# Complex number arithmetic

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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>`

##### Types

imaginary (C99) | imaginary type macro (keyword macro)
complex (C99) | complex type macro (keyword macro)

##### The imaginary constant

_Imaginary_I (C99) | the imaginary unit constant i (macro constant)
_Complex_I (C99) | the complex unit constant i (macro constant)
I (C99) | the complex or imaginary unit constant i (macro constant)

##### Manipulation

CMPLXCMPLXFCMPLXL (C11)(C11)(C11) | constructs a complex number from real and imaginary parts (function macro)
crealcrealfcreall (C99)(C99)(C99) | computes the real part of a complex number (function)
cimagcimagfcimagl (C99)(C99)(C99) | computes the imaginary part a complex number (function)
cabscabsfcabsl (C99)(C99)(C99) | computes the magnitude of a complex number (function)
cargcargfcargl (C99)(C99)(C99) | computes the phase angle of a complex number (function)
conjconjfconjl (C99)(C99)(C99) | computes the complex conjugate (function)
cprojcprojfcprojl (C99)(C99)(C99) | computes the projection on Riemann sphere (function)

##### Exponential functions

cexpcexpfcexpl (C99)(C99)(C99) | computes the complex base-e exponential (function)
clogclogfclogl (C99)(C99)(C99) | computes the complex natural logarithm (function)

##### Power functions

cpowcpowfcpowl (C99)(C99)(C99) | computes the complex power function (function)
csqrtcsqrtfcsqrtl (C99)(C99)(C99) | computes the complex square root (function)

##### Trigonometric functions

csincsinfcsinl (C99)(C99)(C99) | computes the complex sine (function)
ccosccosfccosl (C99)(C99)(C99) | computes the complex cosine (function)
ctanctanfctanl (C99)(C99)(C99) | computes the complex tangent (function)
casincasinfcasinl (C99)(C99)(C99) | computes the complex arc sine (function)
cacoscacosfcacosl (C99)(C99)(C99) | computes the complex arc cosine (function)
catancatanfcatanl (C99)(C99)(C99) | computes the complex arc tangent (function)

##### Hyperbolic functions

csinhcsinhfcsinhl (C99)(C99)(C99) | computes the complex hyperbolic sine (function)
ccoshccoshfccoshl (C99)(C99)(C99) | computes the complex hyperbolic cosine (function)
ctanhctanhfctanhl (C99)(C99)(C99) | computes the complex hyperbolic tangent (function)
casinhcasinhfcasinhl (C99)(C99)(C99) | computes the complex arc hyperbolic sine (function)
cacoshcacoshfcacoshl (C99)(C99)(C99) | computes the complex arc hyperbolic cosine (function)
catanhcatanhfcatanhl (C99)(C99)(C99) | computes the complex arc hyperbolic tangent (function)

### Notes

The following function names are 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`, and `ctgamma`, 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 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)

### See also

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