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gawk: Computer Arithmetic

 15.1 A General Description of Computer Arithmetic
 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
      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
    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.  Floating-point
 values are always signed.  The possible ranges of values are shown in
 ⇒Table 15.1 table-numeric-ranges.
 Numeric 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
 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
 Single-precision         1.175494e-38             3.402823e38
 floating point
 Double-precision         2.225074e-308            1.797693e308
 floating point
 Table 15.1: Value ranges for different numeric representations
    ---------- Footnotes ----------
    (1) We don't know why they expect this, but they do.
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