Complex number arithmetic

The C programming language, as of C99, supports complex number math with the three built-in types double _Complex, float _Complex, and long double _Complex (see _Complex). When the header <complex.h> is included, the three complex number types are also accessible as double complex, float complex, long double complex.

In addition to the complex types, the three imaginary types may be supported: double _Imaginary, float _Imaginary, and long double _Imaginary (see _Imaginary). When the header <complex.h> is included, the three imaginary types are also accessible as double imaginary, float imaginary, and long double imaginary.

Standard arithmetic operators +, -, *, / can be used with real, complex, and imaginary types in any combination.

Notes

The following function names are potentially(since C23) reserved for future addition to complex.h and are not available for use in the programs that include that header: cerf, cerfc, cexp2, cexpm1, clog10, clog1p, clog2, clgamma, ctgamma, csinpi, ccospi, ctanpi, casinpi, cacospi, catanpi, ccompoundn, cpown, cpowr, crootn, crsqrt, cexp10m1, cexp10, cexp2m1, clog10p1, clog2p1, clogp1(since C23), along with their -f and -l suffixed variants.

Although the C standard names the inverse hyperbolics with "complex arc hyperbolic sine" etc., the inverse functions of the hyperbolic functions are the area functions. Their argument is the area of a hyperbolic sector, not an arc. The correct names are "complex inverse hyperbolic sine" etc. Some authors use "complex area hyperbolic sine" etc.

A complex or imaginary number is infinite if one of its parts is infinite, even if the other part is NaN.

A complex or imaginary number is finite if both parts are neither infinities nor NaNs.

A complex or imaginary number is a zero if both parts are positive or negative zeroes.

Example

#include <stdio.h>
#include <complex.h>
#include <tgmath.h>
 
int main(void)
{
    double complex z1 = I * I;     // imaginary unit squared
    printf("I * I = %.1f%+.1fi\n", creal(z1), cimag(z1));
 
    double complex z2 = pow(I, 2); // imaginary unit squared
    printf("pow(I, 2) = %.1f%+.1fi\n", creal(z2), cimag(z2));
 
    double PI = acos(-1);
    double complex z3 = exp(I * PI); // Euler's formula
    printf("exp(I*PI) = %.1f%+.1fi\n", creal(z3), cimag(z3));
 
    double complex z4 = 1+2*I, z5 = 1-2*I; // conjugates
    printf("(1+2i)*(1-2i) = %.1f%+.1fi\n", creal(z4*z5), cimag(z4*z5));
}

I * I = -1.0+0.0i
pow(I, 2) = -1.0+0.0i
exp(I*PI) = -1.0+0.0i
(1+2i)*(1-2i) = 5.0+0.0i

References

• C17 standard (ISO/IEC 9899:2018):
• 6.10.8.3/1/2 __STDC_NO_COMPLEX__ (p: 128)
• 6.10.8.3/1/2 __STDC_IEC_559_COMPLEX__ (p: 128)
• 7.3 Complex arithmetic <complex.h> (p: 136-144)
• 7.25 Type-generic math <tgmath.h> (p: 272-273)
• 7.31.1 Complex arithmetic <complex.h> (p: 391)
• Annex G (normative) IEC 60559-compatible complex arithmetic (p: 469-479)
• C11 standard (ISO/IEC 9899:2011):
• 6.10.8.3/1/2 __STDC_NO_COMPLEX__ (p: 177)
• 6.10.8.3/1/2 __STDC_IEC_559_COMPLEX__ (p: 177)
• 7.3 Complex arithmetic <complex.h> (p: 188-199)
• 7.25 Type-generic math <tgmath.h> (p: 373-375)
• 7.31.1 Complex arithmetic <complex.h> (p: 455)
• Annex G (normative) IEC 60559-compatible complex arithmetic (p: 532-545)
• C99 standard (ISO/IEC 9899:1999):
• 6.10.8/2 __STDC_IEC_559_COMPLEX__ (p: 161)
• 7.3 Complex arithmetic <complex.h> (p: 170-180)
• 7.22 Type-generic math <tgmath.h> (p: 335-337)
• 7.26.1 Complex arithmetic <complex.h> (p: 401)
• Annex G (informative) IEC 60559-compatible complex arithmetic (p: 467-480)

See also