36 Complex Numbers

A tiny primer on Complex numbers¹⁸³ stolen directly from Wikipedia:

A complex number is a number that can be expressed in the form \(a+bi\), where \(a\) and \(b\) are real numbers [i.e. floating point types in C], and \(i\) represents the imaginary unit, satisfying the equation \(i^2=−1\). Because no real number satisfies this equation, \(i\) is called an imaginary number. For the complex number \(a+bi\), \(a\) is called the real part, and \(b\) is called the imaginary part.

But that’s as far as I’m going to go. We’ll assume that if you’re reading this chapter, you know what a complex number is and what you want to do with them.

And all we need to cover is C’s faculties for doing so.

Turns out, though, that complex number support in a compiler is an optional feature. Not all compliant compilers can do it. And the ones that do, might do it to various degrees of completeness.

You can test if your system supports complex numbers with:

#ifdef __STDC_NO_COMPLEX__
#error Complex numbers not supported!
#endif

Furthermore, there is a macro that indicates adherence to the ISO 60559 (IEEE 754) standard for floating point math with complex numbers, as well as the presence of the _Imaginary type.

#if __STDC_IEC_559_COMPLEX__ != 1
#error Need IEC 60559 complex support!
#endif

More details on that are spelled out in Annex G in the C11 spec.

36.1 Complex Types

To use complex numbers, #include <complex.h>.

With that, you get at least two types:

_Complex
complex

Those both mean the same thing, so you might as well use the prettier complex.

You also get some types for imaginary numbers if you implementation is IEC 60559-compliant:

_Imaginary
imaginary

These also both mean the same thing, so you might as well use the prettier imaginary.

You also get values for the imaginary number \(i\), itself:

I
_Complex_I
_Imaginary_I

The macro I is set to _Imaginary_I (if available), or _Complex_I. So just use I for the imaginary number.

One aside: I’ve said that if a compiler has __STDC_IEC_559_COMPLEX__ set to 1, it must support _Imaginary types to be compliant. That’s my read of the spec. However, I don’t know of a single compiler that actually supports _Imaginary even though they have __STDC_IEC_559_COMPLEX__ set. So I’m going to write some code with that type in here I have no way of testing. Sorry!

OK, so now we know there’s a complex type, how can we use it?

36.2 Assigning Complex Numbers

Since the complex number has a real and imaginary part, but both of them rely on floating point numbers to store values, we need to also tell C what precision to use for those parts of the complex number.

We do that by just pinning a float, double, or long double to the complex, either before or after it.

Let’s define a complex number that uses float for its components:

float complex c;   // Spec prefers this way
complex float c;   // Same thing--order doesn't matter

So that’s great for declarations, but how do we initialize them or assign to them?

Turns out we get to use some pretty natural notation. Example!

double complex x = 5 + 2*I;
double complex y = 10 + 3*I;

For \(5+2i\) and \(10+3i\), respectively.

36.3 Constructing, Deconstructing, and Printing

We’re getting there…

We’ve already seen one way to write a complex number:

double complex x = 5 + 2*I;

There’s also no problem using other floating point numbers to build it:

double a = 5;
double b = 2;
double complex x = a + b*I;

There is also a set of macros to help build these. The above code could be written using the CMPLX() macro, like so:

double complex x = CMPLX(5, 2);

As far as I can tell in my research, these are almost equivalent:

double complex x = 5 + 2*I;
double complex x = CMPLX(5, 2);

But the CMPLX() macro will handle negative zeros in the imaginary part correctly every time, whereas the other way might convert them to positive zeros. I think¹⁸⁴ This seems to imply that if there’s a chance the imaginary part will be zero, you should use the macro… but someone should correct me on this if I’m mistaken!

The CMPLX() macro works on double types. There are two other macros for float and long double: CMPLXF() and CMPLXL(). (These “f” and “l” suffixes appear in virtually all the complex-number-related functions.)

Now let’s try the reverse: if we have a complex number, how do we break it apart into its real and imaginary parts?

Here we have a couple functions that will extract the real and imaginary parts from the number: creal() and cimag():

double complex x = 5 + 2*I;
double complex y = 10 + 3*I;

printf("x = %f + %fi\n", creal(x), cimag(x));
printf("y = %f + %fi\n", creal(y), cimag(y));

for the output:

x = 5.000000 + 2.000000i
y = 10.000000 + 3.000000i

Note that the i I have in the printf() format string is a literal i that gets printed—it’s not part of the format specifier. Both return values from creal() and cimag() are double.

And as usual, there are float and long double variants of these functions: crealf(), cimagf(), creall(), and cimagl().

36.4 Complex Arithmetic and Comparisons

Arithmetic can be performed on complex numbers, though how this works mathematically is beyond the scope of the guide.

#include <stdio.h>
#include <complex.h>

int main(void)
{
    double complex x = 1 + 2*I;
    double complex y = 3 + 4*I;
    double complex z;

    z = x + y;
    printf("x + y = %f + %fi\n", creal(z), cimag(z));

    z = x - y;
    printf("x - y = %f + %fi\n", creal(z), cimag(z));

    z = x * y;
    printf("x * y = %f + %fi\n", creal(z), cimag(z));

    z = x / y;
    printf("x / y = %f + %fi\n", creal(z), cimag(z));
}

for a result of:

x + y = 4.000000 + 6.000000i
x - y = -2.000000 + -2.000000i
x * y = -5.000000 + 10.000000i
x / y = 0.440000 + 0.080000i

You can also compare two complex numbers for equality (or inequality):

#include <stdio.h>
#include <complex.h>

int main(void)
{
    double complex x = 1 + 2*I;
    double complex y = 3 + 4*I;

    printf("x == y = %d\n", x == y);  // 0
    printf("x != y = %d\n", x != y);  // 1
}

with the output:

x == y = 0
x != y = 1

They are equal if both components test equal. Note that as with all floating point, they could be equal if they’re close enough due to rounding error¹⁸⁵.

36.5 Complex Math

But wait! There’s more than just simple complex arithmetic!

Here’s a summary table of all the math functions available to you with complex numbers.

I’m only going to list the double version of each function, but for all of them there is a float version that you can get by appending f to the function name, and a long double version that you can get by appending l.

For example, the cabs() function for computing the absolute value of a complex number also has cabsf() and cabsl() variants. I’m omitting them for brevity.

36.5.1 Trigonometry Functions

36.5.2 Exponential and Logarithmic Functions

36.5.3 Power and Absolute Value Functions

36.5.4 Manipulation Functions