If you test if a not-a-number is not equal to itself, the result is also false.

The reason is that the result of comparing a not-number with itself is neither "equal" nor "not equal", but "unordered".

This should make perfect intuitive sense, because a NaN encodes the fact that "an undefined operation has happened".

For instance, you have 2 instances of the same NaN, both having been obtained by multiplying zero with infinity.

However it may happen that one was obtained while computing values of a sequence that converges to 10 and the other may have been obtained by computing values of a sequence that converges to 100.

Then there is no doubt that equality is not true for this 2 NaNs.

However, those 2 NaNs may have been generated while computing values of sequences that both converge to 10, which shows that neither non-equality may be trued for these 2 NaNs.

When you have NaNs, that means that it is unknown which value, if any, the invalid operations should have produced.

When comparing 2 unknown values, you cannot know whether they are equal or not equal, so both testing for equality and for non-equality must fail.