Historical implementations of awk did not support floating-point infinities and NaNs in numeric strings; e.g., "-INF" and "NaN". 
However, implementations that use the atof() or strtod() functions to do the conversion picked up support for these values 
if they used a ISO/IEC 9899:1999 standard version of the function instead of a ISO/IEC 9899:1990 standard version. 
Due to an oversight, the 2001 through 2004 editions of this standard did not allow support for infinities and NaNs, 
but in this revision support is allowed (but not required). This is a silent change to the behavior of awk programs; 
for example, in the POSIX locale the expression:
("-INF" + 0 < 0)
formerly had the value 0 because "-INF" converted to 0, but now it may have the value 0 or 1.

strtod recognizes four special input strings. The strings "inf" and "infinity" are converted to ∞, 
or to the largest representable value if the floating-point format doesn’t support infinities. 
You can prepend a "+" or "-" to specify the sign. Case is ignored when scanning these strings.
The strings "nan" and "nan(chars…)" are converted to NaN. Again, case is ignored. 
If chars… are provided, they are used in some unspecified fashion to select a particular representation of NaN (there can be several).

When a math function suffers a domain error, it raises the invalid exception and returns NaN....

A valid floating point number for strtod using the "C" locale is formed by an optional sign character (+ or -), followed by one of:
...........
- INF or INFINITY (ignoring case).
- NAN or NANsequence (ignoring case), where sequence is a sequence of characters, where each character is either an alphanumeric character

- NAN and INF expressions are defined in 'math.h' as double/floating/long double number to C locale *(C99/C11 standarts)

# define FP_NAN FP_NAN
# define FP_INFINITE FP_INFINITE

 -- nawk uses "strtod" for convert the strings to double type

 - executes "strtod" function and
   + returns 0.000000 double for "strings" and "strings/numbers" mix char sequence
   + returns number.000000 double for "numbers" and "numbers/strings" mix char sequence (cares only first numbers from the beginning )
   + returns NaN double for beginning the "nan" (ignore case) string
   + returns Inf double for beginning the "inf" (ignore case) string

 - executes summ operations 
   
   see results : 

   'NaN + number' =  NaN (Not a Number)
 ----------------------------------------------------------
# echo nAN1 | nawk '{print $1+1}'
NaN

# echo nAN1 | nawk '{print $1*1}'
NaN

# echo nAN1 | nawk '{print $^1}'
NaN

# echo nAN1 | nawk '{print $1-1}'
NaN

# echo nAN1 | nawk '{print $1/1}'
NaN


   'Inf + number' =  Inf (infinity)
 ----------------------------------------------------------
# echo Inf1 | nawk '{print $1+1}'
Inf

# echo Inf1 | nawk '{print $1*1}'
Inf

# echo Inf1 | nawk '{print $1^1}'
Inf

# echo Inf1 | nawk '{print $1-1}'
Inf

# echo Inf1 | nawk '{print $1/1}'
Inf

Note : tested nawk on the SunOS 5.1 11.1 sun4v sparc