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16.4.1.4 Floating Point Values They Didn't Talk About In School
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Both IEEE 754 floating-point hardware, and MPFR, support two kinds of
values that you probably didn't learn about in school.  The first is
"infinity", a special value, that can be either negative or positive,
and which is either smaller than any other value (negative infinity), or
larger than any other value (positive infinity).  When such values are
generated, 'gawk' prints them as either '-inf' or '+inf', respectively.
It accepts those strings as data input and converts them to the proper
floating-point values internally.

   Infinity values of the same sign compare as equal to each other.
Otherwise, operations (addition, subtraction, etc.)  involving another
number and infinity produce mathematically reasonable results.

   The second kind of value is "not a number", or NaN for short.(1)
This is a special value that results from attempting a calculation that
has no answer as a real number.  In such a case, programs can either
receive a floating-point exception, or get NaN back as the result.  The
IEEE 754 standard recommends that systems return NaN. Some examples:

'sqrt(-1)'
     This makes sense in the range of complex numbers, but not in the
     range of real numbers, so the result is NaN.

'log(-8)'
     -8 is out of the domain of 'log()', so the result is NaN.

   NaN values are strange.  In particular, they cannot be compared with
other floating point values; any such comparison, except for "is not
equal to", returns false.  NaN values are so much unequal to other
values that even comparing two identical NaN values with '!=' returns
true!

   NaN values can also be signed, although it depends upon the
implementation as to which sign you get for any operation that returns a
NaN. For example, on some systems, 'sqrt(-1)' returns a negative NaN. On
others, it returns a positive NaN.

   When such values are generated, 'gawk' prints them as either '-nan'
or '+nan', respectively.  Here too, 'gawk' accepts those strings as data
input and converts them to the proper floating-point values internally.

   If you want to dive more deeply into this topic, you can find test
programs in C, 'awk' and Python in the directory
'awklib/eg/test-programs' in the 'gawk' distribution.  These programs
enable comparison among programming languages as to how they handle NaN
and infinity values.

   ---------- Footnotes ----------

   (1) Thanks to Michael Brennan for this description, which we have
paraphrased, and for the examples.