# sorting list of complex numbers

skip at pobox.com skip at pobox.com
Sun Nov 9 15:44:15 CET 2008

```The thread on sorting in Python 3 got me to thinking.  How could I sort a
list of complex numbers using key?

>>> lst = [random.random()+random.random()*1j for i in range(10)]
>>> lst
[(0.32672251849959244+0.41428983433288791j), (0.35238056484609881+0.92758203977208264j), (0.19337824038125528+0.16527285180541951j), (0.47569307114525849+0.72381960418815128j), (0.21498813135082351+0.2046308266222292j), (0.30142745756937939+0.35216751451102601j), (0.77355676386939132+0.0023447924287695043j), (0.2547736124606309+0.52234837788936905j), (0.38349189081350132+0.62017617694427096j), (0.58362096773561245+0.87702443273108477j)]

As expected:

>>> sorted(lst)
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
TypeError: no ordering relation is defined for complex numbers

This works:

>>> sorted(lst, key=lambda x: x.real)
[(0.19337824038125528+0.16527285180541951j), (0.21498813135082351+0.2046308266222292j), (0.2547736124606309+0.52234837788936905j), (0.30142745756937939+0.35216751451102601j), (0.32672251849959244+0.41428983433288791j), (0.35238056484609881+0.92758203977208264j), (0.38349189081350132+0.62017617694427096j), (0.47569307114525849+0.72381960418815128j), (0.58362096773561245+0.87702443273108477j), (0.77355676386939132+0.0023447924287695043j)]

but what if I want to sort by real, then imaginary parts?  Here's a longer
list with 20 elements where there are only 10 distinct reals but 20 distinct
imaginaries:

>>> pprint.pprint(lst)
[(1+2.73j),
(9+3.77j),
(7+27j),
(8+28j),
(2+2.8600000000000003j),
(4+3.1200000000000001j),
(2+22j),
(9+29j),
(3+2.9900000000000002j),
(6+26j),
2.6000000000000001j,
(8+3.6400000000000001j),
(3+23j),
(5+3.25j),
(1+21j),
(5+25j),
20j,
(6+3.3799999999999999j),
(7+3.5100000000000002j),
(4+24j)]

I can sort by the real parts just fine:

>>> lst.sort(key=lambda x: x.real)
>>> pprint.pprint(lst)
[2.6000000000000001j,
20j,
(1+2.73j),
(1+21j),
(2+2.8600000000000003j),
(2+22j),
(3+2.9900000000000002j),
(3+23j),
(4+3.1200000000000001j),
(4+24j),
(5+3.25j),
(5+25j),
(6+26j),
(6+3.3799999999999999j),
(7+27j),
(7+3.5100000000000002j),
(8+28j),
(8+3.6400000000000001j),
(9+3.77j),
(9+29j)]

but how do I then do a secondary sort by the imaginary part, preserving the
existing ordering on the real parts?  Seems like I have to resort to a
Schwartzian transform and map the complex numbers to tuples, sort that, then
map them back.  With the cmp key it would have been a fairly trivial task to
define the desired compare-real-then-imag function.

Is there a way to do this using just the key arg, no extra data structures?

Skip

```