Is Python's == an equivalence relation on the floats?

Question

In native Python, without using NumPy (for which numpy.nan != numpy.nan) there is no NaN, so am I right in thinking that Python's floating point == is reflexive? Then since it is symmetric (a == b implies b == a) and transitive (if a==b and b==c then a==c), can we say that Python's == is an equivalence relation on the floats?

EDIT: OK, so I learned that there is a NaN: float('nan') (thanks @unutbu) which will propagate through various operations, but does any native Python method return it (rather than raising an Exception) without me introducing it by this assignment?

I did not know this: thank you. Is `nan` actually returned by any native Python operation (instead of an Exception being raised)? — xnx, Jan 02 '15 at 14:47
See also [PEP 754](https://www.python.org/dev/peps/pep-0754/) for some more background on this. — Daniel Pryden, Jan 02 '15 at 20:05

score 46 · Accepted Answer · edited Jan 02 '15 at 20:38

== is reflexive for all numbers, zero, -zero, ininity, and -infinity, but not for nan.

You can get inf, -inf, and nan in native Python just by arithmetic operations on literals, like below.

These behave correctly, as in IEEE 754 and without math domain exception:

>>> 1e1000 == 1e1000
True
>>> 1e1000/1e1000 == 1e1000/1e1000
False

1e1000 is a very big number, so float and double represent it as an infinity.

infinity is equal to infinity
infinity divided by infinity is not a number
not a number != not a number

Floating-point arithmetic in Python also works OK for infinity minus infinity etc.:

>>> x = 1e1000
>>> x
inf
>>> x+x
inf
>>> x-x
nan
>>> x*2
inf
>>> x == x
True
>>> x-x == x-x
False
>>>

And for the zero and minus zero case:

>>> inf = float("inf")
>>> 1/inf
0.0
>>> -1/inf
-0.0
>>> -1/inf == 1/inf
True
>>>

For completeness, it might be helpful to include the (short) list of axioms that makes an equivalence relation. Also, you never actually explicitly say it is not an equivalence relation, you simply say that it is not reflexive for `nan`. — Logan, Jan 03 '15 at 02:04

score 34 · Answer 2 · answered Jan 02 '15 at 14:50

34

float('nan') exists in native Python and float('nan') != float('nan'). So no, == is not an equivalence relation since it lacks reflexivity:

In [40]: float('nan') == float('nan')
Out[40]: False

answered Jan 02 '15 at 14:50

unutbu

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Is Python's == an equivalence relation on the floats?

2 Answers2