SIMPLE SOLUTIONS

# COMPLEX - reference manual online

Basics of complex mathematics.

Chapter
2011-09-16
```COMPLEX(7)                          Linux Programmer's Manual                          COMPLEX(7)

NAME
complex - basics of complex mathematics

SYNOPSIS
#include <complex.h>

DESCRIPTION
Complex numbers are numbers of the form z = a+b*i, where a and b are real numbers and i =
sqrt(-1), so that i*i = -1.
There are other ways to represent that number.  The pair (a,b) of real numbers may be
viewed as a point in the plane, given by X- and Y-coordinates.  This same point may also
be described by giving the pair of real numbers (r,phi), where r is the distance to the
origin O, and phi the angle between the X-axis and the line Oz.  Now z = r*exp(i*phi) =
r*(cos(phi)+i*sin(phi)).

The basic operations are defined on z = a+b*i and w = c+d*i as:

addition: z+w = (a+c) + (b+d)*i

multiplication: z*w = (a*c - b*d) + (a*d + b*c)*i

division: z/w = ((a*c + b*d)/(c*c + d*d)) + ((b*c - a*d)/(c*c + d*d))*i

Nearly all math function have a complex counterpart but there are some complex-only func‐
tions.

EXAMPLE
Your C-compiler can work with complex numbers if it supports the C99 standard.  Link with
-lm.  The imaginary unit is represented by I.

/* check that exp(i * pi) == -1 */
#include <math.h>        /* for atan */
#include <stdio.h>
#include <complex.h>

int
main(void)
{
double pi = 4 * atan(1.0);
double complex z = cexp(I * pi);
printf("%f + %f * i\n", creal(z), cimag(z));
}