Question: In my opinion, things like float('nan') should be optimized, but apparently they aren't in Python. >>> NaN = float('nan') >>> a = [ 1, 2, 3, NaN ] >>> NaN in a True >>> float('nan') in a False  Does it have any meaning with not optimizing nan like other things? In my thought, nan is only nan. As well as this, when you use sorted on these things, they give weird results: >>> sorted([3, nan, 4, 2, nan, 1]) [3, nan, 1, 2, 4, nan] >>> 3 > float('nan') False >>> 3 < float('nan') False  The comparison on nan is defined like this, but it doesn't seems 'pythonic' to me. Why doesn't it raise an error?

Answer: Membership testing Two different instances of float('nan') are not equal to each other. They are "Not a Number" so it makes sense that they shouldn't also have to be equal. They are different instances of objects which are not numbers: print(float('nan') == float('nan')) # False  As documented here:  For container types such as list, tuple, set, frozenset, dict, or collections.deque, the expression x in y is equivalent to any(x is e or x == e for e in y).  There is a checking for identity! that's why you see that behavior in your question and why NaN in a returns True and float('nan') in a doesn't.  Sorting in Python Python uses the Timsort algorithm for its sorted() function. (Also see this for a textual explanation.) I'm not going to go into that. I just want to demonstrate a simple example: This is my class A. It's going to be our float('nan') object. It acts like float('nan') in that it returns False for all comparison operations: class A: def __init__(self, n): self.n = n def __lt__(self, other): print(self, 'lt is calling', other) return False def __gt__(self, other): print(self, 'gt is calling', other) return False def __repr__(self): return f'A({self.n})' class B: def __init__(self, n): self.n = n def __lt__(self, other): print(self, 'lt is calling', other) return False def __gt__(self, other): print(self, 'gt is calling', other) return False def __repr__(self): return f'B({self.n})'  When we use the sorted() function (or the .sort() method of a list) without the reverse=True argument, we're asking for the iterable to be sorted in ascending order. To do this, Python tries to call the __lt__ method successively, starting from the second object in the list to see if it is less than its previous object and so on: lst = [A(1), B(2), A(3), B(4)] print(sorted(lst))  output : B(2) lt is calling A(1) A(3) lt is calling B(2) B(4) lt is calling A(3) [A(1), B(2), A(3), B(4)]  Now, switching back to your example: lst = [3, A(1), 4, 2, A(1), 1] print(sorted(lst))  output: A(1) lt is calling 3 A(1) gt is calling 4 A(1) gt is calling 2 A(1) lt is calling 2 A(1) lt is calling 4 A(1) gt is calling 1 [3, A(1), 1, 2, 4, A(1)]   A(1).__lt__(3) will return False. This means A(1) is not less than 3 or This means 3 is in correct position relative to A(1). Then here int.__lt__(4, A(1)) gets called and because it returns NotImplemented object, Python checks to see if A(1) has implemented __gt__ and yes, so A(1).__gt__(4) will return False again and this means the A(1) object is in correct place relative to 4. (Etc.)  This is why the result of sorted() seems to be weird, but it's predictable. A(1) object in both cases, I mean when int class returns NotImplemented and when __lt__ gets called from A(1), will return False. It's better to check the Timsort algorithm and consider those points. I would include the remaining steps if I read Timsort algorithm carefully.