Complex number arithmetic

If the macro constant __STDC_NO_COMPLEX__(C11) 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)

If the #pragma STDC CX_LIMITED_RANGE on is used, complex multiplication, division, and absolute value may use simplified mathematical formulas $(x+iy)\times(u+iv) = (xu-yv)+i(yu+xv)$ , $(x+iy)/(u+iv) = [(xu+yv)+i(yu-xv)]/(u 2 +v 2)$ , and $|x+iy| = \sqrt x 2 +y 2$ , despite the possibility of intermediate overflow.

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

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Common mathematical functions
Floating-point environment (C99)
Pseudo-random number generation
Complex number arithmetic (C99)
Type-generic math (C99)

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Complex number arithmetic

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The imaginary constant

Manipulation

Exponential functions

Power functions

Trigonometric functions

Hyperbolic functions

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Example

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