File: gawk.info,  Node: Computer Arithmetic,  Next: Math Definitions,  Up: Arbitrary Precision Arithmetic

16.1 A General Description of Computer Arithmetic
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Until now, we have worked with data as either numbers or strings.
Ultimately, however, computers represent everything in terms of "binary
digits", or "bits".  A decimal digit can take on any of 10 values: zero
through nine.  A binary digit can take on any of two values, zero or
one.  Using binary, computers (and computer software) can represent and
manipulate numerical and character data.  In general, the more bits you
can use to represent a particular thing, the greater the range of
possible values it can take on.

   Modern computers support at least two, and often more, ways to do
arithmetic.  Each kind of arithmetic uses a different representation
(organization of the bits) for the numbers.  The kinds of arithmetic
that interest us are:

Decimal arithmetic
     This is the kind of arithmetic you learned in elementary school,
     using paper and pencil (and/or a calculator).  In theory, numbers
     can have an arbitrary number of digits on either side (or both
     sides) of the decimal point, and the results of a computation are
     always exact.

     Some modern systems can do decimal arithmetic in hardware, but
     usually you need a special software library to provide access to
     these instructions.  There are also libraries that do decimal
     arithmetic entirely in software.

     Despite the fact that some users expect 'gawk' to be performing
     decimal arithmetic,(1) it does not do so.

Integer arithmetic
     In school, integer values were referred to as "whole" numbers--that
     is, numbers without any fractional part, such as 1, 42, or -17.
     The advantage to integer numbers is that they represent values
     exactly.  The disadvantage is that their range is limited.

     In computers, integer values come in two flavors: "signed" and
     "unsigned".  Signed values may be negative or positive, whereas
     unsigned values are always greater than or equal to zero.

     In computer systems, integer arithmetic is exact, but the possible
     range of values is limited.  Integer arithmetic is generally faster
     than floating-point arithmetic.

Floating-point arithmetic
     Floating-point numbers represent what were called in school "real"
     numbers (i.e., those that have a fractional part, such as
     3.1415927).  The advantage to floating-point numbers is that they
     can represent a much larger range of values than can integers.  The
     disadvantage is that there are numbers that they cannot represent
     exactly.

     Modern systems support floating-point arithmetic in hardware, with
     a limited range of values.  There are software libraries that allow
     the use of arbitrary-precision floating-point calculations.

     POSIX 'awk' uses "double-precision" floating-point numbers, which
     can hold more digits than "single-precision" floating-point
     numbers.  'gawk' has facilities for performing arbitrary-precision
     floating-point arithmetic, which we describe in more detail
     shortly.

   Computers work with integer and floating-point values of different
ranges.  Integer values are usually either 32 or 64 bits in size.
Single-precision floating-point values occupy 32 bits, whereas
double-precision floating-point values occupy 64 bits.
(Quadruple-precision floating point values also exist.  They occupy 128
bits, but such numbers are not available in 'awk'.)  Floating-point
values are always signed.  The possible ranges of values are shown in
*note Table 16.1: table-numeric-ranges. and *note Table 16.2:
table-floating-point-ranges.


Representation           Minimum value            Maximum value
---------------------------------------------------------------------------
32-bit signed integer    -2,147,483,648           2,147,483,647
32-bit unsigned          0                        4,294,967,295
integer
64-bit signed integer    -9,223,372,036,854,775,8089,223,372,036,854,775,807
64-bit unsigned          0                        18,446,744,073,709,551,615
integer

Table 16.1: Value ranges for integer representations


Representation              Minimum          Minimum finite   Maximum finite
                            positive         value            value
                            nonzero value
--------------------------------------------------------------------------------
Single-precision            1.175494e-38     -3.402823e+38    3.402823e+38
floating-point
Double-precision            2.225074e-308    -1.797693e+308   1.797693e+308
floating-point
Quadruple-precision         3.362103e-4932   -1.189731e+4932  1.189731e+4932
floating-point

Table 16.2: Approximate value ranges for floating-point number
representations

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

   (1) We don't know why they expect this, but they do.