Format Of Complex Number In Python

I am wondering about the way Python (3.3.0) prints complex numbers. I am looking for an explanation, not a way to change the print. Example: >>> complex(1,1)-complex(1,1)

Solution 1:

It prints 0j to indicate that it's still a complex value. You can also type it back in that way:

>>> 0j
0j

The rest is probably the result of the magic of IEEE 754 floating point representation, which makes a distinction between 0 and -0, the so-called signed zero. Basically, there's a single bit that says whether the number is positive or negative, regardless of whether the number happens to be zero. This explains why 1j * -1 gives something with a negative zero real part: the positive zero got multiplied by -1.

-0 is required by the standard to compare equal to +0, which explains why (1j * -1).real == 0.0 still holds.

The reason that Python still decides to print the -0, is that in the complex world these make a difference for branch cuts, for instance in the phase function:

>>> phase(complex(-1.0, 0.0))
3.141592653589793
>>> phase(complex(-1.0, -0.0))
-3.141592653589793

This is about the imaginary part, not the real part, but it's easy to imagine situations where the sign of the real part would make a similar difference.

Solution 2:

The answer lies in the Python source code itself.

I'll work with one of your examples. Let

a = complex(0,1)
b = complex(-1, 0)

When you doa*b you're calling this function:

real_part = a.real*b.real - a.imag*b.imag
imag_part = a.real*b.imag + a.imag*b.real

And if you do that in the python interpreter, you'll get

>>> real_part
-0.0
>>> imag_part
-1.0

From IEEE754, you're getting a negative zero, and since that's not +0, you get the parens and the real part when printing it.

if (v->cval.real == 0. && copysign(1.0, v->cval.real)==1.0) {
    /* Real part is +0: just output the imaginary part and do not
       include parens. */
...
else {
    /* Format imaginary part with sign, real part without. Include
       parens in the result. */
...

I guess (but I don't know for sure) that the rationale comes from the importance of that sign when calculating with elementary complex functions (there's a reference for this in the wikipedia article on signed zero).

Solution 3:

• 0j is an imaginary literal which indeed indicates a complex number rather than an integer or floating-point one.

• The +-0 ("signed zero") is a result of Python's conformance to IEEE 754 floating point representation since in Python, complex is by definition a pair of floating point numbers. Due to the latter, there's no need to print or specify zero fraction parts for a complex too.

• The -0 part is printed in order to accurately represent the contents as repr()'s documentation demands (repr() is implicitly called whenever an operation's result is output to the console).

• Regarding the question why (-0+1j) = 1j but (1j*-1) = (-0+1j). Note that (-0+0j) or (-0.0+0j) aren't single complex numbers but expressions - an int/float added to a complex. To compute the result, first the first number is converted to a complex (-0-> (0.0,0.0) since integers don't have signed zeros, -0.0-> (-0.0,0.0)). Then its .real and .imag are added to the corresponding ones of 1j which are (+0.0,1.0). The result is (+0.0,1.0) :^) . To construct a complex directly, use complex(-0.0,1).

Solution 4:

As far as the first question is concerned: if it just printed 0 it would be mathematically correct, but you wouldn't know you were dealing with a complex object vs an int. As long as you don't specify .real you will always get a J component.

I'm not sure why you would ever get -0; it's not technically incorrect (-1 * 0 = 0) but it's syntactically odd.

As far as the rest goes, it's strange that it isn't consistent, however none are technically correct, just an artifact of the implementation.