Complex Numbers in C++ Programming

COMPLEX_NUMBERS is a C++ program which demonstrates very briefly some of the features of using complex numbers in a C++ program.

The first issue is how to declare a complex variable, including the choice of single precision or double precision, whether the variable is a scalar, vector, or array, and whether the variable is initialized with a value, or assigned one.

A second issue concerns the question of how a complex variable is to be printed out.

Another issue concerns how a complex variable is to operated on by the arithmetic operators of addition, subtraction, multiplication, division, and exponentiation.

The language also provides a number of intrinsic functions that can be applied to a complex variable. The names of these functions can sometimes be easy to forget. Moreover, it is occasionally true that there may be a selection of functions with similar names (say, "exp", "cexp" and "dcexp") which may or may not produce the desired results.

Another issue concerns the details of double precision calculation. Even a single accidental use of a single precision function name in a double precision computation can result in the loss of half the digits of accuracy. Thus, it sometimes really matters whether you use "cmplx" or "dcmplx" to assign values to a double precision complex variable.

An unusual feature of the C++ implementation of complex numbers is that "complex <float>" and "complex <double>" are not as easily used as an ordinary type when trying to "cast" values. That is, it may be very tempting to try to assign a complex value with a statement such as

a = ( complex <double> ) ( 1.0, 2.0 ); (illegal!)
but this is wrong because (1.0,2.0) is not a legitimate numeric type that is to be converted to another type, which is what a cast operator requires. The statement, unfortunately, is not necessarily illegal, so the compiler won't warn you, and my compiler will actually end up setting a to 2, which is not what I wanted at all. The correct assignment is
a = complex <double> ( 1.0, 2.0 );
and you can read this as a call to a function whose arguments are the real and imaginary parts of the desired result. People who are used to the idea that parentheses never hurt should take note of this counterexample!

A second, and more peculiar, feature to me is that it seems as though the complex numbers created in C++ cannot easily interoperate with the real and integer values. In particular, assuming the declaration

a = complex <double> ( 1.0, 2.0 );
my compiler complained about statements like these:
        a = a + 1; (illegal!)
        a = a * 4; (illegal!)
        a = a / 8; (illegal!)
        a = 1 / a; (illegal!)
        a = pow ( 2, a ); (illegal!)
        a = pow ( 2.1, a ); (illegal!)
all of which became legal when I replaced the arithmetic operands by corresponding complex <double> variables of the same value.

A third peculiarity of the C++ version of complex numbers is that the function norm ( z ) is stated to return the norm of the complex number z, but in fact, returns the square of the norm of the number. The use of the word "norm" is a misuse, since the norm has a commonly accepted mathematical definition as the square root of the sum of the squares of the real and imaginary parts. Thus, the C++ function returns the "norm" of 4+4i as 32 whereas the norm of 4+4i is sqrt(32). As ever in computer languages, once a bad choice is made, it's never rescinded. Be prepared for needless and inevitable confusion, though, when using this function.


The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.


COMPLEX_NUMBERS is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.

Related Data and Programs:

C4LIB, a C++ library which implements certain elementary functions for "C4" or single precision complex variables;

C8LIB, a C++ library which implements certain elementary functions for "C8" or double precision complex variables;

Source Code:

Examples and Tests:

You can go up one level to the C++ source codes.

Last revised on 20 June 2018.