Editing NaN (section)

=== Comparison with NaN ===
Comparisons are specified by the [[IEEE 754]] standard to take into account possible NaN operands.<ref>{{Harvnb|IEEE 754|2019|loc=§5.11}}</ref> When comparing two real numbers, or [[Extended real number line|extended real numbers]] (as in the IEEE 754 floating-point formats), the first number may be either less than, equal to, or greater than the second number. This gives three possible relations. But when at least one operand of a comparison is NaN, this trichotomy does not apply, and a fourth relation is needed: ''unordered''. In particular, two NaN values compare as unordered, not as equal.

As specified, the predicates associated with the <, ≤, =, ≥, > mathematical symbols (or equivalent notation in programming languages) return false on an unordered relation. So, for instance, {{nobr|1=[[Negation|NOT]](''x'' < ''y'')}} is not logically equivalent to {{nobr|1=''x'' ≥ ''y''}}: on unordered, i.e. when ''x'' or ''y'' is NaN, the former returns true while the latter returns false. However, ≠ is defined as the negation of =, thus it returns true on unordered.

{| class="wikitable" style="text-align: center"
|+ Comparison between NaN and any floating-point value ''x''<br />(including NaN and ±∞)
! Comparison
| NaN ≥ ''x''
| NaN ≤ ''x''
| NaN > ''x''
| NaN < ''x''
| NaN = ''x''
| NaN ≠ ''x''
|-
! Result
| False
| False
| False
| False
| False
| True
|}

From these rules, comparing ''x'' with itself, {{nobr|1=''x'' ≠ ''x''}} or {{nobr|1=''x'' = ''x''}}, can be used to test whether ''x'' is NaN or non-NaN.

The comparison predicates are either signaling or non-signaling on quiet NaN operands; the signaling versions signal the invalid-operation [[IEEE 754#Exception handling|exception]] for such comparisons (i.e., by default, this just sets the corresponding status flag in addition to the behavior of the non-signaling versions). The equality and inequality predicates are non-signaling. The other standard comparison predicates associated with the above mathematical symbols are all signaling if they receive a NaN operand. The standard also provides non-signaling versions of these other predicates. The predicate <code>isNaN(''x'')</code> determines whether a value is a NaN and never signals an exception, even if ''x'' is a signaling NaN.

The [[IEEE 754|IEEE floating-point]] standard requires that ''NaN ≠ NaN'' hold. In contrast, the 2022 [[technical standard|private standard]] of [[posit (number format)|posit]] arithmetic has a similar concept, NaR (Not a Real), where ''NaR = NaR'' holds.<ref>[https://posithub.org/docs/posit_standard-2.pdf Standard for Posit Arithmetic (2022)]</ref>