
#include <complex.h>
#include <stdio.h>
void cprint(double complex c)
{
printf("%f%+fI", creal(c), cimag(c));
}
void complex_operations() {
double complex a = 1.0 + 1.0I;
double complex b = 3.14159 + 1.2I;
double complex c;
printf("\na="); cprint(a);
printf("\nb="); cprint(b);
// addition
c = a + b;
printf("\na+b="); cprint(c);
// multiplication
c = a * b;
printf("\na*b="); cprint(c);
// inversion
c = 1.0 / a;
printf("\n1/c="); cprint(c);
// negation
c = -a;
printf("\n-a="); cprint(c);
// conjugate
c = conj(a);
printf("\nconj a="); cprint(c); printf("\n");
}
User-defined type:
typedef struct{
double real;
double imag;
} Complex;
Complex add(Complex a, Complex b){
Complex ans;
ans.real = a.real + b.real;
ans.imag = a.imag + b.imag;
return ans;
}
Complex mult(Complex a, Complex b){
Complex ans;
ans.real = a.real * b.real - a.imag * b.imag;
ans.imag = a.real * b.imag + a.imag * b.real;
return ans;
}
/* it's arguable that things could be better handled if either
a.real or a.imag is +/-inf, but that's much work */
Complex inv(Complex a){
Complex ans;
double denom = a.real * a.real + a.imag * a.imag;
ans.real = a.real / denom;
ans.imag = -a.imag / denom;
return ans;
}
Complex neg(Complex a){
Complex ans;
ans.real = -a.real;
ans.imag = -a.imag;
return ans;
}
Complex conj(Complex a){
Complex ans;
ans.real = a.real;
ans.imag = -a.imag;
return ans;
}
void put(Complex c)
{
printf("%lf%+lfI", c.real, c.imag);
}
void complex_ops(void)
{
Complex a = { 1.0, 1.0 };
Complex b = { 3.14159, 1.2 };
printf("\na="); put(a);
printf("\nb="); put(b);
printf("\na+b="); put(add(a,b));
printf("\na*b="); put(mult(a,b));
printf("\n1/a="); put(inv(a));
printf("\n-a="); put(neg(a));
printf("\nconj a="); put(conj(a)); printf("\n");
}
C#
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);
}
}
}
}
using System;
public struct ComplexNumber
{
public static readonly ComplexNumber i = new ComplexNumber(0.0, 1.0);
public static readonly ComplexNumber Zero = new ComplexNumber(0.0, 0.0);
public double Re;
public double Im;
public ComplexNumber(double re)
{
this.Re = re;
this.Im = 0;
}
public ComplexNumber(double re, double im)
{
this.Re = re;
this.Im = im;
}
public static ComplexNumber operator *(ComplexNumber n1, ComplexNumber n2)
{
return new ComplexNumber(n1.Re * n2.Re - n1.Im * n2.Im,
n1.Im * n2.Re + n1.Re * n2.Im);
}
public static ComplexNumber operator *(double n1, ComplexNumber n2)
{
return new ComplexNumber(n1 * n2.Re, n1 * n2.Im);
}
public static ComplexNumber operator /(ComplexNumber n1, ComplexNumber n2)
{
double n2Norm = n2.Re * n2.Re + n2.Im * n2.Im;
return new ComplexNumber((n1.Re * n2.Re + n1.Im * n2.Im) / n2Norm,
(n1.Im * n2.Re - n1.Re * n2.Im) / n2Norm);
}
public static ComplexNumber operator /(ComplexNumber n1, double n2)
{
return new ComplexNumber(n1.Re / n2, n1.Im / n2);
}
public static ComplexNumber operator +(ComplexNumber n1, ComplexNumber n2)
{
return new ComplexNumber(n1.Re + n2.Re, n1.Im + n2.Im);
}
public static ComplexNumber operator -(ComplexNumber n1, ComplexNumber n2)
{
return new ComplexNumber(n1.Re - n2.Re, n1.Im - n2.Im);
}
public static ComplexNumber operator -(ComplexNumber n)
{
return new ComplexNumber(-n.Re, -n.Im);
}
public static implicit operator ComplexNumber(double n)
{
return new ComplexNumber(n, 0.0);
}
public static explicit operator double(ComplexNumber n)
{
return n.Re;
}
public static bool operator ==(ComplexNumber n1, ComplexNumber n2)
{
return n1.Re == n2.Re && n1.Im == n2.Im;
}
public static bool operator !=(ComplexNumber n1, ComplexNumber n2)
{
return n1.Re != n2.Re || n1.Im != n2.Im;
}
public override bool Equals(object obj)
{
return this == (ComplexNumber)obj;
}
public override int GetHashCode()
{
return Re.GetHashCode() ^ Im.GetHashCode();
}
public override string ToString()
{
return String.Format("{0}+{1}*i", Re, Im);
}
}
public static class ComplexMath
{
public static double Abs(ComplexNumber a)
{
return Math.Sqrt(Norm(a));
}
public static double Norm(ComplexNumber a)
{
return a.Re * a.Re + a.Im * a.Im;
}
public static double Arg(ComplexNumber a)
{
return Math.Atan2(a.Im, a.Re);
}
public static ComplexNumber Inverse(ComplexNumber a)
{
double norm = Norm(a);
return new ComplexNumber(a.Re / norm, -a.Im / norm);
}
public static ComplexNumber Conjugate(ComplexNumber a)
{
return new ComplexNumber(a.Re, -a.Im);
}
public static ComplexNumber Exp(ComplexNumber a)
{
double e = Math.Exp(a.Re);
return new ComplexNumber(e * Math.Cos(a.Im), e * Math.Sin(a.Im));
}
public static ComplexNumber Log(ComplexNumber a)
{
return new ComplexNumber(0.5 * Math.Log(Norm(a)), Arg(a));
}
public static ComplexNumber Power(ComplexNumber a, ComplexNumber power)
{
return Exp(power * Log(a));
}
public static ComplexNumber Power(ComplexNumber a, int power)
{
bool inverse = false;
if (power < 0)
{
inverse = true; power = -power;
}
ComplexNumber result = 1.0;
ComplexNumber multiplier = a;
while (power > 0)
{
if ((power & 1) != 0) result *= multiplier;
multiplier *= multiplier;
power >>= 1;
}
if (inverse)
return Inverse(result);
else
return result;
}
public static ComplexNumber Sqrt(ComplexNumber a)
{
return Exp(0.5 * Log(a));
}
public static ComplexNumber Sin(ComplexNumber a)
{
return Sinh(ComplexNumber.i * a) / ComplexNumber.i;
}
public static ComplexNumber Cos(ComplexNumber a)
{
return Cosh(ComplexNumber.i * a);
}
public static ComplexNumber Sinh(ComplexNumber a)
{
return 0.5 * (Exp(a) - Exp(-a));
}
public static ComplexNumber Cosh(ComplexNumber a)
{
return 0.5 * (Exp(a) + Exp(-a));
}
}
class Program
{
static void Main(string[] args)
{
// usage
ComplexNumber i = 2;
ComplexNumber j = new ComplexNumber(1, -2);
Console.WriteLine(i * j);
Console.WriteLine(ComplexMath.Power(j, 2));
Console.WriteLine((double)ComplexMath.Sin(i) + " vs " + Math.Sin(2));
Console.WriteLine(ComplexMath.Power(j, 0) == 1.0);
}
}
C++
#include <iostream>
#include <complex>
using std::complex;
void complex_operations() {
complex<double> a(1.0, 1.0);
complex<double> b(3.14159, 1.25);
// addition
std::cout << a + b << std::endl;
// multiplication
std::cout << a * b << std::endl;
// inversion
std::cout << 1.0 / a << std::endl;
// negation
std::cout << -a << std::endl;
// conjugate
std::cout << std::conj(a) << std::endl;
}
Clojure
Clojure on the JVM has no native support for Complex numbers. Therefore, we use defrecord and the multimethods in clojure.algo.generic.arithmetic to make a Complex number type.
(ns rosettacode.arithmetic.cmplx
(:require [clojure.algo.generic.arithmetic :as ga])
(:import [java.lang Number]))
(defrecord Complex [^Number r ^Number i]
Object
(toString [{:keys [r i]}]
(apply str
(cond
(zero? r) [(if (= i 1) "" i) "i"]
(zero? i) [r]
:else [r (if (neg? i) "-" "+") i "i"]))))
(defmethod ga/+ [Complex Complex]
[x y] (map->Complex (merge-with + x y)))
(defmethod ga/+ [Complex Number] ; reals become y + 0i
[{:keys [r i]} y] (->Complex (+ r y) i))
(defmethod ga/- Complex
[x] (->> x vals (map -) (apply ->Complex)))
(defmethod ga/* [Complex Complex]
[x y] (map->Complex (merge-with * x y)))
(defmethod ga/* [Complex Number]
[{:keys [r i]} y] (->Complex (* r y) (* i y)))
(ga/defmethod* ga / Complex
[x] (->> x vals (map /) (apply ->Complex)))
(defn conj [^Complex {:keys [r i]}]
(->Complex r (- i)))
(defn inv [^Complex {:keys [r i]}]
(let [m (+ (* r r) (* i i))]
(->Complex (/ r m) (- (/ i m)))))
COBOL
The following is in the Managed COBOL dialect.
.NET Complex class
$SET SOURCEFORMAT "FREE"
$SET ILUSING "System"
$SET ILUSING "System.Numerics"
class-id Prog.
method-id. Main static.
procedure division.
declare num as type Complex = type Complex::ImaginaryOne()
declare results as type Complex occurs any
set content of results to ((num + num), (num * num), (- num), (1 / num), type Complex::Conjugate(num))
perform varying result as type Complex thru results
display result
end-perform
end method.
end class.
Implementation
$SET SOURCEFORMAT "FREE"
class-id Prog.
method-id. Main static.
procedure division.
declare a as type Complex = new Complex(1, 1)
declare b as type Complex = new Complex(3.14159, 1.25)
display "a = " a
display "b = " b
display space
declare result as type Complex = a + b
display "a + b = " result
move (a - b) to result
display "a - b = " result
move (a * b) to result
display "a * b = " result
move (a / b) to result
display "a / b = " result
move (- b) to result
display "-b = " result
display space
display "Inverse of b: " type Complex::Inverse(b)
display "Conjugate of b: " type Complex::Conjugate(b)
end method.
end class.
class-id Complex.
01 Real float-long property.
01 Imag float-long property.
method-id new.
set Real, Imag to 0
end method.
method-id new.
procedure division using value real-val as float-long, imag-val as float-long.
set Real to real-val
set Imag to imag-val
end method.
method-id Norm static.
procedure division using value a as type Complex returning ret as float-long.
compute ret = a::Real ** 2 + a::Imag ** 2
end method.
method-id Inverse static.
procedure division using value a as type Complex returning ret as type Complex.
declare norm as float-long = type Complex::Norm(a)
set ret to new Complex(a::Real / norm, (0 - a::Imag) / norm)
end method.
method-id Conjugate static.
procedure division using value a as type Complex returning c as type Complex.
set c to new Complex(a::Real, 0 - a::Imag)
end method.
method-id ToString override.
procedure division returning str as string.
set str to type String::Format("{0}{1:+#0;-#}i", Real, Imag)
end method.
operator-id + .
procedure division using value a as type Complex, b as type Complex
returning c as type Complex.
set c to new Complex(a::Real + b::Real, a::Imag + b::Imag)
end operator.
operator-id - .
procedure division using value a as type Complex, b as type Complex
returning c as type Complex.
set c to new Complex(a::Real - b::Real, a::Imag - b::Imag)
end operator.
operator-id * .
procedure division using value a as type Complex, b as type Complex
returning c as type Complex.
set c to new Complex(a::Real * b::Real - a::Imag * b::Imag,
a::Real * b::Imag + a::Imag * b::Real)
end operator.
operator-id / .
procedure division using value a as type Complex, b as type Complex
returning c as type Complex.
set c to new Complex()
declare b-norm as float-long = type Complex::Norm(b)
compute c::Real = (a::Real * b::Real + a::Imag * b::Imag) / b-norm
compute c::Imag = (a::Imag * b::Real - a::Real * b::Imag) / b-norm
end operator.
operator-id - .
procedure division using value a as type Complex returning ret as type Complex.
set ret to new Complex(- a::Real, 0 - a::Imag)
end operator.
end class.
CoffeeScript
# create an immutable Complex type
class Complex
constructor: (@r=0, @i=0) ->
@magnitude = @r*@r + @i*@i
plus: (c2) ->
new Complex(
@r + c2.r,
@i + c2.i
)
times: (c2) ->
new Complex(
@r*c2.r - @i*c2.i,
@r*c2.i + @i*c2.r
)
negation: ->
new Complex(
-1 * @r,
-1 * @i
)
inverse: ->
throw Error "no inverse" if @magnitude is 0
new Complex(
@r / @magnitude,
-1 * @i / @magnitude
)
toString: ->
return "#{@r}" if @i == 0
return "#{@i}i" if @r == 0
if @i > 0
"#{@r} + #{@i}i"
else
"#{@r} - #{-1 * @i}i"
# test
do ->
a = new Complex(5, 3)
b = new Complex(4, -3)
sum = a.plus b
console.log "(#{a}) + (#{b}) = #{sum}"
product = a.times b
console.log "(#{a}) * (#{b}) = #{product}"
negation = b.negation()
console.log "-1 * (#{b}) = #{negation}"
diff = a.plus negation
console.log "(#{a}) - (#{b}) = #{diff}"
inverse = b.inverse()
console.log "1 / (#{b}) = #{inverse}"
quotient = product.times inverse
console.log "(#{product}) / (#{b}) = #{quotient}"
- Output:
> coffee complex.coffee (5 + 3i) + (4 - 3i) = 9 (5 + 3i) * (4 - 3i) = 29 - 3i -1 * (4 - 3i) = -4 + 3i (5 + 3i) - (4 - 3i) = 1 + 6i 1 / (4 - 3i) = 0.16 + 0.12i (29 - 3i) / (4 - 3i) = 5 + 3i
Common Lisp
Complex numbers are a built-in numeric type in Common Lisp. The literal syntax for a complex number is #C(real imaginary). The components of a complex number may be integers, ratios, or floating-point. Arithmetic operations automatically return complex (or real) numbers when appropriate:
> (sqrt -1)
#C(0.0 1.0)
> (expt #c(0 1) 2)
-1
Here are some arithmetic operations on complex numbers:
> (+ #c(0 1) #c(1 0))
#C(1 1)
> (* #c(1 1) 2)
#C(2 2)
> (* #c(1 1) #c(0 2))
#C(-2 2)
> (- #c(1 1))
#C(-1 -1)
> (/ #c(0 2))
#C(0 -1/2)
> (conjugate #c(1 1))
#C(1 -1)
Complex numbers can be constructed from real and imaginary parts using the complex function, and taken apart using the realpart and imagpart functions.
> (complex 64 (/ 3 4))
#C(64 3/4)
> (realpart #c(5 5))
5
> (imagpart (complex 0 pi))
3.141592653589793d0
Component Pascal
BlackBox Component Builder
MODULE Complex;
IMPORT StdLog;
TYPE
Complex* = POINTER TO ComplexDesc;
ComplexDesc = RECORD
r-,i-: REAL;
END;
VAR
r,x,y: Complex;
PROCEDURE New(x,y: REAL): Complex;
VAR
r: Complex;
BEGIN
NEW(r);r.r := x;r.i := y;
RETURN r
END New;
PROCEDURE (x: Complex) Add*(y: Complex): Complex,NEW;
BEGIN
RETURN New(x.r + y.r,x.i + y.i)
END Add;
PROCEDURE ( x: Complex) Sub*( y: Complex): Complex, NEW;
BEGIN
RETURN New(x.r - y.r,x.i - y.i)
END Sub;
PROCEDURE ( x: Complex) Mul*( y: Complex): Complex, NEW;
BEGIN
RETURN New(x.r*y.r - x.i*y.i,x.r*y.i + x.i*y.r)
END Mul;
PROCEDURE ( x: Complex) Div*( y: Complex): Complex, NEW;
VAR
d: REAL;
BEGIN
d := y.r * y.r + y.i * y.i;
RETURN New((x.r*y.r + x.i*y.i)/d,(x.i*y.r - x.r*y.i)/d)
END Div;
(* Reciprocal *)
PROCEDURE (x: Complex) Rec*(): Complex,NEW;
VAR
d: REAL;
BEGIN
d := x.r * x.r + x.i * x.i;
RETURN New(x.r/d,(-1.0 * x.i)/d);
END Rec;
(* Conjugate *)
PROCEDURE (x: Complex) Con*(): Complex,NEW;
BEGIN
RETURN New(x.r, (-1.0) * x.i);
END Con;
PROCEDURE (x: Complex) Out(),NEW;
BEGIN
StdLog.String("Complex(");
StdLog.Real(x.r);StdLog.String(',');StdLog.Real(x.i);
StdLog.String("i );")
END Out;
PROCEDURE Do*;
BEGIN
x := New(1.5,3);
y := New(1.0,1.0);
StdLog.String("x: ");x.Out();StdLog.Ln;
StdLog.String("y: ");y.Out();StdLog.Ln;
r := x.Add(y);
StdLog.String("x + y: ");r.Out();StdLog.Ln;
r := x.Sub(y);
StdLog.String("x - y: ");r.Out();StdLog.Ln;
r := x.Mul(y);
StdLog.String("x * y: ");r.Out();StdLog.Ln;
r := x.Div(y);
StdLog.String("x / y: ");r.Out();StdLog.Ln;
r := y.Rec();
StdLog.String("1 / y: ");r.Out();StdLog.Ln;
r := x.Con();
StdLog.String("x': ");r.Out();StdLog.Ln;
END Do;
END Complex.
Execute: ^Q Complex.Do
- Output:
x: Complex( 1.5, 3.0i ); y: Complex( 1.0, 1.0i ); x + y: Complex( 2.5, 4.0i ); x - y: Complex( 0.5, 2.0i ); x * y: Complex( -1.5, 4.5i ); x / y: Complex( 2.25, 0.75i ); 1 / y: Complex( 0.5, -0.5i ); x': Complex( 1.5, -3.0i );
D
Built-in complex numbers are now deprecated in D, to simplify the language.
import std.stdio, std.complex;
void main() {
auto x = complex(1, 1); // complex of doubles on default
auto y = complex(3.14159, 1.2);
writeln(x + y); // addition
writeln(x * y); // multiplication
writeln(1.0 / x); // inversion
writeln(-x); // negation
}
- Output:
4.14159+2.2i 1.94159+4.34159i 0.5-0.5i -1-1i
Dart
class complex {
num real=0;
num imag=0;
complex(num r,num i){
this.real=r;
this.imag=i;
}
complex add(complex b){
return new complex(this.real + b.real, this.imag + b.imag);
}
complex mult(complex b){
//FOIL of (a+bi)(c+di) with i*i = -1
return new complex(this.real * b.real - this.imag * b.imag, this.real * b.imag + this.imag * b.real);
}
complex inv(){
//1/(a+bi) * (a-bi)/(a-bi) = 1/(a+bi) but it's more workable
num denom = real * real + imag * imag;
double r =real/denom;
double i= -imag/denom;
return new complex( r,-i);
}
complex neg(){
return new complex(-real, -imag);
}
complex conj(){
return new complex(real, -imag);
}
String toString(){
return this.real.toString()+' + '+ this.imag.toString()+'*i';
}
}
void main() {
var cl= new complex(1,2);
var cl2= new complex(3,-1);
print(cl.toString());
print(cl2.toString());
print(cl.inv().toString());
print(cl2.mult(cl).toString());
}
Delphi
program Arithmetic_Complex;
{$APPTYPE CONSOLE}
uses
System.SysUtils,
System.VarCmplx;
var
a, b: Variant;
begin
a := VarComplexCreate(5, 3);
b := VarComplexCreate(0.5, 6.0);
writeln(format('(%s) + (%s) = %s',[a,b, a+b]));
writeln(format('(%s) * (%s) = %s',[a,b, a*b]));
writeln(format('-(%s) = %s',[a,- a]));
writeln(format('1/(%s) = %s',[a,1/a]));
writeln(format('conj(%s) = %s',[a,VarComplexConjugate(a)]));
Readln;
end.
- Output:
(5 + 3i) + (0,5 + 6i) = 5,5 + 9i (5 + 3i) * (0,5 + 6i) = -15,5 + 31,5i -(5 + 3i) = -5 - 3i 1/(5 + 3i) = 0,147058823529412 - 0,0882352941176471i conj(5 + 3i) = 5 - 3i
EchoLisp
Complex numbers are part of the language. No special library is needed.
(define a 42+666i) → a
(define b 1+i) → b
(- a) → -42-666i ; negate
(+ a b) → 43+667i ; add
(* a b) → -624+708i ; multiply
(/ b) → 0.5-0.5i ; invert
(conjugate b) → 1-i
(angle b) → 0.7853981633974483 ; = PI/4
(magnitude b) → 1.4142135623730951 ; = sqrt(2)
(exp (* I PI)) → -1+0i ; Euler = e^(I*PI) = -1
Elixir
defmodule Complex do
import Kernel, except: [abs: 1, div: 2]
defstruct real: 0, imag: 0
def new(real, imag) do
%__MODULE__{real: real, imag: imag}
end
def add(a, b) do
{a, b} = convert(a, b)
new(a.real + b.real, a.imag + b.imag)
end
def sub(a, b) do
{a, b} = convert(a, b)
new(a.real - b.real, a.imag - b.imag)
end
def mul(a, b) do
{a, b} = convert(a, b)
new(a.real*b.real - a.imag*b.imag, a.imag*b.real + a.real*b.imag)
end
def div(a, b) do
{a, b} = convert(a, b)
divisor = abs2(b)
new((a.real*b.real + a.imag*b.imag) / divisor,
(a.imag*b.real - a.real*b.imag) / divisor)
end
def neg(a) do
a = convert(a)
new(-a.real, -a.imag)
end
def inv(a) do
a = convert(a)
divisor = abs2(a)
new(a.real / divisor, -a.imag / divisor)
end
def conj(a) do
a = convert(a)
new(a.real, -a.imag)
end
def abs(a) do
:math.sqrt(abs2(a))
end
defp abs2(a) do
a = convert(a)
a.real*a.real + a.imag*a.imag
end
defp convert(a) when is_number(a), do: new(a, 0)
defp convert(%__MODULE__{} = a), do: a
defp convert(a, b), do: {convert(a), convert(b)}
def task do
a = new(1, 3)
b = new(5, 2)
IO.puts "a = #{a}"
IO.puts "b = #{b}"
IO.puts "add(a,b): #{add(a, b)}"
IO.puts "sub(a,b): #{sub(a, b)}"
IO.puts "mul(a,b): #{mul(a, b)}"
IO.puts "div(a,b): #{div(a, b)}"
IO.puts "div(b,a): #{div(b, a)}"
IO.puts "neg(a) : #{neg(a)}"
IO.puts "inv(a) : #{inv(a)}"
IO.puts "conj(a) : #{conj(a)}"
end
end
defimpl String.Chars, for: Complex do
def to_string(%Complex{real: real, imag: imag}) do
if imag >= 0, do: "#{real}+#{imag}j",
else: "#{real}#{imag}j"
end
end
Complex.task
- Output:
a = 1+3j b = 5+2j add(a,b): 6+5j sub(a,b): -4+1j mul(a,b): -1+17j div(a,b): 0.3793103448275862+0.4482758620689655j div(b,a): 1.1-1.3j neg(a) : -1-3j inv(a) : 0.1-0.3j conj(a) : 1-3j
Erlang
%% Task: Complex Arithmetic
%% Author: Abhay Jain
-module(complex_number).
-export([calculate/0]).
-record(complex, {real, img}).
calculate() ->
A = #complex{real=1, img=3},
B = #complex{real=5, img=2},
Sum = add (A, B),
print (Sum),
Product = multiply (A, B),
print (Product),
Negation = negation (A),
print (Negation),
Inversion = inverse (A),
print (Inversion),
Conjugate = conjugate (A),
print (Conjugate).
add (A, B) ->
RealPart = A#complex.real + B#complex.real,
ImgPart = A#complex.img + B#complex.img,
#complex{real=RealPart, img=ImgPart}.
multiply (A, B) ->
RealPart = (A#complex.real * B#complex.real) - (A#complex.img * B#complex.img),
ImgPart = (A#complex.real * B#complex.img) + (B#complex.real * A#complex.img),
#complex{real=RealPart, img=ImgPart}.
negation (A) ->
#complex{real=-A#complex.real, img=-A#complex.img}.
inverse (A) ->
C = conjugate (A),
Mod = (A#complex.real * A#complex.real) + (A#complex.img * A#complex.img),
RealPart = C#complex.real / Mod,
ImgPart = C#complex.img / Mod,
#complex{real=RealPart, img=ImgPart}.
conjugate (A) ->
RealPart = A#complex.real,
ImgPart = -A#complex.img,
#complex{real=RealPart, img=ImgPart}.
print (A) ->
if A#complex.img < 0 ->
io:format("Ans = ~p~pi~n", [A#complex.real, A#complex.img]);
true ->
io:format("Ans = ~p+~pi~n", [A#complex.real, A#complex.img])
end.
- Output:
Ans = 6+5i
Ans = -1+17i
Ans = -1-3i
Ans = 0.1-0.3i
Ans = 1-3i
ERRE
PROGRAM COMPLEX_ARITH
TYPE COMPLEX=(REAL#,IMAG#)
DIM X:COMPLEX,Y:COMPLEX,Z:COMPLEX
!
! complex arithmetic routines
!
DIM A:COMPLEX,B:COMPLEX,C:COMPLEX
PROCEDURE ADD(A.,B.->C.)
C.REAL#=A.REAL#+B.REAL#
C.IMAG#=A.IMAG#+B.IMAG#
END PROCEDURE
PROCEDURE INV(A.->B.)
LOCAL DENOM#
DENOM#=A.REAL#^2+A.IMAG#^2
B.REAL#=A.REAL#/DENOM#
B.IMAG#=-A.IMAG#/DENOM#
END PROCEDURE
PROCEDURE MULT(A.,B.->C.)
C.REAL#=A.REAL#*B.REAL#-A.IMAG#*B.IMAG#
C.IMAG#=A.REAL#*B.IMAG#+A.IMAG#*B.REAL#
END PROCEDURE
PROCEDURE NEG(A.->B.)
B.REAL#=-A.REAL#
B.IMAG#=-A.IMAG#
END PROCEDURE
BEGIN
PRINT(CHR$(12);) !CLS
X.REAL#=1
X.IMAG#=1
Y.REAL#=2
Y.IMAG#=2
ADD(X.,Y.->Z.)
PRINT(Z.REAL#;" + ";Z.IMAG#;"i")
MULT(X.,Y.->Z.)
PRINT(Z.REAL#;" + ";Z.IMAG#;"i")
INV(X.->Z.)
PRINT(Z.REAL#;" + ";Z.IMAG#;"i")
NEG(X.->Z.)
PRINT(Z.REAL#;" + ";Z.IMAG#;"i")
END PROGRAM
Note: Adapted from QuickBasic source code
- Output:
3 + 3 i 0 + 4 i .5 +-.5 i -1 +-1 i
Euler Math Toolbox
>a=1+4i; b=5-3i;
>a+b
6+1i
>a-b
-4+7i
>a*b
17+17i
>a/b
-0.205882352941+0.676470588235i
>fraction a/b
-7/34+23/34i
>conj(a)
1-4iEuphoria
constant REAL = 1, IMAG = 2
type complex(sequence s)
return length(s) = 2 and atom(s[REAL]) and atom(s[IMAG])
end type
function add(complex a, complex b)
return a + b
end function
function mult(complex a, complex b)
return {a[REAL] * b[REAL] - a[IMAG] * b[IMAG],
a[REAL] * b[IMAG] + a[IMAG] * b[REAL]}
end function
function inv(complex a)
atom denom
denom = a[REAL] * a[REAL] + a[IMAG] * a[IMAG]
return {a[REAL] / denom, -a[IMAG] / denom}
end function
function neg(complex a)
return -a
end function
function scomplex(complex a)
sequence s
if a[REAL] != 0 then
s = sprintf("%g",a)
else
s = {}
end if
if a[IMAG] != 0 then
if a[IMAG] = 1 then
s &= "+i"
elsif a[IMAG] = -1 then
s &= "-i"
else
s &= sprintf("%+gi",a[IMAG])
end if
end if
if length(s) = 0 then
return "0"
else
return s
end if
end function
complex a, b
a = { 1.0, 1.0 }
b = { 3.14159, 1.2 }
printf(1,"a = %s\n",{scomplex(a)})
printf(1,"b = %s\n",{scomplex(b)})
printf(1,"a+b = %s\n",{scomplex(add(a,b))})
printf(1,"a*b = %s\n",{scomplex(mult(a,b))})
printf(1,"1/a = %s\n",{scomplex(inv(a))})
printf(1,"-a = %s\n",{scomplex(neg(a))})- Output:
a = 1+i b = 3.14159+1.2i a+b = 4.14159+2.2i a*b = 1.94159+4.34159i 1/a = 0.5-0.5i -a = -1-i
Excel
Take 7 cells, say A1 to G1. Type in :
C1:
=IMSUM(A1;B1)D1:
=IMPRODUCT(A1;B1)E1:
=IMSUB(0;D1)F1:
=IMDIV(1;E28)G1:
=IMCONJUGATE(C28)E1 will have the negation of D1's value
1+2i 3+5i 4+7i -7+11i 7-11i 0,0411764705882353+0,0647058823529412i 4-7i
F#
Entered into an interactive session to show the results:
> open Microsoft.FSharp.Math;;
> let a = complex 1.0 1.0;;
val a : complex = 1r+1i
> let b = complex 3.14159 1.25;;
val b : complex = 3.14159r+1.25i
> a + b;;
val it : Complex = 4.14159r+2.25i {Conjugate = 4.14159r-2.25i;
ImaginaryPart = 2.25;
Magnitude = 4.713307515;
Phase = 0.497661247;
RealPart = 4.14159;
i = 2.25;
r = 4.14159;}
> a * b;;
val it : Complex = 1.89159r+4.39159i {Conjugate = 1.89159r-4.39159i;
ImaginaryPart = 4.39159;
Magnitude = 4.781649868;
Phase = 1.164082262;
RealPart = 1.89159;
i = 4.39159;
r = 1.89159;}
> a / b;;
val it : Complex =
0.384145932435901r+0.165463215905043i
{Conjugate = 0.384145932435901r-0.165463215905043i;
ImaginaryPart = 0.1654632159;
Magnitude = 0.418265673;
Phase = 0.4067140652;
RealPart = 0.3841459324;
i = 0.1654632159;
r = 0.3841459324;}
> -a;;
val it : complex = -1r-1i {Conjugate = -1r+1i;
ImaginaryPart = -1.0;
Magnitude = 1.414213562;
Phase = -2.35619449;
RealPart = -1.0;
i = -1.0;
r = -1.0;}
Factor
USING: combinators kernel math math.functions prettyprint ;
C{ 1 2 } C{ 0.9 -2.78 } {
[ + . ] ! addition
[ - . ] ! subtraction
[ * . ] ! multiplication
[ / . ] ! division
[ ^ . ] ! power
} 2cleave
C{ 1 2 } {
[ neg . ] ! negation
[ recip . ] ! multiplicative inverse
[ conjugate . ] ! complex conjugate
[ sin . ] ! sine
[ log . ] ! natural logarithm
[ sqrt . ] ! square root
} cleave
Forth
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
type complex = (f64,f64)
fun complexAdd((a,b): complex) ((c,d): complex): complex =
(a + c,
b + d)
fun complexMult((a,b): complex) ((c,d): complex): complex =
(a*c - b * d,
a*d + b * c)
fun complexInv((r,i): complex): complex =
let denom = r*r + i * i
in (r / denom,
-i / denom)
fun complexNeg((r,i): complex): complex =
(-r, -i)
fun complexConj((r,i): complex): complex =
(r, -i)
fun main (o: int) (a: complex) (b: complex): complex =
if o == 0 then complexAdd a b
else if o == 1 then complexMult a b
else if o == 2 then complexInv a
else if o == 3 then complexNeg a
else complexConj a
GAP
# GAP knows gaussian integers, gaussian rationals (i.e. Q[i]), and cyclotomic fields. Here are some examples.
# E(n) is an nth primitive root of 1
i := Sqrt(-1);
# E(4)
(3 + 2*i)*(5 - 7*i);
# 29-11*E(4)
1/i;
# -E(4)
Sqrt(-3);
# E(3)-E(3)^2
i in GaussianIntegers;
# true
i/2 in GaussianIntegers;
# false
i/2 in GaussianRationals;
# true
Sqrt(-3) in Cyclotomics;
# true
Go
Go has complex numbers built in, with the complex conjugate in the standard library.
package main
import (
"fmt"
"math/cmplx"
)
func main() {
a := 1 + 1i
b := 3.14159 + 1.25i
fmt.Println("a: ", a)
fmt.Println("b: ", b)
fmt.Println("a + b: ", a+b)
fmt.Println("a * b: ", a*b)
fmt.Println("-a: ", -a)
fmt.Println("1 / a: ", 1/a)
fmt.Println("a̅: ", cmplx.Conj(a))
}
- Output:
a: (1+1i) b: (3.14159+1.25i) a + b: (4.14159+2.25i) a * b: (1.8915899999999999+4.39159i) -a: (-1-1i) 1 / a: (0.5-0.5i) a̅: (1-1i)
Groovy
Groovy does not provide any built-in facility for complex arithmetic. However, it does support arithmetic operator overloading. Thus it is not too hard to build a fairly robust, complete, and intuitive complex number class, such as the following:
class Complex {
final Number real, imag
static final Complex i = [0,1] as Complex
Complex(Number r, Number i = 0) { (real, imag) = [r, i] }
Complex(Map that) { (real, imag) = [that.real ?: 0, that.imag ?: 0] }
Complex plus (Complex c) { [real + c.real, imag + c.imag] as Complex }
Complex plus (Number n) { [real + n, imag] as Complex }
Complex minus (Complex c) { [real - c.real, imag - c.imag] as Complex }
Complex minus (Number n) { [real - n, imag] as Complex }
Complex multiply (Complex c) { [real*c.real - imag*c.imag , imag*c.real + real*c.imag] as Complex }
Complex multiply (Number n) { [real*n , imag*n] as Complex }
Complex div (Complex c) { this * c.recip() }
Complex div (Number n) { this * (1/n) }
Complex negative () { [-real, -imag] as Complex }
/** the complex conjugate of this complex number. Overloads the bitwise complement (~) operator. */
Complex bitwiseNegate () { [real, -imag] as Complex }
/** the magnitude of this complex number. */
// could also use Math.sqrt( (this * (~this)).real )
Number getAbs() { Math.sqrt( real*real + imag*imag ) }
/** the magnitude of this complex number. */
Number abs() { this.abs }
/** the reciprocal of this complex number. */
Complex getRecip() { (~this) / (ρ**2) }
/** the reciprocal of this complex number. */
Complex recip() { this.recip }
/** derived polar angle θ (theta) for polar form. Normalized to 0 ≤ θ < 2π. */
Number getTheta() {
def θ = Math.atan2(imag,real)
θ = θ < 0 ? θ + 2 * Math.PI : θ
}
/** derived polar angle θ (theta) for polar form. Normalized to 0 ≤ θ < 2π. */
Number getΘ() { this.theta } // this is greek uppercase theta
/** derived polar magnitude ρ (rho) for polar form. */
Number getRho() { this.abs }
/** derived polar magnitude ρ (rho) for polar form. */
Number getΡ() { this.abs } // this is greek uppercase rho, not roman P
/** Runs Euler's polar-to-Cartesian complex conversion,
* converting [ρ, θ] inputs into a [real, imag]-based complex number */
static Complex fromPolar(Number ρ, Number θ) {
[ρ * Math.cos(θ), ρ * Math.sin(θ)] as Complex
}
/** Creates new complex with same magnitude ρ, but different angle θ */
Complex withTheta(Number θ) { fromPolar(this.rho, θ) }
/** Creates new complex with same magnitude ρ, but different angle θ */
Complex withΘ(Number θ) { fromPolar(this.rho, θ) }
/** Creates new complex with same angle θ, but different magnitude ρ */
Complex withRho(Number ρ) { fromPolar(ρ, this.θ) }
/** Creates new complex with same angle θ, but different magnitude ρ */
Complex withΡ(Number ρ) { fromPolar(ρ, this.θ) } // this is greek uppercase rho, not roman P
static Complex exp(Complex c) { fromPolar(Math.exp(c.real), c.imag) }
static Complex log(Complex c) { [Math.log(c.rho), c.theta] as Complex }
Complex power(Complex c) {
def zero = [0] as Complex
(this == zero && c != zero) \
? zero \
: c == 1 \
? this \
: exp( log(this) * c )
}
Complex power(Number n) { this ** ([n, 0] as Complex) }
boolean equals(that) {
that != null && (that instanceof Complex \
? [this.real, this.imag] == [that.real, that.imag] \
: that instanceof Number && [this.real, this.imag] == [that, 0])
}
int hashCode() { [real, imag].hashCode() }
String toString() {
def realPart = "${real}"
def imagPart = imag.abs() == 1 ? "i" : "${imag.abs()}i"
real == 0 && imag == 0 \
? "0" \
: real == 0 \
? (imag > 0 ? '' : "-") + imagPart \
: imag == 0 \
? realPart \
: realPart + (imag > 0 ? " + " : " - ") + imagPart
}
}
The following ComplexCategory class allows for modification of regular Number behavior when interacting with Complex.
import org.codehaus.groovy.runtime.DefaultGroovyMethods
class ComplexCategory {
static Complex getI (Number a) { [0, a] as Complex }
static Complex plus (Number a, Complex b) { b + a }
static Complex minus (Number a, Complex b) { -b + a }
static Complex multiply (Number a, Complex b) { b * a }
static Complex div (Number a, Complex b) { ([a] as Complex) / b }
static Complex power (Number a, Complex b) { ([a] as Complex) ** b }
static <N extends Number,T> T asType (N a, Class<T> type) {
type == Complex \
? [a as Number] as Complex
: DefaultGroovyMethods.asType(a, type)
}
}
Notice also that this solution takes liberal advantage of Groovy's full Unicode support, including support for non-English alphabets used in identifiers.
Test Program (mixes the ComplexCategory methods into the Number class):
import static Complex.*
Number.metaClass.mixin ComplexCategory
Integer.metaClass.mixin ComplexCategory
def ε = 0.000000001 // tolerance (epsilon): acceptable "wrongness" to account for rounding error
println 'Demo 1: functionality as requested'
def a = [5,3] as Complex
def a1 = [real:5, imag:3] as Complex
def a2 = 5 + 3.i
def a3 = 5 + 3*i
assert a == a1 && a == a2 && a == a3
println 'a == ' + a
def b = [0.5,6] as Complex
println 'b == ' + b
println "a + b == (${a}) + (${b}) == " + (a + b)
println "a * b == (${a}) * (${b}) == " + (a * b)
assert a + (-a) == 0
println "-a == -(${a}) == " + (-a)
assert (a * a.recip - 1).abs < ε
println "1/a == (${a}).recip == " + (a.recip)
println "a * 1/a == " + (a * a.recip)
println()
println 'Demo 2: other functionality not requested, but important for completeness'
def c = 10
def d = 10 as Complex
assert d instanceof Complex && c instanceof Number && d == c
assert a + c == c + a
println "a + 10 == 10 + a == " + (c + a)
assert c - a == -(a - c)
println "10 - a == -(a - 10) == " + (c - a)
println "a - b == (${a}) - (${b}) == " + (a - b)
assert c * a == a * c
println "10 * a == a * 10 == " + (c * a)
assert (c / a - (a / c).recip).abs < ε
println "10 / a == 1 / (a / 10) == " + (c / a)
println "a / b == (${a}) / (${b}) == " + (a / b)
assert (a ** 2 - a * a).abs < ε
println "a ** 2 == a * a == " + (a ** 2)
println "0.9 ** b == " + (0.9 ** b)
println "a ** b == (${a}) ** (${b}) == " + (a ** b)
println 'a.real == ' + a.real
println 'a.imag == ' + a.imag
println '|a| == ' + a.abs
println 'a.rho == ' + a.rho
println 'a.ρ == ' + a.ρ
println 'a.theta == ' + a.theta
println 'a.θ == ' + a.θ
println '~a (conjugate) == ' + ~a
def ρ = 10
def π = Math.PI
def n = 3
def θ = π / n
def fromPolar1 = fromPolar(ρ, θ) // direct polar-to-cartesian conversion
def fromPolar2 = exp(θ.i) * ρ // Euler's equation
println "ρ*cos(θ) + i*ρ*sin(θ) == ${ρ}*cos(π/${n}) + i*${ρ}*sin(π/${n})"
println " == 10*0.5 + i*10*√(3/4) == " + fromPolar1
println "ρ*exp(i*θ) == ${ρ}*exp(i*π/${n}) == " + fromPolar2
assert (fromPolar1 - fromPolar2).abs < ε
- Output:
Demo 1: functionality as requested
a == 5 + 3i
b == 0.5 + 6i
a + b == (5 + 3i) + (0.5 + 6i) == 5.5 + 9i
a * b == (5 + 3i) * (0.5 + 6i) == -15.5 + 31.5i
-a == -(5 + 3i) == -5 - 3i
1/a == (5 + 3i).recip == 0.1470588235 - 0.0882352941i
a * 1/a == 0.9999999998
Demo 2: other functionality not requested, but important for completeness
a + 10 == 10 + a == 15 + 3i
10 - a == -(a - 10) == 5 - 3i
a - b == (5 + 3i) - (0.5 + 6i) == 4.5 - 3i
10 * a == a * 10 == 50 + 30i
10 / a == 1 / (a / 10) == 1.4705882350 - 0.8823529410i
a / b == (5 + 3i) / (0.5 + 6i) == 0.5655172413793104 - 0.7862068965517242i
a ** 2 == a * a == 16.000000000000004 + 30.000000000000007i
0.9 ** b == 0.7653514303676113 - 0.5605686291920475i
a ** b == (5 + 3i) ** (0.5 + 6i) == -0.013750112198456853 - 0.09332524760169052i
a.real == 5
a.imag == 3
|a| == 5.830951894845301
a.rho == 5.830951894845301
a.ρ == 5.830951894845301
a.theta == 0.5404195002705842
a.θ == 0.5404195002705842
~a (conjugate) == 5 - 3i
ρ*cos(θ) + i*ρ*sin(θ) == 10*cos(π/3) + i*10*sin(π/3)
== 10*0.5 + i*10*√(3/4) == 5.000000000000001 + 8.660254037844386i
ρ*exp(i*θ) == 10*exp(i*π/3) == 5.000000000000001 + 8.660254037844386i
Hare
use fmt;
use math::complex::{c128,addc128,mulc128,divc128,negc128,conjc128};
export fn main() void = {
let x: c128 = (1.0, 1.0);
let y: c128 = (3.14159265, 1.2);
// addition
let (re, im) = addc128(x, y);
fmt::printfln("{} + {}i", re, im)!;
// multiplication
let (re, im) = mulc128(x, y);
fmt::printfln("{} + {}i", re, im)!;
// inversion
let (re, im) = divc128((1.0, 0.0), x);
fmt::printfln("{} + {}i", re, im)!;
// negation
let (re, im) = negc128(x);
fmt::printfln("{} + {}i", re, im)!;
// conjugate
let (re, im) = conjc128(x);
fmt::printfln("{} + {}i", re, im)!;
};Haskell
Complex numbers are parameterized in their base type, so you can have Complex Integer for the Gaussian Integers, Complex Float, Complex Double, etc. The operations are just the usual overloaded numeric operations.
import Data.Complex
main = do
let a = 1.0 :+ 2.0 -- complex number 1+2i
let b = 4 -- complex number 4+0i
-- 'b' is inferred to be complex because it's used in
-- arithmetic with 'a' below.
putStrLn $ "Add: " ++ show (a + b)
putStrLn $ "Subtract: " ++ show (a - b)
putStrLn $ "Multiply: " ++ show (a * b)
putStrLn $ "Divide: " ++ show (a / b)
putStrLn $ "Negate: " ++ show (-a)
putStrLn $ "Inverse: " ++ show (recip a)
putStrLn $ "Conjugate:" ++ show (conjugate a)
- Output:
*Main> main Add: 5.0 :+ 2.0 Subtract: (-3.0) :+ 2.0 Multiply: 4.0 :+ 8.0 Divide: 0.25 :+ 0.5 Negate: (-1.0) :+ (-2.0) Inverse: 0.2 :+ (-0.4) Conjugate:1.0 :+ (-2.0)
Icon and Unicon
Icon doesn't provide native support for complex numbers. Support is included in the IPL. Note: see the Unicon section below for a Unicon-specific solution.
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.
To take full advantage of the overloaded 'complex' procedure, the other cpxxxx procedures would need to be rewritten or overloaded.
- Output:
#complexdemo.exe a := (1.0+2.0i) b := (3.0+4.0i) c := (3.141592653589793+1.5i) d := (1.0+0.0i) e := (0.0+1.0i) a+b := (4.0+6.0i) a-b := (-2.0-2.0i) a*b := (-5.0+10.0i) a/b := (0.44+0.08i) neg(a) := (-1.0-2.0i) inv(a) := (0.2+0.4i) conj(a) := (1.0-2.0i) abs(a) := 2.23606797749979 neg(1) := (-1.0+0.0i)
IDL
complex (and dcomplex for double-precision) is a built-in data type in IDL:
x=complex(1,1)
y=complex(!pi,1.2)
print,x+y
( 4.14159, 2.20000)
print,x*y
( 1.94159, 4.34159)
print,-x
( -1.00000, -1.00000)
print,1/x
( 0.500000, -0.500000)
J
Complex numbers are a native numeric data type in J. Although the examples shown here are performed on scalars, all numeric operations naturally apply to arrays of complex numbers.
x=: 1j1
y=: 3.14159j1.2
x+y NB. addition
4.14159j2.2
x*y NB. multiplication
1.94159j4.34159
%x NB. inversion
0.5j_0.5
-x NB. negation
_1j_1
+x NB. (complex) conjugation
1j_1
Java
public class Complex {
public final double real;
public final double imag;
public Complex() {
this(0, 0);
}
public Complex(double r, double i) {
real = r;
imag = i;
}
public Complex add(Complex b) {
return new Complex(this.real + b.real, this.imag + b.imag);
}
public Complex mult(Complex b) {
// FOIL of (a+bi)(c+di) with i*i = -1
return new Complex(this.real * b.real - this.imag * b.imag,
this.real * b.imag + this.imag * b.real);
}
public Complex inv() {
// 1/(a+bi) * (a-bi)/(a-bi) = 1/(a+bi) but it's more workable
double denom = real * real + imag * imag;
return new Complex(real / denom, -imag / denom);
}
public Complex neg() {
return new Complex(-real, -imag);
}
public Complex conj() {
return new Complex(real, -imag);
}
@Override
public String toString() {
return real + " + " + imag + " * i";
}
public static void main(String[] args) {
Complex a = new Complex(Math.PI, -5); //just some numbers
Complex b = new Complex(-1, 2.5);
System.out.println(a.neg());
System.out.println(a.add(b));
System.out.println(a.inv());
System.out.println(a.mult(b));
System.out.println(a.conj());
}
}
JavaScript
function Complex(r, i) {
this.r = r;
this.i = i;
}
Complex.add = function() {
var num = arguments[0];
for(var i = 1, ilim = arguments.length; i < ilim; i += 1){
num.r += arguments[i].r;
num.i += arguments[i].i;
}
return num;
}
Complex.multiply = function() {
var num = arguments[0];
for(var i = 1, ilim = arguments.length; i < ilim; i += 1){
num.r = (num.r * arguments[i].r) - (num.i * arguments[i].i);
num.i = (num.i * arguments[i].r) - (num.r * arguments[i].i);
}
return num;
}
Complex.negate = function (z) {
return new Complex(-1*z.r, -1*z.i);
}
Complex.invert = function(z) {
var denom = Math.pow(z.r,2) + Math.pow(z.i,2);
return new Complex(z.r/denom, -1*z.i/denom);
}
Complex.conjugate = function(z) {
return new Complex(z.r, -1*z.i);
}
// BONUSES!
Complex.prototype.toString = function() {
return this.r === 0 && this.i === 0
? "0"
: (this.r !== 0 ? this.r : "")
+ ((this.r !== 0 || this.i < 0) && this.i !== 0
? (this.i > 0 ? "+" : "-")
: "" ) + ( this.i !== 0 ? Math.abs(this.i) + "i" : "" );
}
Complex.prototype.getMod = function() {
return Math.sqrt( Math.pow(this.r,2) , Math.pow(this.i,2) )
}
jq
For speed and for conformance with the complex plane interpretation, x+iy is represented as [x,y]; for flexibility, all the functions defined here will accept both real and complex numbers; and for uniformity, they are implemented as functions that ignore their input.
Recent versions of jq support modules, so these functions could all be placed in a module to avoid name conflicts, and thus no special prefix is used here.
def real(z): if (z|type) == "number" then z else z[0] end;
def imag(z): if (z|type) == "number" then 0 else z[1] end;
def plus(x; y):
if (x|type) == "number" then
if (y|type) == "number" then [ x+y, 0 ]
else [ x + y[0], y[1]]
end
elif (y|type) == "number" then plus(y;x)
else [ x[0] + y[0], x[1] + y[1] ]
end;
def multiply(x; y):
if (x|type) == "number" then
if (y|type) == "number" then [ x*y, 0 ]
else [x * y[0], x * y[1]]
end
elif (y|type) == "number" then multiply(y;x)
else [ x[0] * y[0] - x[1] * y[1],
x[0] * y[1] + x[1] * y[0]]
end;
def multiply: reduce .[] as $x (1; multiply(.; $x));
def negate(x): multiply(-1; x);
def minus(x; y): plus(x; multiply(-1; y));
def conjugate(z):
if (z|type) == "number" then [z, 0]
else [z[0], -(z[1]) ]
end;
def invert(z):
if (z|type) == "number" then [1/z, 0]
else
( (z[0] * z[0]) + (z[1] * z[1]) ) as $d
# use "0 + ." to convert -0 back to 0
| [ z[0]/$d, (0 + -(z[1]) / $d)]
end;
def divide(x;y): multiply(x; invert(y));
def exp(z):
def expi(x): [ (x|cos), (x|sin) ];
if (z|type) == "number" then z|exp
elif z[0] == 0 then expi(z[1]) # for efficiency
else multiply( (z[0]|exp); expi(z[1]) )
end ;
def test(x;y):
"x = \( x )",
"y = \( y )",
"x+y: \( plus(x;y))",
"x*y: \( multiply(x;y))",
"-x: \( negate(x))",
"1/x: \( invert(x))",
"conj(x): \( conjugate(x))",
"(x/y)*y: \( multiply( divide(x;y) ; y) )",
"e^iπ: \( exp( [0, 4 * (1|atan) ] ) )"
;
test( [1,1]; [0,1] )- Output:
$ jq -n -f complex.jq
"x = [1,1]"
"y = [0,1]"
"x+y: [1,2]"
"x*y: [-1,1]"
"-x: [-1,-1]"
"1/x: [0.5,-0.5]"
"conj(x): [1,-1]"
"(x/y)*y: [1,1]"
"e^iπ: [-1,1.2246467991473532e-16]"Julia
Julia has built-in support for complex arithmetic with arbitrary real types.
julia> z1 = 1.5 + 3im
julia> z2 = 1.5 + 1.5im
julia> z1 + z2
3.0 + 4.5im
julia> z1 - z2
0.0 + 1.5im
julia> z1 * z2
-2.25 + 6.75im
julia> z1 / z2
1.5 + 0.5im
julia> - z1
-1.5 - 3.0im
julia> conj(z1), z1' # two ways to conjugate
(1.5 - 3.0im,1.5 - 3.0im)
julia> abs(z1)
3.3541019662496847
julia> z1^z2
-1.102482955327779 - 0.38306415117199305im
julia> real(z1)
1.5
julia> imag(z1)
3.0
Kotlin
class Complex(private val real: Double, private val imag: Double) {
operator fun plus(other: Complex) = Complex(real + other.real, imag + other.imag)
operator fun times(other: Complex) = Complex(
real * other.real - imag * other.imag,
real * other.imag + imag * other.real
)
fun inv(): Complex {
val denom = real * real + imag * imag
return Complex(real / denom, -imag / denom)
}
operator fun unaryMinus() = Complex(-real, -imag)
operator fun minus(other: Complex) = this + (-other)
operator fun div(other: Complex) = this * other.inv()
fun conj() = Complex(real, -imag)
override fun toString() =
if (imag >= 0.0) "$real + ${imag}i"
else "$real - ${-imag}i"
}
fun main(args: Array<String>) {
val x = Complex(1.0, 3.0)
val y = Complex(5.0, 2.0)
println("x = $x")
println("y = $y")
println("x + y = ${x + y}")
println("x - y = ${x - y}")
println("x * y = ${x * y}")
println("x / y = ${x / y}")
println("-x = ${-x}")
println("1 / x = ${x.inv()}")
println("x* = ${x.conj()}")
}
- Output:
x = 1.0 + 3.0i y = 5.0 + 2.0i x + y = 6.0 + 5.0i x - y = -4.0 + 1.0i x * y = -1.0 + 17.0i x / y = 0.3793103448275862 + 0.4482758620689655i -x = -1.0 - 3.0i 1 / x = 0.1 - 0.3i x* = 1.0 - 3.0i
Lambdatalk
{require lib_complex}
{def z1 {C.new 1 1}}
-> z1 = (1 1)
{C.x {z1}} -> 1
{C.y {z1}} -> 1
{C.mod {z1}} -> 1.4142135623730951
{C.arg {z1}} -> 0.7853981633974483 // 45°
{C.conj {z1}} -> (1 -1)
{C.negat {z1}} -> (-1 -1)
{C.invert {z1}} -> (0.5 -0.4999999999999999)
{C.sqrt {z1}} -> (1.0986841134678098 0.45508986056222733)
{C.exp {z1}} -> (1.4686939399158851 2.2873552871788423)
{C.log {z1}} -> (0.3465735902799727 0.7853981633974483)
{def z2 {C.new 1.5 1.5}}
-> z2 = (1.5 1.5)
{C.add {z1} {z2}} -> (2.5 2.5)
{C.sub {z1} {z2}} -> (-0.5 -0.5)
{C.mul {z1} {z2}} -> (0 3)
{C.div {z1} {z2}} -> (0.6666666666666667 0)
Lang
fp.cprint = ($z) -> fn.printf(%.3f%+.3fi%n, fn.creal($z), fn.cimag($z))
$a = fn.complex(1.5, 3)
$b = fn.complex(1.5, 1.5)
fn.print(a =\s)
fp.cprint($a)
fn.print(b =\s)
fp.cprint($b)
# Addition
fn.print(a + b =\s)
fp.cprint(fn.cadd($a, $b))
# Multiplication
fn.print(a * b =\s)
fp.cprint(fn.cmul($a, $b))
# Inversion
fn.print(1/a =\s)
fp.cprint(fn.cdiv(fn.complex(1, 0), $a))
# Negation
fn.print(-a =\s)
fp.cprint(fn.cinv($a))
# Conjugate
fn.print(conj(a) =\s)
fp.cprint(fn.conj($a))- Output:
a = 1.500+3.000i b = 1.500+1.500i a + b = 3.000+4.500i a * b = -2.250+6.750i 1/a = 0.133-0.267i -a = -1.500-3.000i conj(a) = 1.500-3.000i
LFE
There is no native support for complex numbers in either LFE or Erlang. As such, this example shows how to implement complex support. There is, however, an LFE library that offers a complex number data type and many mathematical functions which support this data type: complex.
A convenient data structure for a complex number is the record:
(defrecord complex
real
img)
Here are the required functions:
(defun add
(((match-complex real r1 img i1)
(match-complex real r2 img i2))
(new (+ r1 r2) (+ i1 i2))))
(defun mult
(((match-complex real r1 img i1)
(match-complex real r2 img i2))
(new (- (* r1 r2) (* i1 i2))
(+ (* r1 i2) (* r2 i1)))))
(defun neg
(((match-complex real r img i))
(new (* -1 r) (* -1 i))))
(defun inv (cmplx)
(div (conj cmplx) (modulus cmplx)))
Bonus:
(defun conj
(((match-complex real r img i))
(new r (* -1 i))))
The functions above are built using the following supporting functions:
(defun new (r i)
(make-complex real r img i))
(defun modulus (cmplx)
(mult cmplx (conj cmplx)))
(defun div (c1 c2)
(let* ((denom (complex-real (modulus c2)))
(c3 (mult c1 (conj c2))))
(new (/ (complex-real c3) denom)
(/ (complex-img c3) denom)))))
Finally, we have some functions for use in the conversion and display of our complex number data structure:
(defun ->str
(((match-complex real r img i)) (when (>= i 0))
(->str r i "+"))
(((match-complex real r img i))
(->str r i "")))
(defun ->str (r i pos)
(io_lib:format "~p ~s~pi" `(,r ,pos ,i)))
(defun print (cmplx)
(io:format (++ (->str cmplx) "~n")))
Usage is as follows:
> (set ans1 (add c1 c2)) #(complex 2.5 4.0) > (set ans2 (mult c1 c2)) #(complex -1.5 4.5) > (set ans3 (inv c2)) #(complex 0.5 -0.5) > (set ans4 (conj c1)) #(complex 1.5 -3.0)
These can be printed in the following manner:
> (progn (lists:map #'print/1 `(,ans1 ,ans2 ,ans3 ,ans4)) 'ok) 2.5 +4.0i -1.5 +4.5i 0.5 -0.5i 1.5 -3.0i ok
Liberty BASIC
mainwin 50 10
print " Adding"
call cprint cadd$( complex$( 1, 1), complex$( 3.14159265, 1.2))
print " Multiplying"
call cprint cmulti$( complex$( 1, 1), complex$( 3.14159265, 1.2))
print " Inverting"
call cprint cinv$( complex$( 1, 1))
print " Negating"
call cprint cneg$( complex$( 1, 1))
end
sub cprint cx$
print "( "; word$( cx$, 1); " + i *"; word$( cx$, 2); ")"
end sub
function complex$( a , bj )
''complex number string-object constructor
complex$ = str$( a ) ; " " ; str$( bj )
end function
function cadd$( a$ , b$ )
ar = val( word$( a$ , 1 ) )
ai = val( word$( a$ , 2 ) )
br = val( word$( b$ , 1 ) )
bi = val( word$( b$ , 2 ) )
cadd$ = complex$( ar + br , ai + bi )
end function
function cmulti$( a$ , b$ )
ar = val( word$( a$ , 1 ) )
ai = val( word$( a$ , 2 ) )
br = val( word$( b$ , 1 ) )
bi = val( word$( b$ , 2 ) )
cmulti$ = complex$( ar * br - ai * bi _
, ar * bi + ai * br )
end function
function cneg$( a$)
ar = val( word$( a$ , 1 ) )
ai = val( word$( a$ , 2 ) )
cneg$ =complex$( 0 -ar, 0 -ai)
end function
function cinv$( a$)
ar = val( word$( a$ , 1 ) )
ai = val( word$( a$ , 2 ) )
D =ar^2 +ai^2
cinv$ =complex$( ar /D , 0 -ai /D )
end functionLua
--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 IOther than that, the task merely asks for
x+y;
x*y;
-x;
1/x;Mathematica / Wolfram Language
Mathematica has fully implemented support for complex numbers throughout the software. Addition, subtraction, division, multiplications and powering need no further syntax than for real numbers:
x=1+2I
y=3+4I
x+y => 4 + 6 I
x-y => -2 - 2 I
y x => -5 + 10 I
y/x => 11/5 - (2 I)/5
x^3 => -11 - 2 I
y^4 => -527 - 336 I
x^y => (1 + 2 I)^(3 + 4 I)
N[x^y] => 0.12901 + 0.0339241 I
Powering to a complex power can in general not be written shorter, so Mathematica leaves it unevaluated if the numbers are exact. An approximation can be acquired using the function N. However Mathematica goes much further, basically all functions can handle complex numbers to arbitrary precision, including (but not limited to!):
Exp Log
Sin Cos Tan Csc Sec Cot
ArcSin ArcCos ArcTan ArcCsc ArcSec ArcCot
Sinh Cosh Tanh Csch Sech Coth
ArcSinh ArcCosh ArcTanh ArcCsch ArcSech ArcCoth
Sinc
Haversine InverseHaversine
Factorial Gamma PolyGamma LogGamma
Erf BarnesG Hyperfactorial Zeta ProductLog RamanujanTauL
and many many more. The documentation states:
Mathematica has fundamental support for both explicit complex numbers and symbolic complex variables. All applicable mathematical functions support arbitrary-precision evaluation for complex values of all parameters, and symbolic operations automatically treat complex variables with full generality.
MATLAB
Complex numbers are a primitive data type in MATLAB. All the typical complex operations can be performed. There are two keywords that specify a number as complex: "i" and "j".
>> a = 1+i
a =
1.000000000000000 + 1.000000000000000i
>> b = 3+7i
b =
3.000000000000000 + 7.000000000000000i
>> a+b
ans =
4.000000000000000 + 8.000000000000000i
>> a-b
ans =
-2.000000000000000 - 6.000000000000000i
>> a*b
ans =
-4.000000000000000 +10.000000000000000i
>> a/b
ans =
0.172413793103448 - 0.068965517241379i
>> -a
ans =
-1.000000000000000 - 1.000000000000000i
>> a'
ans =
1.000000000000000 - 1.000000000000000i
>> a^b
ans =
0.000808197112874 - 0.011556516327187i
>> norm(a)
ans =
1.414213562373095
Maxima
z1: 5 + 2 * %i;
2*%i+5
z2: 3 - 7 * %i;
3-7*%i
carg(z1);
atan(2/5)
cabs(z1);
sqrt(29)
rectform(z1 * z2);
29-29*%i
polarform(z1);
sqrt(29)*%e^(%i*atan(2/5))
conjugate(z1);
5-2*%i
z1 + z2;
8-5*%i
z1 - z2;
9*%i+2
z1 * z2;
(3-7*%i)*(2*%i+5)
z1 * z2, rectform;
29-29*%i
z1 / z2;
(2*%i+5)/(3-7*%i)
z1 / z2, rectform;
(41*%i)/58+1/58
realpart(z1);
5
imagpart(z1);
2
МК-61/52
Instrustion:
Z1 = a + ib; Z2 = c + id;
a С/П b С/П c С/П d С/П
Division: С/П; multiplication: БП 36 С/П; addition: БП 54 С/П; subtraction: БП 63 С/П.
ПA С/П ПB С/П ПC С/П ПD С/П ИПC x^2
ИПD x^2 + П3 ИПA ИПC * ИПB ИПD *
+ ИП3 / П1 ИПB ИПC * ИПA ИПD *
- ИП3 / П2 ИП1 С/П ИПA ИПC * ИПB
ИПD * - П1 ИПB ИПC * ИПA ИПD *
+ П2 ИП1 С/П ИПB ИПD + П2 ИПA ИПC
+ ИП1 С/П ИПB ИПD - П2 ИПA ИПC -
П1 С/П
Modula-2
MODULE complex;
IMPORT InOut;
TYPE Complex = RECORD R, Im : REAL END;
VAR z : ARRAY [0..3] OF Complex;
PROCEDURE ShowComplex (str : ARRAY OF CHAR; p : Complex);
BEGIN
InOut.WriteString (str); InOut.WriteString (" = ");
InOut.WriteReal (p.R, 6, 2);
IF p.Im >= 0.0 THEN InOut.WriteString (" + ") ELSE InOut.WriteString (" - ") END;
InOut.WriteReal (ABS (p.Im), 6, 2); InOut.WriteString (" i ");
InOut.WriteLn; InOut.WriteBf
END ShowComplex;
PROCEDURE AddComplex (x1, x2 : Complex; VAR x3 : Complex);
BEGIN
x3.R := x1.R + x2.R;
x3.Im := x1.Im + x2.Im
END AddComplex;
PROCEDURE SubComplex (x1, x2 : Complex; VAR x3 : Complex);
BEGIN
x3.R := x1.R - x2.R;
x3.Im := x1.Im - x2.Im
END SubComplex;
PROCEDURE MulComplex (x1, x2 : Complex; VAR x3 : Complex);
BEGIN
x3.R := x1.R * x2.R - x1.Im * x2.Im;
x3.Im := x1.R * x2.Im + x1.Im * x2.R
END MulComplex;
PROCEDURE InvComplex (x1 : Complex; VAR x2 : Complex);
BEGIN
x2.R := x1.R / (x1.R * x1.R + x1.Im * x1.Im);
x2.Im := -1.0 * x1.Im / (x1.R * x1.R + x1.Im * x1.Im)
END InvComplex;
PROCEDURE NegComplex (x1 : Complex; VAR x2 : Complex);
BEGIN
x2.R := - x1.R; x2.Im := - x1.Im
END NegComplex;
BEGIN
InOut.WriteString ("Enter two complex numbers : ");
InOut.WriteBf;
InOut.ReadReal (z[0].R); InOut.ReadReal (z[0].Im);
InOut.ReadReal (z[1].R); InOut.ReadReal (z[1].Im);
ShowComplex ("z1", z[0]); ShowComplex ("z2", z[1]);
InOut.WriteLn;
AddComplex (z[0], z[1], z[2]); ShowComplex ("z1 + z2", z[2]);
SubComplex (z[0], z[1], z[2]); ShowComplex ("z1 - z2", z[2]);
MulComplex (z[0], z[1], z[2]); ShowComplex ("z1 * z2", z[2]);
InvComplex (z[0], z[2]); ShowComplex ("1 / z1", z[2]);
NegComplex (z[0], z[2]); ShowComplex (" - z1", z[2]);
InOut.WriteLn
END complex.
- Output:
Enter two complex numbers : 5 3 0.5 6 z1 = 5.00 + 3.00 i z2 = 0.50 + 6.00 i z1 + z2 = 5.50 + 9.00 i z1 - z2 = 4.50 - 3.00 i z1 * z2 = -15.50 + 31.50 i 1 / z1 = 0.15 - 0.09 i - z1 = -5.00 - 3.00 i
Nanoquery
This is a translation of the Java version, but it uses operator redefinition where possible.
import math
class Complex
declare real
declare imag
def Complex()
real = 0.0
imag = 0.0
end
def Complex(r, i)
real = double(r)
imag = double(i)
end
def operator-(b)
return new(Complex, this.real - b.real, this.imag - b.imag)
end
def operator+(b)
return new(Complex, this.real + b.real, this.imag + b.imag)
end
def operator*(b)
// FOIL of (a+bi)(c+di) with i*i = -1
return new(Complex, this.real * b.real - this.imag * b.imag,\
this.real * b.imag + this.imag * b.real)
end
def inv()
// 1/(a+bi) * (a-bi)/(a-bi) = 1/(a+bi) but it's more workable
denom = this.real * this.real + this.imag * this.imag
return new(Complex, real/denom, -imag/denom)
end
def neg()
return new(Complex, -this.real, -this.imag)
end
def conj()
return new(Complex, this.real, -this.imag)
end
def toString()
return this.real + " + " + this.imag + " * i"
end
end
a = new(Complex, math.pi, -5)
b = new(Complex, -1, 2.5)
println a.neg()
println a + b
println a.inv()
println a * b
println a.conj()Nemerle
using System;
using System.Console;
using System.Numerics;
using System.Numerics.Complex;
module RCComplex
{
PrettyPrint(this c : Complex) : string
{
mutable sign = '+';
when (c.Imaginary < 0) sign = '-';
$"$(c.Real) $sign $(Math.Abs(c.Imaginary))i"
}
Main() : void
{
def complex1 = Complex(1.0, 1.0);
def complex2 = Complex(3.14159, 1.2);
WriteLine(Add(complex1, complex2).PrettyPrint());
WriteLine(Multiply(complex1, complex2).PrettyPrint());
WriteLine(Negate(complex2).PrettyPrint());
WriteLine(Reciprocal(complex2).PrettyPrint());
WriteLine(Conjugate(complex2).PrettyPrint());
}
}
- Output:
4.14159 + 2.2i 1.94159 + 4.34159i -3.14159 - 1.2i 0.277781124787984 - 0.106104663481097i 3.14159 - 1.2i
Nim
import complex
var a: Complex = (1.0,1.0)
var b: Complex = (3.1415,1.2)
echo("a : " & $a)
echo("b : " & $b)
echo("a + b: " & $(a + b))
echo("a * b: " & $(a * b))
echo("1/a : " & $(1/a))
echo("-a : " & $(-a))
- Output:
a : (1.0000000000000000e+00, 1.0000000000000000e+00) b : (3.1415000000000002e+00, 1.2000000000000000e+00) a + b: (4.1415000000000006e+00, 2.2000000000000002e+00) a * b: (1.9415000000000002e+00, 4.3414999999999999e+00) 1/a : (5.0000000000000000e-01, -5.0000000000000000e-01) -a : (-1.0000000000000000e+00, -1.0000000000000000e+00)
Oberon-2
Oxford Oberon Compiler
MODULE Complex;
IMPORT Files,Out;
TYPE
Complex* = POINTER TO ComplexDesc;
ComplexDesc = RECORD
r-,i-: REAL;
END;
PROCEDURE (CONST x: Complex) Add*(CONST y: Complex): Complex;
BEGIN
RETURN New(x.r + y.r,x.i + y.i)
END Add;
PROCEDURE (CONST x: Complex) Sub*(CONST y: Complex): Complex;
BEGIN
RETURN New(x.r - y.r,x.i - y.i)
END Sub;
PROCEDURE (CONST x: Complex) Mul*(CONST y: Complex): Complex;
BEGIN
RETURN New(x.r*y.r - x.i*y.i,x.r*y.i + x.i*y.r)
END Mul;
PROCEDURE (CONST x: Complex) Div*(CONST y: Complex): Complex;
VAR
d: REAL;
BEGIN
d := y.r * y.r + y.i * y.i;
RETURN New((x.r*y.r + x.i*y.i)/d,(x.i*y.r - x.r*y.i)/d)
END Div;
(* Reciprocal *)
PROCEDURE (CONST x: Complex) Rec*(): Complex;
VAR
d: REAL;
BEGIN
d := x.r * x.r + y.i * y.i;
RETURN New(x.r/d,(-1.0 * x.i)/d);
END Rec;
(* Conjugate *)
PROCEDURE (x: Complex) Con*(): Complex;
BEGIN
RETURN New(x.r, (-1.0) * x.i);
END Con;
PROCEDURE (x: Complex) Out(out : Files.File);
BEGIN
Files.WriteString(out,"(");
Files.WriteReal(out,x.r);
Files.WriteString(out,",");
Files.WriteReal(out,x.i);
Files.WriteString(out,"i)")
END Out;
PROCEDURE New(x,y: REAL): Complex;
VAR
r: Complex;
BEGIN
NEW(r);r.r := x;r.i := y;
RETURN r
END New;
VAR
r,x,y: Complex;
BEGIN
x := New(1.5,3);
y := New(1.0,1.0);
Out.String("x: ");x.Out(Files.stdout);Out.Ln;
Out.String("y: ");y.Out(Files.stdout);Out.Ln;
r := x.Add(y);
Out.String("x + y: ");r.Out(Files.stdout);Out.Ln;
r := x.Sub(y);
Out.String("x - y: ");r.Out(Files.stdout);Out.Ln;
r := x.Mul(y);
Out.String("x * y: ");r.Out(Files.stdout);Out.Ln;
r := x.Div(y);
Out.String("x / y: ");r.Out(Files.stdout);Out.Ln;
r := y.Rec();
Out.String("1 / y: ");r.Out(Files.stdout);Out.Ln;
r := x.Con();
Out.String("x': ");r.Out(Files.stdout);Out.Ln;
END Complex.
- Output:
x: (1.50000,3.00000i) y: (1.00000,1.00000i) x + y: (2.50000,4.00000i) x - y: (0.500000,2.00000i) x * y: (-1.50000,4.50000i) x / y: (2.25000,0.750000i) 1 / y: (0.500000,-0.500000i) x': (1.50000,-3.00000i)
OCaml
The Complex module from the standard library provides the functionality of complex numbers:
open Complex
let print_complex z =
Printf.printf "%f + %f i\n" z.re z.im
let () =
let a = { re = 1.0; im = 1.0 }
and b = { re = 3.14159; im = 1.25 } in
print_complex (add a b);
print_complex (mul a b);
print_complex (inv a);
print_complex (neg a);
print_complex (conj a)
Using Delimited Overloading, the syntax can be made closer to the usual one:
let () =
Complex.(
let print txt z = Printf.printf "%s = %s\n" txt (to_string z) in
let a = 1 + I
and b = 3 + 7I in
print "a + b" (a + b);
print "a - b" (a - b);
print "a * b" (a * b);
print "a / b" (a / b);
print "-a" (- a);
print "conj a" (conj a);
print "a^b" (a**b);
Printf.printf "norm a = %g\n" (float(abs a));
)
Octave
GNU Octave handles naturally complex numbers:
z1 = 1.5 + 3i;
z2 = 1.5 + 1.5i;
disp(z1 + z2); % 3.0 + 4.5i
disp(z1 - z2); % 0.0 + 1.5i
disp(z1 * z2); % -2.25 + 6.75i
disp(z1 / z2); % 1.5 + 0.5i
disp(-z1); % -1.5 - 3i
disp(z1'); % 1.5 - 3i
disp(abs(z1)); % 3.3541 = sqrt(z1*z1')
disp(z1 ^ z2); % -1.10248 - 0.38306i
disp( exp(z1) ); % -4.43684 + 0.63246i
disp( imag(z1) ); % 3
disp( real(z2) ); % 1.5
%...
Oforth
Object Class new: Complex(re, im)
Complex method: re @re ;
Complex method: im @im ;
Complex method: initialize := im := re ;
Complex method: << '(' <<c @re << ',' <<c @im << ')' <<c ;
0 1 Complex new const: I
Complex method: ==(c -- b )
c re @re == c im @im == and ;
Complex method: norm -- f
@re sq @im sq + sqrt ;
Complex method: conj -- c
@re @im neg Complex new ;
Complex method: +(c -- d )
c re @re + c im @im + Complex new ;
Complex method: -(c -- d )
c re @re - c im @im - Complex new ;
Complex method: *(c -- d)
c re @re * c im @im * - c re @im * @re c im * + Complex new ;
Complex method: inv
| n |
@re sq @im sq + >float ->n
@re n / @im neg n / Complex new
;
Complex method: /( c -- d )
c self inv * ;
Integer method: >complex self 0 Complex new ;
Float method: >complex self 0 Complex new ;Usage :
3.2 >complex I * 2 >complex + .cr
2 3 Complex new 1.2 >complex + .cr
2 3 Complex new 1.2 >complex * .cr
2 >complex 2 3 Complex new / .cr- Output:
(2,3.2) (3.2,3) (2.4,3.6) (0.307692307692308,-0.461538461538462)
Ol
Ol supports complex numbers by default. Numbers must be entered manually in form A+Bi without spaces between elements, where A and B - numbers (can be rational), i - imaginary unit; or in functional form using function `complex`.
(define A 0+1i) ; manually entered numbers
(define B 1+0i)
(print (+ A B))
; <== 1+i
(print (- A B))
; <== -1+i
(print (* A B))
; <== 0+i
(print (/ A B))
; <== 0+i
(define C (complex 2/7 -3)) ; functional way
(print "real part of " C " is " (car C))
; <== real part of 2/7-3i is 2/7
(print "imaginary part of " C " is " (cdr C))
; <== imaginary part of 2/7-3i is -3
ooRexx
c1 = .complex~new(1, 2)
c2 = .complex~new(3, 4)
r = 7
say "c1 =" c1
say "c2 =" c2
say "r =" r
say "-c1 =" (-c1)
say "c1 + r =" c1 + r
say "c1 + c2 =" c1 + c2
say "c1 - r =" c1 - r
say "c1 - c2 =" c1 - c2
say "c1 * r =" c1 * r
say "c1 * c2 =" c1 * c2
say "inv(c1) =" c1~inv
say "conj(c1) =" c1~conjugate
say "c1 / r =" c1 / r
say "c1 / c2 =" c1 / c2
say "c1 == c1 =" (c1 == c1)
say "c1 == c2 =" (c1 == c2)
::class complex
::method init
expose r i
use strict arg r, i = 0
-- complex instances are immutable, so these are
-- read only attributes
::attribute r GET
::attribute i GET
::method negative
expose r i
return self~class~new(-r, -i)
::method add
expose r i
use strict arg other
if other~isa(.complex) then
return self~class~new(r + other~r, i + other~i)
else return self~class~new(r + other, i)
::method subtract
expose r i
use strict arg other
if other~isa(.complex) then
return self~class~new(r - other~r, i - other~i)
else return self~class~new(r - other, i)
::method times
expose r i
use strict arg other
if other~isa(.complex) then
return self~class~new(r * other~r - i * other~i, r * other~i + i * other~r)
else return self~class~new(r * other, i * other)
::method inv
expose r i
denom = r * r + i * i
return self~class~new(r/denom,-i/denom)
::method conjugate
expose r i
return self~class~new(r, -i)
::method divide
use strict arg other
-- this is easier if everything is a complex number
if \other~isA(.complex) then other = .complex~new(other)
-- division is multiplication with the inversion
return self * other~inv
::method "=="
expose r i
use strict arg other
if \other~isa(.complex) then return .false
-- Note: these are numeric comparisons, so we're using the "="
-- method so those are handled correctly
return r = other~r & i = other~i
::method "\=="
use strict arg other
return \self~"\=="(other)
::method "="
-- this is equivalent of "=="
forward message("==")
::method "\="
-- this is equivalent of "\=="
forward message("\==")
::method "<>"
-- this is equivalent of "\=="
forward message("\==")
::method "><"
-- this is equivalent of "\=="
forward message("\==")
-- some operator overrides -- these only work if the left-hand-side of the
-- subexpression is a quaternion
::method "*"
forward message("TIMES")
::method "/"
forward message("DIVIDE")
::method "-"
-- need to check if this is a prefix minus or a subtract
if arg() == 0 then
forward message("NEGATIVE")
else
forward message("SUBTRACT")
::method "+"
-- need to check if this is a prefix plus or an addition
if arg() == 0 then
return self -- we can return this copy since it is immutable
else
forward message("ADD")
::method string
expose r i
return r self~formatnumber(i)"i"
::method formatnumber private
use arg value
if value > 0 then return "+" value
else return "-" value~abs
-- override hashcode for collection class hash uses
::method hashCode
expose r i
return r~hashcode~bitxor(i~hashcode)
- Output:
c1 = 1 + 2i c2 = 3 + 4i r = 7 -c1 = -1 - 2i c1 + r = 8 + 2i c1 + c2 = 4 + 6i c1 - r = -6 + 2i c1 - c2 = -2 - 2i c1 * r = 7 + 14i c1 * c2 = -5 + 10i inv(c1) = 0.2 - 0.4i conj(c1) = 1 - 2i c1 / r = 0.142857143 + 0.285714286i c1 / c2 = 0.44 + 0.08i c1 == c1 = 1 c1 == c2 = 0
OxygenBasic
Implementation of a complex numbers class with arithmetical operations, and powers using DeMoivre's theorem (polar conversion).
'COMPLEX OPERATIONS
'=================
type tcomplex double x,y
class Complex
'============
has tcomplex
static sys i,pp
static tcomplex accum[32]
def operands
tcomplex*a,*b
@a=@accum+i
if pp then
@b=@a+sizeof accum
pp=0
else
@b=@this
end if
end def
method "load"()
operands
a.x=b.x
a.y=b.y
end method
method "push"()
i+=sizeof accum
end method
method "pop"()
pp=1
i-=sizeof accum
end method
method "="()
operands
b.x=a.x
b.y=a.y
end method
method "+"()
operands
a.x+=b.x
a.y+=b.y
end method
method "-"()
operands
a.x-=b.x
a.y-=b.y
end method
method "*"()
operands
double d
d=a.x
a.x = a.x * b.x - a.y * b.y
a.y = a.y * b.x + d * b.y
end method
method "/"()
operands
double d,v
v=1/(b.x * b.x + b.y * b.y)
d=a.x
a.x = (a.x * b.x + a.y * b.y) * v
a.y = (a.y * b.x - d * b.y) * v
end method
method power(double n)
operands
'Using DeMoivre theorem
double r,an,mg
r = hypot(b.x,b.y)
mg = r^n
if b.x=0 then
ay=.5*pi
if b.y<0 then ay=-ay
else
an = atan(b.y,b.x)
end if
an *= n
a.x = mg * cos(an)
a.y = mg * sin(an)
end method
method show() as string
return str(x,14) ", " str(y,14)
end method
end class
'#recordof complexop
'====
'TEST
'====
complex z1,z2,z3,z4,z5
'ENTER VALUES
z1 <= 0, 0
z2 <= 2, 1
z3 <= -2, 1
z4 <= 2, 4
z5 <= 1, 1
'EVALUATE COMPLEX EXPRESSIONS
z1 = z2 * z3
print "Z1 = "+z1.show 'RESULT -5.0, 0
z1 = z3+(z2.power(2))
print "Z1 = "+z1.show 'RESULT 1.0, 5.0
z1 = z5/z4
print "Z1 = "+z1.show 'RESULT 0.3, 0.1
z1 = z5/z1
print "Z1 = "+z1.show 'RESULT 2.0, 4.0
z1 = z2/z4
print "Z1 = "+z1.show 'RESULT -0.4, -0.3
z1 = z1*z4
print "Z1 = "+z1.show 'RESULT 2.0, 1.0PARI/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
The simple data type complex is part of Extended Pascal, ISO standard 10206.
program complexDemo(output);
const
{ I experienced some hiccups with -1.0 using GPC (GNU Pascal Compiler) }
negativeOne = -1.0;
type
line = string(80);
{ as per task requirements wrap arithmetic operations into separate functions }
function sum(protected x, y: complex): complex;
begin
sum := x + y
end;
function product(protected x, y: complex): complex;
begin
product := x * y
end;
function negative(protected x: complex): complex;
begin
negative := -x
end;
function inverse(protected x: complex): complex;
begin
inverse := x ** negativeOne
end;
{ only this function is not covered by Extended Pascal, ISO 10206 }
function conjugation(protected x: complex): complex;
begin
conjugation := cmplx(re(x), im(x) * negativeOne)
end;
{ --- test suite ------------------------------------------------------------- }
function asString(protected x: complex): line;
const
totalWidth = 5;
fractionDigits = 2;
var
result: line;
begin
writeStr(result, '(', re(x):totalWidth:fractionDigits, ', ',
im(x):totalWidth:fractionDigits, ')');
asString := result
end;
{ === MAIN =================================================================== }
var
x: complex;
{ for demonstration purposes: how to initialize complex variables }
y: complex value cmplx(1.0, 4.0);
z: complex value polar(exp(1.0), 3.14159265358979);
begin
x := cmplx(-3, 2);
writeLn(asString(x), ' + ', asString(y), ' = ', asString(sum(x, y)));
writeLn(asString(x), ' * ', asString(z), ' = ', asString(product(x, z)));
writeLn;
writeLn(' −', asString(z), ' = ', asString(negative(z)));
writeLn(' inverse(', asString(z), ') = ', asString(inverse(z)));
writeLn(' conjugation(', asString(y), ') = ', asString(conjugation(y)));
end.
- Output:
(-3.00, 2.00) + ( 1.00, 4.00) = (-2.00, 6.00)
(-3.00, 2.00) * (-2.72, 0.00) = ( 8.15, -5.44)
−(-2.72, 0.00) = ( 2.72, -0.00)
inverse((-2.72, 0.00)) = (-0.37, -0.00)
conjugation(( 1.00, 4.00)) = ( 1.00, -4.00)
The GPC, GNU Pascal Compiler, supports Extended Pascal’s complex data type and operations as shown. Furthermore, the GPC defines a function conjugate so there is no need for writing such a custom function. The PXSC, Pascal eXtensions for scientific computing, define a standard data type similar to Free Pascal’s ucomplex data type.
Perl
The Math::Complex module implements complex arithmetic.
use Math::Complex;
my $a = 1 + 1*i;
my $b = 3.14159 + 1.25*i;
print "$_\n" foreach
$a + $b, # addition
$a * $b, # multiplication
-$a, # negation
1 / $a, # multiplicative inverse
~$a; # complex conjugate
Phix
-- demo\rosetta\ArithComplex.exw with javascript_semantics include complex.e complex a = complex_new(1,1), -- (or just {1,1}) b = complex_new(3.14159,1.25), c = complex_new(1,0), d = complex_new(0,1) printf(1,"a = %s\n",{complex_sprint(a)}) printf(1,"b = %s\n",{complex_sprint(b)}) printf(1,"c = %s\n",{complex_sprint(c)}) printf(1,"d = %s\n",{complex_sprint(d)}) printf(1,"a+b = %s\n",{complex_sprint(complex_add(a,b))}) printf(1,"a*b = %s\n",{complex_sprint(complex_mul(a,b))}) printf(1,"1/a = %s\n",{complex_sprint(complex_inv(a))}) printf(1,"c/a = %s\n",{complex_sprint(complex_div(c,a))}) printf(1,"c-a = %s\n",{complex_sprint(complex_sub(c,a))}) printf(1,"d-a = %s\n",{complex_sprint(complex_sub(d,a))}) printf(1,"-a = %s\n",{complex_sprint(complex_neg(a))}) printf(1,"conj a = %s\n",{complex_sprint(complex_conjugate(a))})
- Output:
a = 1+i b = 3.14159+1.25i c = 1 d = i a+b = 4.14159+2.25i a*b = 1.89159+4.39159i 1/a = 0.5-0.5i c/a = 0.5-0.5i c-a = -i d-a = -1 -a = -1-i conj a = 1-i
PicoLisp
(load "@lib/math.l")
(de addComplex (A B)
(cons
(+ (car A) (car B)) # Real
(+ (cdr A) (cdr B)) ) ) # Imag
(de mulComplex (A B)
(cons
(-
(*/ (car A) (car B) 1.0)
(*/ (cdr A) (cdr B) 1.0) )
(+
(*/ (car A) (cdr B) 1.0)
(*/ (cdr A) (car B) 1.0) ) ) )
(de invComplex (A)
(let Denom
(+
(*/ (car A) (car A) 1.0)
(*/ (cdr A) (cdr A) 1.0) )
(cons
(*/ (car A) 1.0 Denom)
(- (*/ (cdr A) 1.0 Denom)) ) ) )
(de negComplex (A)
(cons (- (car A)) (- (cdr A))) )
(de fmtComplex (A)
(pack
(round (car A) (dec *Scl))
(and (gt0 (cdr A)) "+")
(round (cdr A) (dec *Scl))
"i" ) )
(let (A (1.0 . 1.0) B (cons pi 1.2))
(prinl "A = " (fmtComplex A))
(prinl "B = " (fmtComplex B))
(prinl "A+B = " (fmtComplex (addComplex A B)))
(prinl "A*B = " (fmtComplex (mulComplex A B)))
(prinl "1/A = " (fmtComplex (invComplex A)))
(prinl "-A = " (fmtComplex (negComplex A))) )- Output:
A = 1.00000+1.00000i B = 3.14159+1.20000i A+B = 4.14159+2.20000i A*B = 1.94159+4.34159i 1/A = 0.50000-0.50000i -A = -1.00000-1.00000i
PL/I
/* PL/I complex numbers may be integer or floating-point. */
/* In this example, the variables are floating-pint. */
/* For integer variables, change 'float' to 'fixed binary' */
declare (a, b) complex float;
a = 2+5i;
b = 7-6i;
put skip list (a+b);
put skip list (a - b);
put skip list (a*b);
put skip list (a/b);
put skip list (a**b);
put skip list (1/a);
put skip list (conjg(a)); /* gives the conjugate of 'a'. */
/* Functions exist for extracting the real and imaginary parts */
/* of a complex number. */
/* As well, trigonometric functions may be used with complex */
/* numbers, such as SIN, COS, TAN, ATAN, and so on. */Pop11
Complex numbers are a built-in data type in Pop11. Real and imaginary part of complex numbers can be floating point or exact (integer or rational) value (both part must be of the same type). Operations on floating point complex numbers always produce complex numbers. Operations on exact complex numbers give real result (integer or rational) if imaginary part of the result is 0. The '+:' and '-:' operators create complex numbers: '1 -: 3' is '1 - 3i' in mathematical notation.
lvars a = 1.0 +: 1.0, b = 2.0 +: 5.0 ;
a+b =>
a*b =>
1/a =>
a-b =>
a-a =>
a/b =>
a/a =>
;;; The same, but using exact values
1 +: 1 -> a;
2 +: 5 -> b;
a+b =>
a*b =>
1/a =>
a-b =>
a-a =>
a/b =>
a/a =>PostScript
Complex numbers can be represented as 2 element vectors ( arrays ). Thus, a+bi can be written as [a b] in PostScript.
%Adding two complex numbers
/addcomp{
/x exch def
/y exch def
/z [0 0] def
z 0 x 0 get y 0 get add put
z 1 x 1 get y 1 get add put
z pstack
}def
%Subtracting one complex number from another
/subcomp{
/x exch def
/y exch def
/z [0 0] def
z 0 x 0 get y 0 get sub put
z 1 x 1 get y 1 get sub put
z pstack
}def
%Multiplying two complex numbers
/mulcomp{
/x exch def
/y exch def
/z [0 0] def
z 0 x 0 get y 0 get mul x 1 get y 1 get mul sub put
z 1 x 1 get y 0 get mul x 0 get y 1 get mul add put
z pstack
}def
%Negating a complex number
/negcomp{
/x exch def
/z [0 0] def
z 0 x 0 get neg put
z 1 x 1 get neg put
z pstack
}def
%Inverting a complex number
/invcomp{
/x exch def
/z [0 0] def
z 0 x 0 get x 0 get 2 exp x 1 get 2 exp add div put
z 0 x 1 get neg x 0 get 2 exp x 1 get 2 exp add div put
z pstack
}def
PowerShell
Implementation
class Complex {
[Double]$x
[Double]$y
Complex() {
$this.x = 0
$this.y = 0
}
Complex([Double]$x, [Double]$y) {
$this.x = $x
$this.y = $y
}
[Double]abs2() {return $this.x*$this.x + $this.y*$this.y}
[Double]abs() {return [math]::sqrt($this.abs2())}
static [Complex]add([Complex]$m,[Complex]$n) {return [Complex]::new($m.x+$n.x, $m.y+$n.y)}
static [Complex]mul([Complex]$m,[Complex]$n) {return [Complex]::new($m.x*$n.x - $m.y*$n.y, $m.x*$n.y + $n.x*$m.y)}
[Complex]mul([Double]$k) {return [Complex]::new($k*$this.x, $k*$this.y)}
[Complex]negate() {return $this.mul(-1)}
[Complex]conjugate() {return [Complex]::new($this.x, -$this.y)}
[Complex]inverse() {return $this.conjugate().mul(1/$this.abs2())}
[String]show() {
if(0 -ge $this.y) {
return "$($this.x)+$($this.y)i"
} else {
return "$($this.x)$($this.y)i"
}
}
static [String]show([Complex]$other) {
return $other.show()
}
}
$m = [complex]::new(3, 4)
$n = [complex]::new(7, 6)
"`$m: $($m.show())"
"`$n: $($n.show())"
"`$m + `$n: $([complex]::show([complex]::add($m,$n)))"
"`$m * `$n: $([complex]::show([complex]::mul($m,$n)))"
"negate `$m: $($m.negate().show())"
"1/`$m: $([complex]::show($m.inverse()))"
"conjugate `$m: $([complex]::show($m.conjugate()))"
Output:
$m: 3+4i $n: 7+6i $m + $n: 10+10i $m * $n: -3+46i negate $m: -3-4i 1/$m: 0.12-0.16i conjugate $m: 3-4i
Library
function show([System.Numerics.Complex]$c) {
if(0 -le $c.Imaginary) {
return "$($c.Real)+$($c.Imaginary)i"
} else {
return "$($c.Real)$($c.Imaginary)i"
}
}
$m = [System.Numerics.Complex]::new(3, 4)
$n = [System.Numerics.Complex]::new(7, 6)
"`$m: $(show $m)"
"`$n: $(show $n)"
"`$m + `$n: $(show ([System.Numerics.Complex]::Add($m,$n)))"
"`$m * `$n: $(show ([System.Numerics.Complex]::Multiply($m,$n)))"
"negate `$m: $(show ([System.Numerics.Complex]::Negate($m)))"
"1/`$m: $(show ([System.Numerics.Complex]::Reciprocal($m)))"
"conjugate `$m: $(show ([System.Numerics.Complex]::Conjugate($m)))"
Output:
$m: 3+4i $n: 7+6i $m + $n: 10+10i $m * $n: -3+46i negate $m: -3-4i 1/$m: 0.12-0.16i conjugate $m: 3-4i
PureBasic
Structure Complex
real.d
imag.d
EndStructure
Procedure Add_Complex(*A.Complex, *B.Complex)
Protected *R.Complex=AllocateMemory(SizeOf(Complex))
If *R
*R\real=*A\real+*B\real
*R\imag=*A\imag+*B\imag
EndIf
ProcedureReturn *R
EndProcedure
Procedure Inv_Complex(*A.Complex)
Protected *R.Complex=AllocateMemory(SizeOf(Complex)), denom.d
If *R
denom = *A\real * *A\real + *A\imag * *A\imag
*R\real= *A\real / denom
*R\imag=-*A\imag / denom
EndIf
ProcedureReturn *R
EndProcedure
Procedure Mul_Complex(*A.Complex, *B.Complex)
Protected *R.Complex=AllocateMemory(SizeOf(Complex))
If *R
*R\real=*A\real * *B\real - *A\imag * *B\imag
*R\imag=*A\real * *B\imag + *A\imag * *B\real
EndIf
ProcedureReturn *R
EndProcedure
Procedure Neg_Complex(*A.Complex)
Protected *R.Complex=AllocateMemory(SizeOf(Complex))
If *R
*R\real=-*A\real
*R\imag=-*A\imag
EndIf
ProcedureReturn *R
EndProcedure
Procedure ShowAndFree(Header$, *Complex.Complex)
If *Complex
Protected.d i=*Complex\imag, r=*Complex\real
Print(LSet(Header$,7))
Print("= "+StrD(r,3))
If i>=0: Print(" + ")
Else: Print(" - ")
EndIf
PrintN(StrD(Abs(i),3)+"i")
FreeMemory(*Complex)
EndIf
EndProcedure
If OpenConsole()
Define.Complex a, b, *c
a\real=1.0: a\imag=1.0
b\real=#PI: b\imag=1.2
*c=Add_Complex(a,b): ShowAndFree("a+b", *c)
*c=Mul_Complex(a,b): ShowAndFree("a*b", *c)
*c=Inv_Complex(a): ShowAndFree("Inv(a)", *c)
*c=Neg_Complex(a): ShowAndFree("-a", *c)
Print(#CRLF$+"Press ENTER to exit"):Input()
EndIf
Python
>>> z1 = 1.5 + 3j
>>> z2 = 1.5 + 1.5j
>>> z1 + z2
(3+4.5j)
>>> z1 - z2
1.5j
>>> z1 * z2
(-2.25+6.75j)
>>> z1 / z2
(1.5+0.5j)
>>> - z1
(-1.5-3j)
>>> z1.conjugate()
(1.5-3j)
>>> abs(z1)
3.3541019662496847
>>> z1 ** z2
(-1.1024829553277784-0.38306415117199333j)
>>> z1.real
1.5
>>> z1.imag
3.0
>>>
R
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)
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)
1RPL
- 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.conjugatereturns 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
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(orreal-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+ior-iat the end. - polar coordinates:
r@theta, where r is the absolute value (magnitude) and theta is the angle
(define a 1+i)
(define b 3.14159+1.25i)
(define c (+ a b))
(define c (* a b))
(define c (/ 1 a))
(define c (- a))
Seed7
$ include "seed7_05.s7i";
include "float.s7i";
include "complex.s7i";
const proc: main is func
local
var complex: a is complex(1.0, 1.0);
var complex: b is complex(3.14159, 1.2);
begin
writeln("a=" <& a digits 5);
writeln("b=" <& b digits 5);
# addition
writeln("a+b=" <& a + b digits 5);
# multiplication
writeln("a*b=" <& a * b digits 5);
# inversion
writeln("1/a=" <& complex(1.0) / a digits 5);
# negation
writeln("-a=" <& -a digits 5);
end func;Sidef
var a = 1:1 # Complex(1, 1)
var b = 3.14159:1.25 # Complex(3.14159, 1.25)
[ a + b, # addition
a * b, # multiplication
-a, # negation
a.inv, # multiplicative inverse
a.conj, # complex conjugate
a.abs, # abs
a.sqrt, # sqrt
b.re, # real
b.im, # imaginary
].each { |c| say c }
- Output:
4.14159+2.25i 1.89159+4.39159i -1-i 0.5-0.5i 1-i 1.4142135623730950488016887242097 1.09868411346780996603980119524068+0.45508986056222734130435775782247i 3.14159 1.25
Slate
[| a b |
a: 1 + 1 i.
b: Pi + 1.2 i.
print: a + b.
print: a * b.
print: a / b.
print: a reciprocal.
print: a conjugated.
print: a abs.
print: a negated.
].Smalltalk
PackageLoader fileInPackage: 'Complex'.
|a b|
a := 1 + 1 i.
b := 3.14159 + 1.2 i.
(a + b) displayNl.
(a * b) displayNl.
(a / b) displayNl.
a reciprocal displayNl.
a conjugate displayNl.
a abs displayNl.
a real displayNl.
a imaginary displayNl.
a negated displayNl.
Complex is already in the basic class library. Multiples of imaginary are created by sending an "i" message to a number, which can be added to another number. Thus 5i => (0+5i), 1+(1/3)I => (1+1/3i) and (1.0+2i) => (1.0+2i). Notice that the real and imaginary components can be arbitrary integers, fractions or floating point numbers. And the results will be exact (i.e. have fractions or integer) if possible.
|a b|
a := 1 + 1i.
b := 3.14159 + 1.2i.
Transcript show:'a => '; showCR:a.
Transcript show:'b => '; showCR:b.
Transcript show:'a+b => '; showCR:(a + b).
Transcript show:'a-b => '; showCR:(a - b).
Transcript show:'a*b => '; showCR:(a * b).
Transcript show:'a/b => '; showCR:(a / b).
Transcript show:'a reciprocal => '; showCR:a reciprocal.
Transcript show:'a conjugated => '; showCR:a conjugated.
Transcript show:'a abs => '; showCR:a abs.
Transcript show:'a real => '; showCR:a real.
Transcript show:'a imaginary => '; showCR:a imaginary.
Transcript show:'a negated => '; showCR:a negated.
Transcript show:'a sqrt => '; showCR:a sqrt.
a2 := (1/2) + 1i.
b2 := (2/3) + 2i.
Transcript show:'a2+b2 => '; showCR:(a2 + b2).
Transcript show:'a2-b2 => '; showCR:(a2 - b2).
Transcript show:'a2*b2 => '; showCR:(a2 * b2).
Transcript show:'a2/b2 => '; showCR:(a2 / b2).
Transcript show:'a2 reciprocal => '; showCR:a2 reciprocal.
- Output:
a => (1+1i) b => (3.14159+1.2i) a+b => (4.14159+2.2i) a-b => (-2.14159-0.2i) a*b => (1.94159+4.34159i) a/b => (0.383885788269082+0.171676461306887i) a reciprocal => ((1/2)-(1/2)i) a conjugated => (1-1i) a abs => 1.4142135623731 a real => 1 a imaginary => 1 a negated => (-1-1i) a sqrt => (1.09868411346781+0.455089860562227i) a2+b2 => ((7/6)+3i) a2-b2 => ((-1/6)-1i) a2*b2 => ((-5/3)+(5/3)i) a2/b2 => ((21/40)-(3/40)i) a2 reciprocal => ((2/5)-(4/5)i)
smart BASIC
Original author unknown {:o(
' complex numbers are native for "smart BASIC"
A=1+2i
B=3-5i
' all math operations and functions work with complex numbers
C=A*B
PRINT SQR(-4)
' example of solving quadratic equation with complex roots
' x^2+2x+5=0
a=1 ! b=2 ! c=5
x1=(-b+sqr(b^2-4*a*c))/(2*a)
x2=(-b-sqr(b^2-4*a*c))/(2*a)
print x1,x2
' gives output
-1+2i -1-2i
SNOBOL4
* # Define complex datatype
data('complex(r,i)')
* # Addition
define('addx(x1,x2)a,b,c,d') :(addx_end)
addx a = r(x1); b = i(x1); c = r(x2); d = i(x2)
addx = complex(a + c, b + d) :(return)
addx_end
* # Multiplication
define('multx(x1,x2)a,b,c,d') :(multx_end)
multx a = r(x1); b = i(x1); c = r(x2); d = i(x2)
multx = complex(a * c - b * d, b * c + a * d) :(return)
multx_end
* # Negation
define('negx(x)') :(negx_end)
negx negx = complex(-r(x), -i(x)) :(return)
negx_end
* # Inverse
define('invx(x)d') :(invx_end)
invx d = (r(x) * r(x)) + (i(x) * i(x))
invx = complex(1.0 * r(x) / d, 1.0 * -i(x) / d) :(return)
invx_end
* # Print compex number: a+bi / a-bi
define('printx(x)sign') :(printx_end)
printx sign = ge(i(x),0) '+'
printx = r(x) sign i(x) 'i' :(return)
printx_end
* # Test and display
a = complex(1,1)
b = complex(3.14159, 1.2)
output = printx( addx(a,b) )
output = printx( multx(a,b) )
output = printx( negx(a) ) ', ' printx( negx(b) )
output = printx( invx(a) ) ', ' printx( invx(b) )
end- Output:
4.14159+2.2i 1.94159+4.34159i -1-1i, -3.14159-1.2i 0.5-0.5i, 0.277781125-0.106104663i
Standard ML
(* Signature for complex numbers *)
signature COMPLEX = sig
type num
val complex : real * real -> num
val negative : num -> num
val plus : num -> num -> num
val minus : num -> num -> num
val times : num -> num -> num
val invert : num -> num
val print_number : num -> unit
end;
(* Actual implementation *)
structure Complex :> COMPLEX = struct
type num = real * real
fun complex (a, b) = (a, b)
fun negative (a, b) = (Real.~a, Real.~b)
fun plus (a1, b1) (a2, b2) = (Real.+ (a1, a2), Real.+(b1, b2))
fun minus i1 i2 = plus i1 (negative i2)
fun times (a1, b1) (a2, b2)= (Real.*(a1, a2) - Real.*(b1, b2), Real.*(a1, b2) + Real.*(a2, b1))
fun invert (a, b) =
let
val denom = a * a + b * b
in
(a / denom, ~b / denom)
end
fun print_number (a, b) =
print (Real.toString(a) ^ " + " ^ Real.toString(b) ^ "i\n")
end;
val i1 = Complex.complex(1.0,2.0); (* 1 + 2i *)
val i2 = Complex.complex(3.0,4.0); (* 3 + 4i *)
Complex.print_number(Complex.negative(i1)); (* -1 - 2i *)
Complex.print_number(Complex.plus i1 i2); (* 4 + 6i *)
Complex.print_number(Complex.minus i2 i1); (* 2 + 2i *)
Complex.print_number(Complex.times i1 i2); (* -5 + 10i *)
Complex.print_number(Complex.invert i1); (* 1/5 - 2i/5 *)Stata
mata
C(2,3)
2 + 3i
a=2+3i
b=1-2*i
a+b
-5 + 3i
a-b
9 + 3i
a*b
-14 - 21i
a/b
-.285714286 - .428571429i
-a
-2 - 3i
1/a
.153846154 - .230769231i
conj(a)
2 - 3i
abs(a)
3.605551275
arg(a)
.9827937232
exp(a)
-7.31511009 + 1.04274366i
log(a)
1.28247468 + .982793723i
end
Swift
Use a struct to create a complex number type in Swift. Math Operations can be added using operator overloading
public struct Complex {
public let real : Double
public let imaginary : Double
public init(real inReal:Double, imaginary inImaginary:Double) {
real = inReal
imaginary = inImaginary
}
public static var i : Complex = Complex(real:0, imaginary: 1)
public static var zero : Complex = Complex(real: 0, imaginary: 0)
public var negate : Complex {
return Complex(real: -real, imaginary: -imaginary)
}
public var invert : Complex {
let d = (real*real + imaginary*imaginary)
return Complex(real: real/d, imaginary: -imaginary/d)
}
public var conjugate : Complex {
return Complex(real: real, imaginary: -imaginary)
}
}
public func + (left: Complex, right: Complex) -> Complex {
return Complex(real: left.real+right.real, imaginary: left.imaginary+right.imaginary)
}
public func * (left: Complex, right: Complex) -> Complex {
return Complex(real: left.real*right.real - left.imaginary*right.imaginary,
imaginary: left.real*right.imaginary+left.imaginary*right.real)
}
public prefix func - (right:Complex) -> Complex {
return right.negate
}
// Checking equality is almost necessary for a struct of this type to be useful
extension Complex : Equatable {}
public func == (left:Complex, right:Complex) -> Bool {
return left.real == right.real && left.imaginary == right.imaginary
}
Make the Complex Number struct printable and easier to debug by adding making it conform to CustomStringConvertible
extension Complex : CustomStringConvertible {
public var description : String {
guard real != 0 || imaginary != 0 else { return "0" }
let rs : String = real != 0 ? "\(real)" : ""
let iS : String
let sign : String
let iSpace = real != 0 ? " " : ""
switch imaginary {
case let i where i < 0:
sign = "-"
iS = i == -1 ? "i" : "\(-i)i"
case let i where i > 0:
sign = real != 0 ? "+" : ""
iS = i == 1 ? "i" : "\(i)i"
default:
sign = ""
iS = ""
}
return "\(rs)\(iSpace)\(sign)\(iSpace)\(iS)"
}
}
Explicitly support subtraction and division
public func - (left:Complex, right:Complex) -> Complex {
return left + -right
}
public func / (divident:Complex, divisor:Complex) -> Complex {
let rc = divisor.conjugate
let num = divident * rc
let den = divisor * rc
return Complex(real: num.real/den.real, imaginary: num.imaginary/den.real)
}
Tcl
package require math::complexnumbers
namespace import math::complexnumbers::*
set a [complex 1 1]
set b [complex 3.14159 1.2]
puts [tostring [+ $a $b]] ;# ==> 4.14159+2.2i
puts [tostring [* $a $b]] ;# ==> 1.94159+4.34159i
puts [tostring [pow $a [complex -1 0]]] ;# ==> 0.5-0.4999999999999999i
puts [tostring [- $a]] ;# ==> -1.0-i
TI-83 BASIC
TI-83 BASIC has built in complex number support; the normal arithmetic operators + - * / are used.
The method complex numbers are displayed can be chosen in the "MODE" menu.
Real: Does not show complex numbers, gives an error if a number is imaginary.
a+bi: The classic display for imaginary numbers with the real and imaginary components
re^Θi: Displays imaginary numbers in Polar Coordinates.
TI-89 BASIC
TI-89 BASIC has built-in complex number support; the normal arithmetic operators + - * / are used.
- Character set note: the symbol for the imaginary unit is not the normal "i" but a different character (Unicode: U+F02F "" (private use area); this character should display with the "TI Uni" font). Also, U+3013 EN DASH “–”, displayed on the TI as a superscript minus, is used for the minus sign on numbers, distinct from ASCII "-" used for subtraction.
The choice of examples here is
.
■ √(–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
typeset -T Complex_t=(
float real=0
float imag=0
function to_s {
print -- "${_.real} + ${_.imag} i"
}
function dup {
nameref other=$1
_=( real=${other.real} imag=${other.imag} )
}
function add {
typeset varname
for varname; do
nameref other=$varname
(( _.real += other.real ))
(( _.imag += other.imag ))
done
}
function negate {
(( _.real *= -1 ))
(( _.imag *= -1 ))
}
function conjugate {
(( _.imag *= -1 ))
}
function multiply {
typeset varname
for varname; do
nameref other=$varname
float a=${_.real} b=${_.imag} c=${other.real} d=${other.imag}
(( _.real = a*c - b*d ))
(( _.imag = b*c + a*d ))
done
}
function inverse {
if (( _.real == 0 && _.imag == 0 )); then
print -u2 "division by zero"
return 1
fi
float denom=$(( _.real*_.real + _.imag*_.imag ))
(( _.real = _.real / denom ))
(( _.imag = -1 * _.imag / denom ))
}
)
Complex_t a=(real=1 imag=1)
a.to_s # 1 + 1 i
Complex_t b=(real=3.14159 imag=1.2)
b.to_s # 3.14159 + 1.2 i
Complex_t c
c.add a b
c.to_s # 4.14159 + 2.2 i
c.negate
c.to_s # -4.14159 + -2.2 i
c.conjugate
c.to_s # -4.14159 + 2.2 i
c.dup a
c.multiply b
c.to_s # 1.94159 + 4.34159 i
Complex_t d=(real=2 imag=1)
d.inverse
d.to_s # 0.4 + -0.2 iUrsala
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
@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
import "./complex" for Complex
var x = Complex.new(1, 3)
var y = Complex.new(5, 2)
System.print("x = %(x)")
System.print("y = %(y)")
System.print("x + y = %(x + y)")
System.print("x - y = %(x - y)")
System.print("x * y = %(x * y)")
System.print("x / y = %(x / y)")
System.print("-x = %(-x)")
System.print("1 / x = %(x.inverse)")
System.print("x* = %(x.conj)")- Output:
x = 1 + 3i y = 5 + 2i x + y = 6 + 5i x - y = -4 + 1i x * y = -1 + 17i x / y = 0.37931034482759 + 0.44827586206897i -x = -1 - 3i 1 / x = 0.1 - 0.3i x* = 1 - 3i
XPL0
include c:\cxpl\codes;
func real CAdd(A, B, C); \Return complex sum of two complex numbers
real A, B, C;
[C(0):= A(0) + B(0);
C(1):= A(1) + B(1);
return C;
];
func real CMul(A, B, C); \Return complex product of two complex numbers
real A, B, C;
[C(0):= A(0)*B(0) - A(1)*B(1);
C(1):= A(1)*B(0) + A(0)*B(1);
return C;
];
func real CNeg(A, C); \Return negative of a complex number
real A, C;
[C(0):= -A(0);
C(1):= -A(1);
return C;
];
func real CInv(A, C); \Return inversion (reciprical) of complex number
real A, C;
real D;
[D:= sq(A(0)) + sq(A(1));
C(0):= A(0)/D;
C(1):=-A(1)/D;
return C;
];
func real Conj(A, C); \Return conjugate of a complex number
real A, C;
[C(0):= A(0);
C(1):=-A(1);
return C;
];
proc COut(D, A); \Output a complex number to specified device
int D; real A;
[RlOut(D, A(0));
Text(D, if A(1)>=0.0 then " +" else " -");
RlOut(D, abs(A(1)));
ChOut(D, ^i);
];
real U, V, W(2);
[Format(2,2);
U:= [1.0, 1.0];
V:= [3.14, 1.2];
COut(0, CAdd(U,V,W)); CrLf(0);
COut(0, CMul(U,V,W)); CrLf(0);
COut(0, CNeg(U,W)); CrLf(0);
COut(0, CInv(U,W)); CrLf(0);
COut(0, Conj(U,W)); CrLf(0);
]- Output:
4.14 + 2.20i 1.94 + 4.34i -1.00 - 1.00i 0.50 - 0.50i 1.00 - 1.00i
Yabasic
rem ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
rem CADDI/CADDR addition of complex numbers Z1 + Z2 with Z1 = a1 + b1 *i Z2 = a2 + b2*i
rem CADDI returns imaginary part and CADDR the real part
rem ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
export sub caddi( a1 , b1 , a2 , b2)
return (b1 + b2)
end sub
export sub caddr( a1 , b1 , a2 , b2)
return (a1 + a2)
end sub
rem ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
rem CDIVI/CDIVR division of complex numbers Z1 / Z2 with Z1 = r + s *i Z2 = t + u*i
rem CDIVI returns imaginary part and CDIVR the real part
rem ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
export sub cdivi(r,s,t,u)
return ((s*t- u*r) / (t^2 + u^2))
end sub
export sub cdivr( r , s , t , u)
return ((r*t- s*u) / (t^2 + u^2))
end sub
rem ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
rem CMULI/CMULR multiplication of complex numbers Z1 * Z2, with Z1 = r + s *i Z2 = t + u*i
rem CMULI returns imaginary part and CMULR the real part
rem ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
export sub cmuli( r , s , t , u)
return (r * u + s * t)
end sub
export sub cmulr( r , s , t , u)
return (r * t - s * u)
end sub
rem ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
rem CSUBI/CSUBR subtraction of complex numbers Z1 - Z2 with Z1 = a1 + b1 *i Z2 = a2 + b2*i
rem CSUBI returns imaginary part and CSUBR the real part
rem ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
export sub csubi( a1 , b1 , a2 , b2)
return (b1 - b2)
end sub
export sub csubr( a1 , b1 , a2 , b2)
return (a1 - a2)
end sub
if (peek$("library") = "main") then
print "Example: Z1 + Z2 with Z1 = 3 +2i , Z2 = 1-3i: Z1 + Z2 = 4 -1i"
print caddr(3,2,1,-2), "/", caddi(3,2,1,-3) // 4/-1
end ifzkl
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
5 LET complex=2: LET r=1: LET i=2
10 DIM a(complex): LET a(r)=1.0: LET a(i)=1.0
20 DIM b(complex): LET b(r)=PI: LET b(i)=1.2
30 DIM o(complex)
40 REM add
50 LET o(r)=a(r)+b(r)
60 LET o(i)=a(i)+b(i)
70 PRINT "Result of addition is:": GO SUB 1000
80 REM mult
90 LET o(r)=a(r)*b(r)-a(i)*b(i)
100 LET o(i)=a(i)*b(r)+a(r)*b(i)
110 PRINT "Result of multiplication is:": GO SUB 1000
120 REM neg
130 LET o(r)=-a(r)
140 LET o(i)=-a(i)
150 PRINT "Result of negation is:": GO SUB 1000
160 LET denom=a(r)^2+a(i)^2
170 LET o(r)=a(r)/denom
180 LET o(i)=-a(i)/denom
190 PRINT "Result of inversion is:": GO SUB 1000
200 STOP
1000 IF o(i)>=0 THEN PRINT o(r);" + ";o(i);"i": RETURN
1010 PRINT o(r);" - ";-o(i);"i": RETURN- Output:
Result of addition is: 4.1415927 + 2.2i Result of multiplication is: 1.9415927 + 4.3415927i Result of negation is: -1 - 1i Result of inversion is: 0.5 - 0.5i
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- Zkl
- Zonnon
- ZX Spectrum Basic
- M4/Omit
- Pages with too many expensive parser function calls