[GENERAL] Does NUMERIC lose precision?

Hi everyone. I’m trying to perform some exact precision arithmetic with PostgreSQL’s NUMERIC type. However I can’t seem to get the unparameterized NUMERIC type to perform exact precision arithmetic:

# SELECT 2::NUMERIC ^ 64;
               ?column?                
---------------------------------------
 18446744073709551616.0000000000000000
(1 row)

While the above operation works fine once I divide 1 by that number the result is an inexact decimal number:

# SELECT 1 / (2::NUMERIC ^ 64);
                ?column?                
----------------------------------------
 0.000000000000000000054210108624275222
(1 row)

It doesn't seem to be an issue with the output either as taking the reciprocal yields a different number than I started with:

# SELECT 1 / (1 / (2::NUMERIC ^ 64));
                         ?column?                          
-----------------------------------------------------------
 18446744073709551514.042092759729171265910020841463748922
(1 row)

The only way to get an exact result is by specifying an explicit precision and scale:

# SELECT 1 / (2::NUMERIC(96, 64) ^ 64);
                              ?column?                              
--------------------------------------------------------------------
 0.0000000000000000000542101086242752217003726400434970855712890625
(1 row)

# SELECT 1 / (1 / (2::NUMERIC(96, 64) ^ 64));
                                       ?column?                                        
---------------------------------------------------------------------------------------
 18446744073709551616.0000000000000000000000000000000000000000000000000000000000000000
(1 row)

However this does not seem intuitive from the documentation which states that:

Specifying:

NUMERIC

without any precision or scale creates a column in which numeric values of any precision and scale can be stored, up to the implementation limit on precision. A column of this kind will not coerce input values to any particular scale...