Complex number arithmetic

If the macro constant __STDC_NO_COMPLEX__ is defined by the implementation, the complex types, the header <complex.h> and all of the names listed here are not provided. (since C11)

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.

A compiler that defines __STDC_IEC_559_COMPLEX__ is recommended, but not required to support imaginary numbers. POSIX recommends checking if the macro _Imaginary_I is defined to identify imaginary number support.	(since C99) (until C11)
Imaginary numbers are supported if __STDC_IEC_559_COMPLEX__ is defined.	(since C11)

Defined in header <complex.h>

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 components is infinite, even if the other component is NaN.

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

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

Example

Run this code

#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));
}

Output:

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

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)

7.3.1/2 __STDC_NO_COMPLEX__ (p: 188)

7.25 Type-generic math <tgmath.h> (p: 373-375)

7.31.1 Complex arithmetic <complex.h> (p: 455)

B.2 Complex <complex.h> (p: 475-477)

Annex G (normative) IEC 60559-compatible complex arithmetic (p: 532-545)

G.1/1 __STDC_IEC_559_COMPLEX__ (p: 532)

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)

B.2 Complex <complex.h> (p: 419-420)

Annex G (informative) IEC 60559-compatible complex arithmetic (p: 467-480)

G.1/1 __STDC_IEC_559_COMPLEX__ (p: 467)

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Common mathematical functions
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Complex number arithmetic

Contents

Types

The imaginary constant

Manipulation

Exponential functions

Power functions

Trigonometric functions

Hyperbolic functions

Notes

Example

References

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