complex.h is what you include when you need the maths to work directly with complex numbers, rather than manually tracking real and imaginary parts in separate double variables. Added in C99, it gives you _Complex types, the imaginary unit I, and the full set of functions for magnitude, phase, conjugate, exponential, trig, and so on.

#include <complex.h>
 
double complex z = 3.0 + 4.0 * I;   // I is sqrt(-1)
 
double mag   = cabs(z);       // magnitude: sqrt(3² + 4²) = 5.0
double phase = carg(z);       // phase angle: atan2(4, 3) ≈ 0.9273 rad
double complex c = conj(z);   // conjugate: 3 - 4i

The arithmetic operators work natively on double complex — addition, subtraction, multiplication, and division all just work. The full function set mirrors <math.h>: csin, ccos, cexp, clog, cpow, csqrt, creal, cimag. Float variants have an f suffix (cabsf) and long double variants have an l suffix.

Complex numbers come up in signal processing (FFT, filter design), electrical engineering (impedance), and quantum mechanics (wave functions). For heavy numerical work, libraries like FFTW provide the optimised implementations; <complex.h> is the language-level foundation they sit on top of.

// compile: gcc -o cplex cplex.c -lm
// run: ./cplex
// description: basic complex arithmetic; Euler's formula as a sanity check
 
#include <complex.h>
#include <math.h>
#include <stdio.h>
 
int main(void) {
    double complex z = 3.0 + 4.0 * I;
 
    printf("|z|     = %.1f\n",     cabs(z));            // 5.0
    printf("arg(z)  = %.4f rad\n", carg(z));            // 0.9273
    printf("conj(z) = %.1f%+.1fi\n", creal(conj(z)), cimag(conj(z)));
 
    // Euler's formula: e^(i*pi) should equal -1 + 0i
    double complex euler = cexp(I * M_PI);
    printf("e^(i*pi) = %.6f%+.6fi\n", creal(euler), cimag(euler));
    return 0;
}

The last line demonstrates Euler's formula: $e^{i\pi} \approx -1$. The imaginary part will be a very small number near zero but not exactly zero — floating-point rounding applies to complex arithmetic just as it does to real arithmetic. If you are seeing values like -1.000000+0.000000i, that tiny residue is hidden by the format width; try %.20f to see it.